IHES - Bibliothèque

La bibliothèque N. H. Kuiper propose un fonds documentaire spécialisé dans les domaines de recherche de l’Institut et offre à ses lecteurs des services et des produits documentaires variés.
La revue Les Publications mathématiques de l’IHES publie les articles des principaux résultats de recherche. De diffusion internationale, elle est reconnue comme l’une des meilleures au monde.
Les prépublications rassemblent l’ensemble des travaux menés à l’Institut et témoignent de l’interdisciplinarité et des collaborations entre professeurs et chercheurs invités.

# La bibliothèque Nicolaas Hendrik Kuiper

Inaugurée le 23 mai 2003, la bibliothèque de l’IHES porte le nom du deuxième directeur afin de rendre hommage et d’honorer Nicolaas Hendrik Kuiper. Créée dès l’emménagement de l’Institut à Bures-sur-Yvette en 1962, la bibliothèque se situait initialement dans le pavillon de musique du domaine de Bois-Marie. Aujourd’hui installée au premier étage du bâtiment scientifique, elle offre un cadre propice à l’étude et à l’inspiration des lecteurs, avec une vue sur la nature environnante.

Spécialisée dans les domaines des mathématiques, de la physique théorique et dans toute autre science qui s’y rattache, la bibliothèque Nicolaas Hendrik Kuiper accueille :
– les professeurs permanents,
– les chercheurs invités de l’IHES,
– les étudiants de l’université Paris-Saclay des écoles doctorales ou ayant un niveau supérieur au Master 2,
– toute personne ayant l’autorisation du directeur.

Avec FOCUS, consultez le catalogue et les ressources électroniques de la bibliothèque

La bibliothèque propose un fonds papier de 10 000 ouvrages environ et 170 périodiques en mathématiques et physique. Un fonds russe est également mis à la disposition des usagers.

Les ressources numériques comprennent les accès aux bases de données (MathScinet, Zentralblatt, ArXiv, Numdam…) et à plus de 13 000 revues électroniques.

Pour obtenir la liste de nos abonnements contactez nous

### Informations pratiques

Heures d’ouverture de la bibliothèque :
La bibliothèque est accessible 24h sur 24 aux professeurs permanents et les chercheurs invités de l’IHES.
L’accueil des chercheurs extérieurs et des étudiants de l’université Paris-Saclay est soumis aux horaires de la bibliothécaire, Aurélie Brest.

Horaires et coordonnées de la bibliothécaire :
Lundi-mardi-jeudi-vendredi : 9h-12h et 13h30-17h00
Mercredi : 8h30 – 12h00
Contact : 01 60 92 66 98 – service.bibli@ihes.fr

### Institutions partenaires:

RNBM : Le Réseau National des Bibliothèques de Mathématiques mène une politique documentaire nationale et joue un rôle majeur dans l’accès à la documentation électronique (accords de consortiums) pour la communauté mathématique française. www.rnbm.org

CNRS : La bibliothèque Nicolaas H. Kuiper est une unité mixte de Service (UMS 1786) du CNRS. www.cnrs.fr

Université Paris-Saclay : La bibliothèque Nicolaas H. Kuiper est aussi l’une des bibliothèques membres de l’Université Paris-Saclay.

# Les Publications mathématiques de l’IHES

Avec une distribution trilingue et internationale, la revue Les Publications mathématiques de l’IHES est l’une des plus prestigieuses revues dans ce domaine.

Créée en 1959 sous l’impulsion de Léon Motchane, et de Jean Dieudonné, professeur de l’institut, la revue Les Publications mathématiques de l’IHES paraît au début par fascicules isolés et en quatre langues : français, anglais, allemand et russe (version aujourd’hui abandonnée). Progressivement, l’IHES fait paraître annuellement deux volumes totalisant 800 pages. Depuis 2012, la revue est tirée à 320 exemplaires. Elle est également disponible en ligne et sur numdam.org.

Grâce à sa distribution dans le monde entier (on la trouve dans les bibliothèques de toutes les grandes institutions mathématiques), à sa tradition de publication d’articles historiques et à sa large couverture de la discipline, c’est une revue de haut niveau scientifique qui bénéficie d’une reconnaissance internationale.

Jean Dieudonné est le premier rédacteur en chef de la revue. Jacques Tits lui succède de 1980 à 1998, puis c’est­ Etienne Ghys qui reprend le poste de rédacteur-en chef jusqu’en 2009.

Le comité de rédaction s’élargit au fil des ans à d’autres professeurs de l’Institut : Pierre Deligne, Dennis Sullivan et René Thom l’intègrent en 1980.

En 1982, Mikhail Gromov rejoint le comité, suivi de Jean Bourgain en 1987, puis de Maxim Kontsevich en 1996, de Laurent Lafforgue en 1999 et de Claire Voisin en 2007.

Les Publications mathématiques de l’IHES ont été dirigées en 2010 et 2011 par Claire Voisin et Sergiu Klainerman ainsi que par Claire Voisin de janvier 2012 à juin 2019. Nicolas Bergeron a pris la relève en tant qu’ éditeur en chef depuis juillet 2019.

## Le comité de rédaction actuel

Il est aujourd’hui élargi à des scientifiques extérieurs à l’Institut.

Rédacteur en chef : Nicolas Bergeron

Editeurs : Nicolas Bergeron (ENS), Sébastien Boucksom (CNRS, CMLS, Ecole polytechnique), Pierre Colmez (CNRS, IMJ-PRG), Sylvain Crovisier (CNRS, Université Paris-Sud), Hugo Duminil-Copin (IHES, Université de Genève), Fanny Kassel (CNRS, IHES), Frank Merle (Université Cergy-Pontoise et IHES), Hiraku Nakajima (RIMS, Kyoto, Japon), Olivier Schiffmann (CNRS, Université Paris-Sud)

Comité de rédaction : Denis Auroux (University of California, Berkeley), Alain Connes (IHES et Collège de France), Pierre Deligne (IAS, Princeton, Etats-Unis), Mikhail Gromov (IHES), Maxim Kontsevich (IHES), Dennis Sullivan (CUNY and SUNY, Stony Brook, Etats-Unis)

## Soumettre un article

Pour soumettre un article dans Les Publications mathématiques de l’IHES, merci d’adresser votre manuscrit de préférence en fichier pdf à Nicolas Bergeron (nicolas.bergeron@ens.fr)

Par voie postale : adresser 2 copies du manuscrit et une lettre d’accompagnement à
Nicolas Bergeron
Les Publications Mathématiques de l’IHES
DMA, École normale supérieure
45 rue d’Ulm
F-75230 Paris cedex 5

## Abonnement et vente au numéro

La distribution des abonnements aux Publications mathématiques de l’IHES est assurée par :

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La vente au numéro est également possible et assurée par l’IHES.

Télécharger le bon de commande en ligne et le renvoyer dûment rempli par email ou par courrier postal

Institut des Hautes Études Scientifiques
Le Bois-Marie
35, route de Chartres
F-91440 Bures-sur-Yvette (France)
Tel : +33 1 60 92 66 98
Fax : +33 1 69 28 00 88
publications@ihes.fr

Modes de paiement
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Merci de la retourner dûment complétée, signée et accompagnée de votre paiement :
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Les livres seront expédiés à réception du règlement.

Commandes et informations sur les tarifs :
publications@ihes.fr

La revue est également disponible ­­en ligne et sur numdam.org.

Prépublication Date PDF

On Kontsevich Generalizations of Tian-Todorov Theorem and Applications M/20/12

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Kontsevich recently generalized Tian-Todorov Theorem regarding the structure of the Kuranish space of deformations of a Kahler manifold with trivial canonical bundle. An alternative proof was given using a general result regarding the smoothness of moduli space of formal deformations, based on BV-algebra resolutions. From this, various other generalizations ensue and a conjecture relating the dimension of the tangent space of formal deformations and the first non-trivial Hodge number $h(n-1,1)$. Additional details are provided, together with a proposed explanation regarding the above conjecture. Related considerations regarding mirror symmetry and motives of Calabi-Yau manifolds are included, based on the idea of complexifying TQFTs, modeled after Chow pure motives.

13/11/2020 PDF

A functorial characterization of von Neumann entropy P/20/11

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We classify the von Neumann entropy as a certain concave functor from finite- dimensional non-commutative probability spaces and state-preserving ∗-homomorphisms to real numbers. This is made precise by first showing that the category of non-commutative probability spaces has the structure of a Grothendieck fibration with a fiberwise convex structure. The entropy difference associated to a ∗-homomorphism between probability spaces is shown to be a functor from this fibration to another one involving the real num- bers. Furthermore, the von Neumann entropy difference is classified by a set of axioms similar to those of Baez, Fritz, and Leinster characterizing the Shannon entropy difference. The existence of disintegrations for classical probability spaces plays a crucial role in our classification.

15/09/2020 PDF

Denseness conditions, morphisms and equivalences of toposes M/20/09

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We systematically investigate morphisms and equivalences of toposes from multiple points of view. We establish a dual adjunction between morphisms and comorphisms of sites, introduce the notion of weak morphism of toposes and characterize the functors which induce such morphisms. In particular, we examine continuous comorphism of sites and show that this class of comorphisms notably includes all fibrations as well as morphisms of fibrations. We also establish a characterization theorem for essential geometric morphisms and locally connected morphisms in terms of continuous functors, and a relative version of the comprehensive factorization of a functor. Then we prove a general theorem providing necessary and sufficient explicit conditions for a morphism of sites to induce an equivalence of toposes. This stems from a detailed analysis of arrows in Grothendieck toposes and denseness conditions, which yields results of independent interest. We also derive site characterizations of the property of a geometric morphism to be an inclusion (resp. a surjection, hyperconnected, localic), as well as site-level descriptions of the surjection-inclusion and hyperconnected-localic factorizations of a geometric morphism.

05/08/2020 PDF

A characterization of complex quasi-projective manifolds uniformized by unit balls M/20/08

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In 1988 Simpson extended the Donaldson-Uhlenbeck-Yau theorem to the context of Higgs bundles, and as an application he proved a uniformization theorem which characterizes complex projective manifolds and quasi-projective curves whose universal coverings are complex unit balls. In this paper we give a necessary and sufficient condition for quasi-projective manifolds to be uniformized by complex unit balls. This generalizes the uniformization theorem by Simpson. Several byproducts are also obtained in this paper.

14/07/2020 PDF

A non-commutative Bayes' theorem P/20/10

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Using a diagrammatic reformulation of Bayes' theorem, we provide a necessary and sufficient condition for the existence of Bayesian inference in the setting of finite-dimensional $C^*$-algebras. In other words, we prove an analogue of Bayes' theorem in the joint classical and quantum context. Our analogue is justified by recent advances in categorical probability theory, which have provided an abstract formulation of the classical Bayes' theorem. In the process, we further develop non-commutative almost everywhere equivalence and illustrate its important role in non-commutative Bayesian inversion. The construction of such Bayesian inverses, when they exist, involves solving a positive semidefinite matrix completion problem for the Choi matrix. This gives a solution to the open problem of constructing Bayesian inversion for completely positive unital maps acting on density matrices that do not have full support. We illustrate how the procedure works for several examples relevant to quantum information theory.

08/05/2020 PDF

Backbone free energy estimator applied to viral glycoproteins M/20/07

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Proteins, Backbone Hydrogen Bonds, Backbone Free Energy, Viral Glycoproteins, Antiviral Vaccine/Drug TargetsEarlier analysis of the Protein Data Bank derived the distribution of rotations from the plane of a protein hydrogen bond donor peptide group to the plane of its acceptor peptide group. The quasi Boltzmann formalism of Pohl-Finkelstein is employed to estimate free energies of protein elements with these hydrogen bonds pinpointing residues with high propensity for conformational change. This is applied to viral glycoproteins as well as capsids, where the 90th-plus percentiles of free energies determine residues that correlate well with viral fusion peptides and other functional domains in known cases and thus provide a novel method for predicting these sites of importance as antiviral drug or vaccine targets in general. The method is implemented at https://bion-server.au.dk/hbonds/ from an uploaded Protein Data Bank file.

17/03/2020 PDF

Superconnection in the spin factor approach to particle physics P/20/06

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The notion of superconnection devised by Quillen in 1985 and used in gauge-Higgs field theory in the 1990's is applied to the spin factors (finite-dimensional euclidean Jordan algebras) recently considered as representing the finite quantum geometry of one generation of fermions in the Standard Model of particle physics.

12/03/2020 PDF

Les suites spectrales de Hodge-Tate M/20/05

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This book presents two important results in p-adic Hodge theory following the approach initiated by Faltings, namely (i) his main p-adic comparison theorem, and (ii) the Hodge-Tate spectral sequence. We establish for each of these results two versions, an absolute one and a relative one. While the absolute statements can reasonably be considered as well understood, particularly after their extension to rigid varieties by Scholze, Faltings' initial approach for the relative variants has remained much less studied. Although we follow the same strategy as that used by Faltings to establish his main p-adic comparison theorem, part of our proofs is based on new results. The relative Hodge-Tate spectral sequence is new in this approach.

11/03/2020 PDF

The relative Hodge-Tate spectral sequence - an overview M/20/04

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We give in this note an overview of a recent work leading to a generalization of the Hodge-Tate spectral sequence to morphisms. The latter takes place in Faltings topos, but its construction requires the introduction of a relative variant of this topos which is the main novelty of our work.

11/03/2020 PDF

Graphon Models in Quantum Physics M/20/03

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In this work we explain some new applications of Infinite Combinatorics to Quantum Physics. We investigate the use of the theory of graphons in non-perturbative Quantum Field Theory and Deformation Quantization which lead us to discover some new interrelationships between these fundamental topics. In one direction, we study Dyson--Schwinger equations in the context of the graph function theory of sparse graphs which enables us to analyze non-perturbative parameters of strongly coupled Quantum Field Theories via cut-distance compact topological regions of Feynman diagrams, Kontsevich's $\star$-product and other new mathematical settings. In another direction, we initiate a theory of graph function representations for Kontsevich admissible graphs to formulate a new topological Hopf algebraic formalism for the study of these graphs which brings some new useful mathematical tools to relate Deformation Quantization program with non-perturbative renormalization program in Quantum Field Theory models.

07/02/2020 PDF

Big Picard theorem and algebraic hyperbolicity for varieties admitting a variation of Hodge structures M/20/02

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For a complex smooth log pair $(Y,D)$, if the quasi-projective manifold $U=Y-D$ admits a complex polarized variation of Hodge structures with local unipotent monodromies around $D$ or admits an integral polarized variation of Hodge structures, whose period map is quasi-finite, then we prove that $(Y,D)$ is algebraically hyperbolic in the sense of Demailly, and that the generalized big Picard theorem holds for $U$: any holomorphic map $f:\Delta-\{0\}\to U$ from the punctured unit disk to $U$ extends to a holomorphic map of the unit disk $\Delta$ into $Y$. This result generalizes a recent work by Bakker-Brunebarbe-Tsimerman, in which they proved that if the monodromy group of the above variation of Hodge structures is arithmetic, then $U$ is Borel hyperbolic: any holomorphic map from a quasi-projective variety to $U$ is algebraic.

27/01/2020 PDF

Inverses, disintegrations, and Bayesian inversion in quantum Markov categories M/20/01

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We analyze three successively more general notions of reversibility and statistical inference: ordinary inverses, disintegrations, and Bayesian inferences. We provide purely categorical definitions of these notions and show how each one is a strictly special instance of the latter in the cases of classical and quantum probability. This provides a categorical foundation for Bayesian inference as a generalization of reversing a process. To properly formulate these ideas, we develop quantum Markov categories by extending recent work of Cho–Jacobs and Fritz on classical Markov categories. We unify Cho–Jacobs’ categorical notion of almost everywhere (a.e.) equivalence in a way that is compatible with Parzygnat–Russo’s C∗-algebraic a.e. equivalence in quantum probability.

20/01/2020 PDF

Universal cocycles and the graph complex action on homogeneous Poisson brackets by diffeomorphisms M/19/20

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The graph complex acts on the spaces of Poisson bi-vectors $P$ by infinitesimal symmetries. We prove that whenever a Poisson structure is homogeneous, i.e. $P = L_{\vec{V}}(P)$ w.r.t. the Lie derivative along some vector field $\vec{V}$, but not quadratic (the coefficients of $P$ are not degree-two homogeneous polynomials), and whenever its velocity bi-vector $\dot{P}=Q(P)$, also homogeneous w.r.t. $\vec{V}$ by $L_{\vec{V}}(Q)=n\cdot Q$ whenever $Q(P)= Or(\gamma)(P^{the orientation morphism$Or$from a graph cocycle$\gamma$on$n$vertices and$2n-2$edges in each term, then the$1$-vector$\vec{X}=Or(\gamma)(\vec{V}vectors $P$ satisfying the above assumptions, on all finite-dimensional affine manifolds $M$. Still, if the bi-vector $Q\not\equiv 0$ is exact in the respective Poisson cohomology, so there exists a vector field $\vec{Y}$ such that $Q(P)=[\![\vec{Y},P]\!]$, then the universal cocycle $\vec{\cX}$ does not belong to the coset of $\vec{Y}$ mod $\ker[\![P,\cdot]\!]$. We illustrate the construction using two examples of cubic-coefficient Poisson brackets associated with the $R$-matrices for the Lie algebra $\mathfrak{gl}(2)$.

29/12/2019 PDF

Big Picard theorem for moduli spaces of polarized manifolds M/19/19

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Consider a smooth projective family of complex polarized manifolds with semi-ample canonical sheaf over a quasi-projective manifold $V$. When the associated moduli map $V\to P_h$ from the base to coarse moduli space is quasi-finite, we prove that the generalized big Picard theorem holds for the base manifold $V$: for any projective compactification $Y$ of $V$, any holomorphic map $f:\Delta-\{0\}\to V$ from the punctured unit disk to $V$ extends to a holomorphic map of the unit disk $\Delta$ into $Y$. This result generalizes our previous work on the Brody hyperbolicity of $V$ (\emph{i.e.} there are no entire curves on $V$), as well as a more recent work by Lu-Sun-Zuo on the Borel hyperbolicity of $V$ (\emph{i.e.} any holomorphic map from a quasi-projective variety to $V$ is algebraic). We also obtain generalized big Picard theorem for bases of log Calabi-Yau families.

25/12/2019 PDF

Topological Recursion, Airy structures in the space of cycles P/19/18

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Topological recursion associates to a spectral curve, a sequence of meromorphic differential forms. A tangent space to the "moduli space" of spectral curves (its space of deformations) is locally described by meromorphic 1-forms, and we use form-cycle duality to re-express it in terms of cycles (generalized cycles). This formulation allows to express the ABCD tensors of Quantum Airy Structures acting on the vector space of cycles, in an intrinsic spectral-curve geometric way.

10/12/2019 PDF

Vanishing theorem for tame harmonic bundles via $L^2$-cohomology M/19/17

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Using $L^2$-methods, we prove a vanishing theorem for tame harmonic bundles over quasi-Ka and for parabolic Higgs bundles by Arapura, Li and the second named author. To prove our vanishing theorem, we construct a fine resolution of the Dolbeault complex for tame harmonic bundles via the complex of sheaves of $L^2$-forms, and we establish the H

05/12/2019 PDF

Construction of the classical time crystal Lagrangians from Sisyphus dynamics and duality description with the Liénard type equation P/19/16

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We explore the connection between the equations describing sisyphus dynamics and the generic Liénard type or Liénard II equation from the viewpoint of branched Hamil- tonians. The former provides the appropriate setting for classical time crystal being derivable from a higher order Lagrangian. However it appears the equations of Sisyphus dynamics have a close relation with the Liénard-II equation when expressed in terms of the ‘velocity’ variable. Another interesting feature of the equations of Sisyphus dynamics is the appearance of velocity dependent ”mass function” in contrast to the more com- monly encountered position dependent mass. The consequences of such mass functions seem to have connections to cosmological time crystals .

22/11/2019 PDF

Equivariant connective K-theory M/19/15

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For separated schemes of finite type over a field with an action of an affine group scheme of finite type, we construct the bi-graded equivariant connective K-theory mapping to the equivariant K-homology of Guillot and the equivariant algebraic K-theory of Thomason. It has all the standard basic properties as the homotopy invariance and localization. We also get the equivariant version of the Brown-Gersten-Quillen spectral sequence and study its convergence.

20/11/2019 PDF

Pre-Calabi-Yau algebras and $\xi\partial$-calculus on higher cyclic Hochschild cohomology M/19/14

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We formulate the notion of pre-Calabi-Yau structure via the higher cyclic Hochschild complex and study its cohomology. A small quasi-isomorphic subcomplex in higher cyclic Hochschild complex gives rise to the graphical calculus of $\xi\partial$-monomials. Develop- ing this calculus we are able to give a nice combinatorial formulation of the Lie structure on the corresponding Lie subalgebra. Then using basis of ξ∂-monomials and employ- ing elements of Gr{\"o}bner bases theory we prove homological purity of the higher cyclic Hochschild complex and as a consequence obtain $L_{\infty}$-formality. This construction in particular allows an easy interpretation of a pre-Calabi-Yau structure as a noncommu- tative Poisson structure. We give an explicit formula showing how the double Poisson algebra introduced in [25] appears as a particular part of a pre-Calabi-Yau structure. This result holds for any associative algebra A and emphasizes the special role of the fourth component of a pre-Calabi-Yau structure in this respect.

23/10/2019 PDF

Three-body closed chain of interactive (an)harmonic oscillators and the algebra $sl(4)$ P/19/13

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In this work we study 2- and 3-body oscillators with quadratic and sextic pairwise potentials which depend on relative {\it distances}, $|{\bf r}_i - {\bf r}_j |$, between particles. The two-body harmonic oscillator is two-parametric and can be reduced to a one-dimensional radial Jacobi oscillator, while in the 3-body case such a reduction is not possible in general. Our study is restricted to solutions in the space of relative motion which are functions of mutual (relative) distances only ($S$-states). In general, three-body harmonic oscillator is 7-parametric depending on 3 masses and 3 spring constants, and frequency. It is shown that for certain relations involving masses and spring constants the system becomes maximally (minimally) superintegrable in the case of two (one) relations.

07/10/2019 PDF

Solutions of loop equations are random matrices M/19/12

23/09/2019 PDF

On volume subregion complexity in Vaidya spacetime P/19/11

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We study holographic subregion volume complexity for a line segment in the AdS3 Vaidya geometry. On the field theory side, this gravity background corresponds to a sudden quench which leads to the thermalization of the strongly-coupled dual conformal field theory. We find the time-dependent extremal volume surface by numerically solving a partial differential equation with boundary condition given by the Hubeny-Rangamani-Takayanagi surface, and we use this solution to compute holographic subregion complexity as a function of time. Approximate analytical expressions valid at early and at late times are derived.

29/08/2019 PDF

Action potential solitons and waves in axons P/19/10

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We show that the action potential signals generated inside axons propagate as reaction-diffusion solitons or as reaction-diffusion waves, refuting the Hodgkin and Huxley (HH) hypothesis that action potentials propagate along axons with an elastic wave mechanism. Action potential signals are solitary propagating spikes along the axon, occurring in a type I intermittency regime of the HH model. Reaction-diffusion action potential wave fronts annihilate at collision and at the boundaries of axons with zero flux, in contrast with elastic waves, where amplitudes add up and reflect at boundaries. We calculate numerically the values of the speed of the action potential spikes, as well as the dispersion relations. These findings suggest several experiments as validating and falsifying tests for the HH model.

07/08/2019 PDF

Super McShane Identity M/19/08

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The McShane identity for the once-punctured super torus is derived following Bowditch's proof in the bosonic case using techniques in super Teichmueller theory developed by the two latter-named authors.

06/08/2019 PDF

An electrophysiology model for cells and tissues P/19/09

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We introduce a kinetic model to study the dynamics of ions in aggregates of cells and tissues. Different types of communication channels between adjacent cells and between cells and intracellular space are considered (ion channels, pumps and gap junctions). We shows that stable transmembrane ionic Nernst potentials are due to the coexistence of both specialised ion pumps and channels. Ion pumps or channels alone do not contribute to an equilibrium transmembrane potential drop. The kinetic parameters of the model straightforwardly calibrate with the Nernst potentials and ion concentrations. The model is based on the ATPase enzymatic mechanism for the ions $\hbox{Na}^+$, $\hbox{K}^+$, and it can be generalised for other ion pumps. We extend the model to account for electrochemical effects, where transmembrane gating mechanism are introduced. In this framework, axons can be seen as the evolutionary result of the aggregation of cells through gap junctions, which can be identified as the Ranvier nodes. In this kinetic framework, the injection of current in an axon induces the modification of the potassium equilibrium potential along the axon.

06/08/2019 PDF

McShane identities for Higher Teichmuller theory and the Goncharov-Shen potential M/19/07

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In [GS15], Goncharov and Shen introduce a family of mapping class group invariant regular functions on their A-moduli space to explicitly formulate a particular homological mirror symmetry conjecture. Using these regular functions, we obtain McShane identities for general rank positive surface group representations with loxodromic boundary monodromy and (non-strict) McShane-type inequalities for general rank positive representations with unipotent boundary monodromy. Our identities are expressed in terms of projective invariants, and we study these invariants: we establish boundedness and Fuchsian rigidity results for triple ratios. Moreover, we obtain McShane identities for finite-area cusped convex real projective surfaces by generalizing the Birman--Series geodesic scarcity theorem. We apply our identities to derive the simple spectral discreteness of unipotent bordered positive representations, collar lemmas, and generalizations of the Thurston metric.

25/06/2019 PDF

Crystal Volumes and Monopole Dynamics P/19/06

08/06/2019 PDF

Third Kind Elliptic Integrals and 1-Motives M/19/05

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In [5] we have showed that the Generalized Grothendieck's Conjecture of Periods applied to 1-motives, whose underlying semi-abelian variety is a product of elliptic curves and of tori, is equivalent to a transcendental conjecture involving elliptic integrals of the first and second kind, and logarithms of complex numbers. In this paper we investigate the Generalized Grothendieck's Conjecture of Periods in the case of 1-motives whose underlying semi-abelian variety is a non trivial extension of a product of elliptic curves by a torus. This will imply the introduction of elliptic integrals of the third kind for the computation of the period matrix of M and therefore the Generalized Grothendieck's Conjecture of Periods applied to M will be equivalent to a transcendental conjecture involving elliptic integrals of the first, second and third kind.

21/05/2019 PDF

Large genus behavior of topological recursion M/19/04

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We show that for a rather generic set of regular spectral curves, the {\it Topological--Recursion} invariants $F_g$ grow at most like $O((\beta g)! r^{-g})$ with some $r>0$ and $\beta\leq 5$

13/05/2019 PDF

Standard conjectures in model theory, and categoricity of comparison isomorphisms. A model theory perspective. M/19/03

22/03/2019 PDF

On Phases Of Melonic Quantum Mechanics P/19/02

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We explore in detail the properties of two melonic quantum mechanical theories which can be formulated either as fermionic matrix quantum mechanics in the new large D limit, or as disordered models. Both models have a mass parameter m and the transition from the perturbative large m region to the strongly coupled "black-hole" small m region is associated with several interesting phenomena. One model, with U(n)^2 symmetry and equivalent to complex SYK, has a line of first-order phase transitions terminating, for a strictly positive temperature, at a critical point having non-trivial, non-mean-field critical exponents for standard thermodynamical quantities. Quasi-normal frequencies, as well as Lyapunov exponents associated with out-of-time-ordered four-point functions, are also singular at the critical point, leading to interesting new critical exponents. The other model, with reduced U(n) symmetry, has a quantum critical point at strictly zero temperature and positive critical mass m∗. For 0

18/03/2019 PDF

The lure of conformal symmetry P/19/01

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The Clifford algebra ${\rm Cl} (4,1) \simeq {\mathbb C} [4]$, generated by the real (Majorana) $\gamma$-matrices and by a hermitian $\gamma_5$, gives room to the reductive Lie algebra $u(2,2)$ of the conformal group extended by the $u(1)$ helicity operator. Its unitary positive energy ladder representations, constructed by Gerhard Mack and the author 50 years ago, opened the way to a better understanding of zero-mass particles and fields and their relation to the space of bound states of the hydrogen atom. They became a prototypical example of a minimal representation of a non-compact reductive group introduced during the subsequent decade by Joseph.

22/01/2019 PDF

Hyperbolic endomorphisms and overlap numbers M/18/10

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Hyperbolic endomorphisms and overlap numbers for equilibrium measures are studied on lifts of invariant sets. We look into the structure of Rokhlin conditional measures with respect to various fiber partitions, and find relations between them. We also compute topological overlap numbers in several concrete cases. In particular, we obtain a large class of endomorphisms which asymptotically are irrational-to-1.

13/11/2018 PDF

Some aspects of topological Galois theory M/18/09

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We establish a number of results on the subject of the first author's topos-theoretic generalization of Grothendieck's Galois formalism. In particular, we generalize in this context the existence theorem of algebraic closures, we give a concrete description of the atomic completion of a small category whose opposite satisfies the amalgamation pro- perty, and we explore to which extent a model of a Galois-type theory is determined by its symmetries.

04/09/2018 PDF

Three Hopf algebras from number theory, physics and topology, and their common operadic, simplicial and categorical background M/18/08

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We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. The primary examples are the Hopf algebras of Goncharov for multiple zeta values, that of Connes--Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects, cooperads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretation of known constructions in a large common framework.

18/06/2018 PDF

On the Ramanujan conjecture for automorphic forms over function fields I. Geometry M/18/07

31/05/2018 PDF

Global bifurcations of limit cycles in the Leslie-Gover model with the Allee effect M/18/06

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In this paper, we complete the global qualitative analysis of the Leslie-Gover system with the Allee effect which models the dynamics of the populations of predators and their prey in a given ecological or biomedical system. In particular, studying global bifurcations, we prove that such a system can have at most two limit cycles surrounding one singular point. We also conduct some numerical experiments to illustrate the obtained results.

23/05/2018 PDF

Multi-parameter planar polynomial dynamical systems M/18/05

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In this paper, we study multi-parameter planar dynamical systems and carry out the global bifurcation analysis of such systems. To control the global bifurcations of limit cycle in these systems, it is necessary to know the properties and combine the effects of all their field rotation parameters. It can be done by means of the development of our bifurcational geometric method based on the Wintner-Perko termination principle and application of canonical systems with field rotation parameters. Using this method, we solve, e.g., Hilbert's Sixteenth Problem on the maximum number of limit cycles and their distribution for the general Li\'{e}nard polynomial system and a Holling-type quartic dynamical system. We also conduct some numerical experiments to illustrate the obtained results.

13/04/2018 PDF

Pre-Calabi-Yau algebras as noncommutative Poisson structures M/18/04

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We show how the double Poisson algebra introduced in \cite{VdB} appear as a particular part of a pre-Calabi-Yau structure, i.e. cyclically invariant, with respect to the natural inner form, solution of the Maurer-Cartan equation on $Ac part of this solution is described, which is in one-to-one correspondence with the double Poisson algebra structure. As a consequence we have that appropriate pre-Calabi-Yau structures induce a Poisson bracket on representation spaces$({\rm Rep}_n A)^{Gl_n}$for any associative algebra$A$. 27/03/2018 PDF Introduction to Higher Cubical Operads. Second Part: The Functor of Fundamental Cubical Weak$\infty$-Groupoids for Spaces M/18/03 16/01/2018 PDF Some Aspects of Dynamical Topology: Dynamical Compactness and Slovak Spaces M/18/02 Voir plus > Voir moins > The area of dynamical systems where one investigates dynamical properties that can be described in topological terms is "Topological Dynamics". Investigating the topological properties of spaces and maps that can be described in dynamical terms is in a sense the opposite idea. This area is recently called as "Dynamical Topology". For (discrete) dynamical systems given by compact metric spaces and continuous (surjective) self-maps we (mostly) survey some results on two new notions: "Slovak Space" and "Dynamical Compactness". Slovak space is a dynamical analogue of the rigid space: a nontrivial compact metric space whose homeomorphism group is cyclic and generated by a minimal homeomorphism. Dynamical compactness is a new concept of chaotic dynamics. The omega-limit set of a point is a basic notion in theory of dynamical systems and means the collection of states which "attract" this point while going forward in time. It is always nonempty when the phase space is compact. By changing the time we introduced the notion of the omega-limit set of a point with respect to a Furstenberg family. A dynamical system is called dynamically compact (with respect to a Furstenberg family) if for any point of the phase space this omega-limit set is nonempty. A nice property of dynamical compactness: all dynamical systems are dynamically compact with respect to a Furstenberg family if and only if this family has the finite intersection property. 08/01/2018 PDF From Euler's play with infinite series to the anomalous magnetic moment P/18/01 05/01/2018 PDF Introduction to Higher Cubical Operads. First Part: The Cubical Operad of Cubical Weak$\infty$-Categories M/17/19 21/12/2017 PDF In memoriam: Cécile DeWitt-Morette M/17/15 08/12/2017 PDF Ars combinatoria'' chez Gian-Carlo Rota ou le triomphe du symbolisme M/17/18 Voir plus > Voir moins > Gian-Carlos Rota est l'inventeur d'une nouvelle science : la combinatoire algébrique qui n'était jusque-là qu'un ensemble de questions disparates, dont la solution, parfois ingénieuse, ne laissait entrevoir la méthode. Il s'agit ici de décrire la stratégie mise en oeuvre sous sa forme mathématique dont les enjeux philosophiques ne sont pas négligeables, en poursuivant le geste de Rota, pour sortir la combinatoire de son ghetto. 06/12/2017 PDF Generalized conformal Hamiltonian dynamics and the pattern formation equations M/17/16 Voir plus > Voir moins > We demonstrate the significance of the Jacobi last multiplier in Hamiltonian theory by explicitly constructing the Hamiltonians of certain well known first-order systems of differential equations arising in the activator-inhibitor (AI) systems. We investigate the generalized Hamiltonian dynamics of the AI systems of Turing pattern formation problems, and demonstrate that various subsystems of AI, depending on the choices of parameters, are described either by conformal or contact Hamiltonian dynamics or both. Both these dynamics are subclasses of another dynamics, known as Jacobi mechanics. Furthermore we show that for non Turing pattern formation, like the Gray-Scott model, may actually be described by generalized conformal Hamiltonian dynamics using two Hamiltonians. Finally, we construct a locally defined dissipative Hamiltonian generating function \cite{HHG} of the original system. This generating function coincides with the free energy'' of the associated system if it is a pure conformal class. Examples of pattern formation equation are presented to illustrate the method. 05/12/2017 PDF Study of quasi-integrable and non-holonomic deformation of equations in the NLS and DNLS hierarchy M/17/17 Voir plus > Voir moins > The hierarchy of equations belonging to two different but related integrable systems, the Nonlinear Schrödinger and its derivative variant, DNLS are subjected to two distinct deformation procedures, viz. quasi-integrable deformation (QID) that generally do not preserve the integrability, only asymptotically integrable, and non-holonomic deformation (N HD) that does. QID is carried out generically for the NLS hierarchy while for the DNLS hierarchy, it is first done on the Kaup-Newell system followed by other members of the family. No QI anomaly is observed at the level of EOMs which suggests that at that level the QID may be identified as some integrable deformation. NHD is applied to the NLS hierarchy generally as well as with the specific focus on the NLS equation itself and the coupled KdV type NLS equation. For the DNLS hierarchy, the Kaup-Newell(KN) and Chen-Lee-Liu (CLL) equations are deformed non-holonomically and subsequently, different aspects of the results are discussed. 05/12/2017 PDF Noncommutative Catalan Numbers M/17/14 Voir plus > Voir moins > The goal of this paper is to introduce and study non-commutative Catalan numbers C_n which belong to the free Laurent polynomial algebra in n generators. Our non-commutative numbers admit interesting (commutative and non-commutative) specializations, one of them related to Garsia-Haiman (q,t)-versions, another to solving non-commutative quadratic equations. We also establish total positivity of the corresponding non-commutative Hankel matrices and introduce accompanying nonc-mmutative binomial coefficients. 25/08/2017 PDF Theory of Morphogenesis M/17/13 Voir plus > Voir moins > A model of morphogenesis is proposed based upon seven explicit postulates. The mathematical import and biological significance of the postulates are explored and discussed. 08/06/2017 PDF Spin-orbit precession along eccentric orbits for extreme mass ratio black hole binaries and its effective-one-body transcription P/17/12 07/06/2017 PDF Properties of soliton surfaces associated with integrable$\mathbb{C}P^{N-1}$sigma models P/17/11 Voir plus > Voir moins > We investigate certain properties of$\\mathfrak{su}(N)$-valued two-dimensional soliton surfaces associated with the integrable$\\mathbb{C}P^{N-1}$sigma models constructed by the orthogonal rank-one Hermitian projectors, which are defined on the two-dimensional Riemann sphere with finite action functional. Several new properties of the projectors mapping onto one-dimensional subspaces as well as their relations with three mutually different immersion formulas, namely, the generalized Weierstrass, Sym-Tafel and Fokas-Gel\'fand have been discussed in detail. Explicit connections among these three surfaces are also established by purely analytical descriptions and, it is demonstrated that the three immersion formulas actually correspond to the single surface parametrized by some specific conditions. 06/06/2017 PDF q-deformed quadrature operator and optical tomogram P/17/10 Voir plus > Voir moins > In this paper, we define the homodyne q-deformed quadrature operator and find its eigenstates in terms of the deformed Fock states. We find the quadrature representation of q-deformed Fock states in the process. Furthermore, we calculate the explicit analytical expression for the optical tomogram of the q-deformed coherent states. 22/05/2017 PDF SKEW PRODUCT SMALE ENDOMORPHISMS OVER COUNTABLE SHIFTS OF FINITE TYPE M/17/09 Voir plus > Voir moins > We introduce and study skew product Smale endomorphisms over finitely irreducible topological Markov shifts with countable alphabets. We prove that almost all conditional measures of equilibrium states of summable and locally Holder continuous potentials are dimensionally exact, and that their dimension is equal to the ratio of the (global) entropy and the Lyapunov exponent. We also prove for them a formula of Bowen type for the Hausdorff dimension of all fibers. We develop a version of thermodynamic formalism for finitely irreducible two-sided topological Markov shifts with countable alphabets. We describe then the thermodynamic formalism for Smale skew products over countable-to-1 endomorphisms, and give several applications to measures on natural extensions of endomorphisms. We show that the exact dimensionality of conditional measures on fibers, implies the global exact dimensionality of the measure, in certain cases. We then study equilibrium states for skew products over endomorphisms generated by graph directed Markov systems, in particular for skew products over expanding Markov-Renyi (EMR) maps, and we settle the question of the exact dimensionality of such measures. In particular, this applies to skew products over the continued fractions transformation, and over parabolic maps. We prove next two results related to Diophantine approximation, which make the Doeblin-Lenstra Conjecture more general and more precise, for a different class of measures than in the classical case. In the end, we 14/05/2017 PDF M-theoretic Lichnerowicz formula and supersymmetry P/17/06 Voir plus > Voir moins > A suitable generalisation of the Lichnerowicz formula can relate the squares of supersymmetric operators to the effective action, the Bianchi identities for fluxes, and some equations of motion. Recently, such formulae have also been shown to underlie the (generalised) geometry of supersymmetric theories. In this paper, we derive an M-theoretic Lichnerowicz formula that describes eleven-dimensional supergravity together with its higher-derivative couplings. The first corrections to the action appear at eight-derivative level, and the construction yields two different supersymmetric invariants, each with a free coefficient. We discuss the restriction of our construction to seven-dimensional internal spaces, and implications for compactifications on manifolds of G 2 holonomy. Inclusion of fluxes and computation of contributions with higher than eight derivatives are also discussed. 12/05/2017 PDF Modification of Schrodinger-Newton equation due to braneworld models with minimal length P/17/07 Voir plus > Voir moins > We study the correction of the energy spectrum of a gravitational quantum well due to the combined effect of the braneworld model with infinite extra dimensions and generalized uncertainty principle. The correction terms arise from a natural deformation of a semiclassical theory of quantum gravity governed by the Schrodinger-Newton equation based on a minimal length framework. The two fold correction in the energy yields new values of the spectrum, which are closer to the values obtained in the GRANIT experiment. This raises the possibility that the combined theory of the semiclassical quantum gravity and the generalized uncertainty principle may provide an intermediate theory between the semiclassical and the full theory of quantum gravity. We also prepare a schematic experimental set-up which may guide to the understanding of the phenomena in the laboratory. 12/05/2017 PDF Quantum Supersymmetric Cosmological Billiards and their Hidden Kac-Moody Structure P/17/08 12/05/2017 PDF Software modules and computer-assisted proof schemes in the Kontsevich deformation quantization M/17/05 Voir plus > Voir moins > The Kontsevich deformation quantization combines Poisson dynamics, noncommutative geometry, number theory, and calculus of oriented graphs. To manage the algebra and differential calculus of series of weighted graphs, we present software modules: these allow generating the Kontsevich graphs, expanding the non-commutative ⋆-product by using a priori undetermined coefficients, and deriving linear relations between the weights of graphs. Throughout this text we illustrate the assembly of the Kontsevich ⋆-product up to order 4 in the deformation parameter ħ. Already at this stage, the ⋆-product involves hundreds of graphs; expressing all their coefficients via 149 weights of basic graphs (of which 67 weights are now known exactly), we express the remaining 82 weights in terms of only 10 parameters (more specifically, in terms of only 6 parameters modulo gauge-equivalence). Finally, we outline a scheme for computer-assisted proof of the associativity, modulo o(ħ^4), for the newly built ⋆-product expansion. 11/05/2017 PDF Galois equivariance of critical values of$L$-functions for unitary groups M/17/04 Voir plus > Voir moins > The goal of this paper is to provide a refinement of a formula proved by the first author which expresses some critical values of automorphic$L$-functions on unitary groups as Petersson norms of automorphic forms. Here we provide a Galois equivariant version of the formula. We also give some applications to special values of automorphic representations of$\GL_{n}\times\GL_{1}$. We show that our results are compatible with Deligne's conjecture. 08/04/2017 PDF Deducing the symmetry of the standard model from the automorphism and structure groups of the exceptional Jordan algebra P/17/03 Voir plus > Voir moins > We continue the study undertaken in [DV] of the exceptional Jordan algebra$J = J_3^8$as (part of) the finite-dimensional quantum algebra in an almost classical space-time approach to particle physics. Along with reviewing known properties of$J$and of the associated exceptional Lie groups we argue that the symmetry of the model can be deduced from the Borel-Siebenthal theory of maximal subgroups of simple compact Lie groups. 30/03/2017 PDF Examples of pre-CY structures, associated operads and cohomologies M/17/02 01/02/2017 PDF On the four-loop static contribution to the gravitational interaction potential of two point masses P/17/01 19/01/2017 PDF Three-body problem in 3D space: ground state, (quasi)-exact-solvability P/16/29 22/11/2016 PDF Diffeomorphisms of quantum fields P/16/28 Voir plus > Voir moins > We study field diffeomorphisms$\Phi(x)= F(\rho(x))=a_0\rho(x)+a_1\rho^2(x)+\ldots=\sum_{j+0}^\infty a_j \rho^{j+1}, $for free and interacting quantum fields$\Phi$. We find that the theory is invariant under such diffeomorphisms if and only if kinematic renormalization schemes are used. 07/10/2016 PDF Sur la dualité des topos et de leurs présentations et ses applications : une introduction M/16/26 Voir plus > Voir moins > Ce texte est une version écrite enrichie des notes d'un exposé donné à l'Université de Nantes le 1er avril 2016. Il a été rédigé par le second auteur, à partir de notes succinctes et d'expositions orales du premier auteur. Il peut servir d'introduction à la technique des topos comme ponts et à ses applications, en particulier pour les géomètres peu familiers de la logique catégorique et de la théorie des topos classifiants. 22/09/2016 PDF Le principe de fonctorialité de Langlands comme un problème de généralisation de la loi d'addition M/16/27 Voir plus > Voir moins > Ce texte représente les notes écrites d'un cours donné à l'Université de Nottingham en juin 2016. Dans la continuité des écrits précédents de l'auteur, il étudie le transfert automorphe de Langlands des groupes réductifs vers les groupes linéaires sous la forme équivalente de la définition d'opérateurs de transformation de Fourier locaux et globaux sur les groupes réductifs induits par les représentations de leurs groupes duaux, d'espaces fonctionnels locaux et globaux fixés par ces opérateurs et d'une fonctionnelle linéaire de Poisson globale qui serait invariante par transformation de Fourier. Se fondant sur une étude du cas des tores, il propose dans le cas des groupes réductifs non abéliens généraux une définition conjecturale des espaces fonctionnels recherchés et une caractérisation conjecturale de la fonctionnelle de Poisson associée. Il montre que la définition des opérateurs de transformations de Fourier et l'éventuelle vérification des propriétés attendues de ces espaces fonctionnels et de cette fonctionnelle de Poisson posent la question cruciale de la construction et de l'étude des opérateurs de convolution (transformés de Fourier de l'opérateur de multiplication point par point des fonctions) associés. Le texte propose pour ces opérateurs de convolution une conjecture d'algébricité qui semble plus raisonnable que la conjecture de "stabilité du tore par convolution" qui avait été faite dans la précédente pr 22/09/2016 PDF Algebraic flows on Shimura varieties M/16/21 15/09/2016 PDF Holomorphic curves in compact Shimura varieties M/16/22 Voir plus > Voir moins > We prove a hyperbolic analogue of the We prove an analogue of the Bloch-Ochiai theorem about the Zariski closure of holomorphic curves in abelian varieties. 15/09/2016 PDF Structures spéciales et problème de Zilber-Pink M/16/23 Voir plus > Voir moins > The Manin-Mumford and the André-Oort conjectures as well as the one formulated by Zilber and Pink concern algebraic varieties (algebraic tori, Abelian or semi-Abelian varieties, pure or mixed Shimura varieties) endowed with a natural set of special points and special subvarieties. An axiomatisation, in the spirit of model theory, is presented for a description of algebraic varieties endowed with a natural set of special points and special subvarieties with an emphasis on the bi-algebraic nature of the question. The text also reviews recent results on these conjectures. 15/09/2016 PDF Bi-algebraic geometry and the André-Oort conjecture M/16/24 15/09/2016 PDF O-minimal flows on abelian varieties M/16/25 15/09/2016 PDF Gravitational scattering, post-Minkowskian approximation and Effective One-Body theory P/16/20 09/09/2016 PDF Golod-Shafarevich type theorems and potential algebras M/16/19 Voir plus > Voir moins > Potential algebras feature in the minimal model program and noncommutative resolution of singularities, and the important cases are when they are finite dimensional, or of linear growth. We develop techniques, involving Groebner basis theory and generalized Golod-Shafarevich type theorems for potential algebras, to determine finiteness conditions in terms of the potential. We consider two-generated potential algebras, and prove that they can not have dimension smaller than 8, using Groebner bases arguments, and arguing in terms of associated truncated algebra. We derive from the improved version of the Golod-Shafarevich theorem,that if the potential has only terms of degree 5 or higher, then the potential algebra is infinite dimensional. We prove, that potential algebra for any homogeneous potential of degree n>=3 is infinite dimensional. The proof includes a complete classification of all potentials of degree 3. Then we introduce a certain version of Koszul complex, and prove that in the class P_n of potential algebras with homogeneous potential of degree n+1>= 4, the minimal Hilbert series is H_n=1/1-2t+2t^n-t^{n+1}, so they are all infinite dimensional. Moreover, growth could be polynomial (but at least quadratic) for the potential of degree 4, and is always exponential for potential of degree starting from 5. For one particular type of potential we prove a conjecture by Wemyss, which relates the difference of dimensions of potential algebra and its abelianization with Gopakumar-Vafa invaria 02/07/2016 PDF Divisor Braids M/16/18 Voir plus > Voir moins > We study a novel type of braid groups on a closed orientable surface Σ. These are fundamental groups of certain manifolds that are hybrids between symmetric products and configuration spaces of points on Σ; a class of examples arises naturally in gauge theory, as moduli spaces of vortices in toric fibre bundles over Σ. The elements of these braid groups, which we call divisor braids, have coloured strands that are allowed to intersect according to rules specified by a graph Γ. In situations where there is more than one strand of each colour, we show that the corresponding braid group admits a metabelian presentation as a central extension of the free Abelian group H_1(Σ;Z)^r, where r is the number of colours, and describe its Abelian commutator. This computation relies crucially on producing a link invariant (of closed divisor braids) in the three-manifold S^1×Σ for each graph Γ. We also describe the von Neumann algebras associated to these groups in terms of rings that are familiar from noncommutative geometry. 25/05/2016 PDF One question from the Polishchuk and Positselski book on Quadratic algebras M/16/16 Voir plus > Voir moins > In the book 'Quadratic algebras' due to Polishchuk and Positselski algebras with small number of generators (n=2,3) is considered. For some number of relations r possible Hilbert series are listed, and those appearing as series of Koszul algebras are specified. The first case, where it was not possible to do, namely the case of three generators n=3 and three relations r=6 is formulated as an open problem. We give here a complete answer to this question, namely for quadratic algebra A with dim A_1=dim A_2=3 we list all possible Hilbert series, and find out which of them can come from Koszul algebras, and which can not. 05/05/2016 PDF On the proof of the homology conjecture for monomial non-unital algebras M/16/15 Voir plus > Voir moins > We consider the bar complex of a monomial non-unital associative algebra A=k / (w_1,...,w_t). For any fixed monomial w=x_1..x_n in A one can define certain subcomplex of the Bar complex of A. It was conjectured in [3] that homology of this complex is at most one. We prove here this conjecture, and describe the place where this nontrivial homology appears in terms of length of the Dyck path associated to a given word in A. 04/05/2016 PDF Periodic subvarieties of a projective variety under the action of a maximal rank abelian group of positive entropy M/16/14 19/04/2016 PDF Conservative second-order gravitational self-force on circular orbits and the effective one-body formalism P/16/13 01/04/2016 PDF Do the Kontsevich tetrahedral flows preserve or destroy the space of Poisson bi-vectors? M/16/12 Voir plus > Voir moins > We examine two claims from the paper "Formality Conjecture" (Ascona 1996): specifically, that 1) a certain tetrahedral graph flow preserves the class of (real-analytic) Poisson structures, and that 2) another tetrahedral graph flow vanishes at every such Poisson structure. By using twelve Poisson structures with high-order polynomial coefficients as explicit counterexamples, we show that both the above claims are false: neither does the first flow preserve the property of bi-vectors to be Poisson nor does the second flow vanish identically at the Poisson bi-vectors. The counterexamples at hand themselves suggest a correction to the formula for the "exotic" flow on the space of Poisson bi-vectors; in fact, this flow is encoded by the balanced sum involving both the Kontsevich tetrahedral graphs (that give rise to the flows mentioned above). We reveal that it is only the balance (1:6) for which the flow does preserve the space of Poisson bi-vectors. 31/03/2016 PDF BADLY APPROXIMABLE VECTORS AND FRACTALS DEFINED BY CONFORMAL DYNAMICAL SYSTEMS M/16/10 Voir plus > Voir moins > We prove that if J is the limit set of an irreducible conformal iterated function system (with either finite or countably infinite alphabet), then the badly approximable vectors form a set of full Hausdorff dimension in J. The same is true if J is the radial Julia set of an irreducible meromorphic function (either rational or transcendental). The method of proof is to find subsets of J that support absolutely friendly and Ahlfors regular measures of large dimension. In the appendix to this paper, we answer a question of Broderick, Kleinbock, Reich, Weiss, and the second-named author (’12) by showing that every hyperplane diffuse set supports an absolutely decaying measure. 22/03/2016 PDF Real Analyticity for random dynamics of transcendental functions M/16/11 Voir plus > Voir moins > Analyticity results of expected pressure and invariant densities in the context of random dynamics of transcendental functions are established. These are obtained by a refinement of work by Rugh \cite{Rug08} leading to a simple approach to analyticity. We work under very mild dynamical assumptions. Just the iterates of the Perron-Frobenius operator are assumed to converge. We also provide a Bowen's formula expressing the almost sure Hausdorff dimension of the radial fiberwise Julia sets in terms of the zero of an expected pressure function. Our main application states real analyticity for the variation of this dimension for suitable hyperbolic random systems of entire or meromorphic functions. 22/03/2016 PDF SU(N) transitions in M-theory on Calabi-Yau fourfolds and background fluxes P/16/07 Voir plus > Voir moins > We study M-theory on a Calabi-Yau fourfold with a smooth surface S of AN−1 singularities. The resulting three-dimensional theory has a =2 SU(N) gauge theory sector, which we obtain from a twisted dimensional reduction of a seven-dimensional =1 SU(N) gauge theory on the surface S. A variant of the Vafa-Witten equations governs the moduli space of the gauge theory, which, for a trivial SU(N) principal bundle over S, admits a Coulomb and a Higgs branch. In M-theory these two gauge theory branches arise from a resolution and a deformation to smooth Calabi-Yau fourfolds, respectively. We find that the deformed Calabi-Yau fourfold associated to the Higgs branch requires for consistency a non-trivial four-form background flux in M-theory. The flat directions of the flux-induced superpotential are in agreement with the gauge theory prediction for the moduli space of the Higgs branch. We illustrate our findings with explicit examples that realize the Coulomb and Higgs phase transition in Calabi-Yau fourfolds embedded in weighted projective spaces. We generalize and enlarge this class of examples to Calabi-Yau fourfolds embedded in toric varieties with an AN−1 singularity in codimension two. 09/03/2016 PDF New gravitational self-force analytical results for eccentric orbits around a Schwarzschild black hole P/16/08 09/03/2016 PDF High post-Newtonian order gravitational self-force analytical results for eccentric orbits around a Kerr black hole P/16/09 09/03/2016 PDF Intermittency in the Hodgkin-Huxley model M/16/06 Voir plus > Voir moins > We show that action potentials in the Hodgkin-Huxley neuron model result from a type I intermittency phenomenon that occurs in the proximity of a saddle-node bifurcation of limit cycles. For the Hodgkin-Huxley spatially extended model, describing propagation of action potential along axons, we show the existence of type I intermittency and a new type of chaotic intermittency, as well as space propagating regular and chaotic diffusion waves. Chaotic intermittency occurs in the transition from a turbulent regime to the resting regime of the transmembrane potential and is characterised by the existence of a sequence of action potential spikes occurring at irregular time intervals. 26/02/2016 PDF The Langlands-Shahidi method over function fields: the Ramanujan Conjecture and the Riemann Hypothesis for the unitary groups M/16/05 Voir plus > Voir moins > On étudie la méthode de Langlands-Shahidi sur les corps de fonctions de caractéristique p. On prouve la fonctorialité de Langlands globale et locale des groupes unitaires vers les groupes linéaires pour les représentations génériques. Supposant connue la conjecture de Shahidi pour les L-paquets modérés, on donne une extension de la définition des fonctions L et des facteurs ε. Enfin, utilisant le travail de L. Lafforgue, on établit la conjecture de Ramanujan et on prouve que les fonctions L automorphes de Langlands-Shahidi satisfont l'hypothèse de Riemann. 11/02/2016 PDF Class number problems and Lang conjectures M/16/04 Voir plus > Voir moins > Given a square-free integer d we introduce an affine hypersurface whose integer points are in one-to-one correspondence with ideal classes of the quadratic number field Q(\sqrt{d}). Using this we relate class number problems of Gauss to Lang conjectures. 01/02/2016 PDF On Emergent Geometry from Entanglement Entropy in Matrix Theory P/16/03 Voir plus > Voir moins > Using Matrix theory, we compute the entanglement entropy between a supergravity probe and modes on a spherical membrane. We demonstrate that a membrane stretched between the probe and the sphere entangles these modes and leads to an expression for the entanglement entropy that encodes information about local gravitational geometry seen by the probe. We propose in particular that this entanglement entropy measures the rate of convergence of geodesics at the location of the probe. 29/01/2016 PDF Perturbative quantum field theory meets number theory P/16/02 27/01/2016 PDF On the conservative dynamics of two-body systems at the fourth post-Newtonian approximation of general relativity P/16/01 08/01/2016 PDF Modular Graph Functions P/15/29 Voir plus > Voir moins > We consider properties of modular graph functions, which are non-holomorphic modular functions associated with the Feynman graphs for a conformal scalar field theory on a two-dimensional torus. Such functions arise, for example, in the low energy expansion of genus-one Type II superstring amplitudes. We demonstrate that these functions are sums, with rational coefficients, of special values of single-valued elliptic multiple polylogarithms, which will be introduced in this paper. This insight suggests the many interrelations between these modular graph functions (a few of which were motivated in an earlier paper) may be obtained as a consequence of identities involving elliptic polylogarithms. 22/12/2015 PDF Cutkosky Rules and Outer Space P/15/34 Voir plus > Voir moins > We derive Cutkosky’s theorem starting from Pham’s classical work. We emphasize structural relations to Outer Space. 08/12/2015 PDF Higher Chern classes in Iwasawa theory M/15/33 Voir plus > Voir moins > We begin a study of m-th Chern classes and m-th characteristic symbols for Iwasawa modules which are supported in codimension at least m. This extends the classical theory of characteristic ideals and their generators for Iwasawa modules which are torsion, i.e., supported in codimension at least 1. We apply this to an Iwasawa module constructed from an inverse limit of p-parts of ideal class groups of abelian extensions of an imaginary quadratic field. When this module is pseudo-null, which is conjecturally always the case, we determine its second Chern class and show that it has a characteristic symbol given by the Steinberg symbol of two Katz p-adic L-functions. 05/12/2015 PDF Confirming and improving post-Newtonian and effective-one-body results from self-force computations along eccentric orbits around a Schwarzschild black hole P/15/31 04/12/2015 PDF Minisuperspace quantum supersymmetric cosmology (and its hidden hyperbolic Kac-Moody structures) P/15/32 04/12/2015 PDF Persistent homology and string vacua P/15/30 Voir plus > Voir moins > We use methods from topological data analysis to study the topological features of certain distributions of string vacua. Topological data analysis is a multi-scale approach used to analyze the topological features of a dataset by identifying which homological characteristics persist over a long range of scales. We apply these techniques in several contexts. We analyze N=2 vacua by focusing on certain distributions of Calabi-Yau varieties and Landau-Ginzburg models. We then turn to flux compactifications and discuss how we can use topological data analysis to extract physical informations. Finally we apply these techniques to certain phenomenologically realistic heterotic models. We discuss the possibility of characterizing string vacua using the topological properties of their distributions. 03/12/2015 PDF Catégories syntactiques pour les motifs de Nori M/15/26 13/11/2015 PDF A new effective-one-body Hamiltonian with next-to-leading order spin-spin coupling P/15/27 13/11/2015 PDF Spin-dependent two-body interactions from gravitational self-force computations P/15/28 13/11/2015 PDF BPS spectra, barcodes and walls P/15/25 Voir plus > Voir moins > BPS spectra give important insights into the non-perturbative regimes of supersymmetric theories. Often from the study of BPS states one can infer properties of the geometrical or algebraic structures underlying such theories. In this paper we approach this problem from the perspective of persistent homology. Persistent homology is at the base of topological data analysis, which aims at extracting topological features out of a set of points. We use these techniques to investigate the topological properties which characterize the spectra of several supersymmetric models in field and string theory. We discuss how such features change upon crossing walls of marginal stability in a few examples. Then we look at the topological properties of the distributions of BPS invariants in string compactifications on compact threefolds used to engineer black hole microstates. Finally we discuss the interplay between persistent homology and modularity by considering certain number theoretical functions used to count dyons in string compactifications and by studying equivariant elliptic genera in the context of the Mathieu moonshine. 04/11/2015 PDF Moduli Spaces and Macromolecules M/15/24 Voir plus > Voir moins > Techniques from moduli spaces are applied to biological macromolecules. The first main result provides new a priori constraints on protein geometry discovered empirically and confirmed computationally. The second main result identifies up to homotopy the natural moduli space of several interacting RNA molecules with the Riemann moduli space of a surface with several boundary components in each fixed genus. Applications to RNA folding prediction are discussed. The mathematical and biological frameworks are surveyed and presented from first principles. 30/10/2015 PDF Smoothness and classicality on eigenvarieties M/15/23 Voir plus > Voir moins > Let p be a prime number and f an overconvergent p-adic automorphic form on a definite unitary group which is split at p. Assume that f is of "classical weight" and that its Galois representation is crystalline at places dividing p, then f is conjectured to be a classical automorphic form. We prove new cases of this conjecture in arbitrary dimension by making crucial use of the "patched eigenvariety". 07/10/2015 PDF A program for branching problems in the representation theory of real reductive groups M/15/22 Voir plus > Voir moins > We wish to understand how irreducible representations of a group G behave when restricted to a subgroup G' (the branching problem). Our primary concern is with representations of reductive Lie groups, which involve both algebraic and analytic approaches. We divide branching problems into three stages: (A) abstract features of the restriction; (B) branching laws (irreducible decompositions of the restriction); and (C) construction of symmetry breaking operators on geometric models. We could expect a simple and detailed study of branching problems in Stages B and C in the settings that are {\it{a priori}} known to be "nice" in Stage A, and conversely, new results and methods in Stage C that might open another fruitful direction of branching problems including Stage A. The aim of this article is to give new perspectives on the subjects, to explain the methods based on some recent progress, and to raise some conjectures and open questions. 29/09/2015 PDF Decorated super-Teichmueller space M/15/21 25/09/2015 PDF La suite spectrale de Hodge-Tate M/15/20 Voir plus > Voir moins > The Hodge-Tate spectral sequence for a proper smooth variety over a p-adic field provides a framework for us to revisit Faltings' approach to p-adic Hodge theory and to fill in many details. The spectral sequence is obtained from the Cartan-Leray spectral sequence for the canonical projection from the Faltings topos to the étale topos of an integral model of the variety. Its abutment is computed by Faltings' main comparison theorem from which derive all comparison theorems between p-adic étale cohomology and other p-adic cohomologies, and its initial term is related to the sheaf of differential forms by a construction reminiscent of the Cartier isomorphism. 14/09/2015 PDF On the modular structure of the genus-one Type II superstring low energy expansion P/15/04 Voir plus > Voir moins > The analytic contribution to the low energy expansion of Type II string amplitudes at genus-one is a power series in space-time derivatives with coefficients that are determined by integrals of modular functions over the complex structure modulus of the world-sheet torus. These modular functions are associated with world-sheet vacuum Feynman diagrams and given by multiple sums over the discrete momenta on the torus. In this paper we exhibit exact differential and algebraic relations for a certain infinite class of such modular functions by showing that they satisfy Laplace eigenvalue equations with inhomogeneous terms that are polynomial in non-holomorphic Eisenstein series. Furthermore, we argue that the set of modular functions that contribute to the coefficients of interactions up to order$D^{10} \cR^4$are linear sums of functions in this class and quadratic polynomials in Eisenstein series and odd Riemann zeta values. Integration over the complex structure results in coefficients of the low energy expansion that are rational numbers multiplying monomials in odd Riemann zeta values. 02/09/2015 PDF Proof of a modular relation between 1-, 2- and 3-loop Feynman diagrams on a torus P/15/07 Voir plus > Voir moins > The coefficients of the higher-derivative terms in the low energy expansion of genus-one graviton scattering amplitudes are determined by integrating sums of non-holomorphic modular functions over the complex structure modulus of a torus. In the case of the four-graviton amplitude, each of these modular functions is a multiple sum associated with a Feynman diagram for a free massless scalar field on the torus. The lines in each diagram join pairs of vertex insertion points and the number of lines defines its weight$w$, which corresponds to its order in the low energy expansion. Previous results concerning the low energy expansion of the genus-one four-graviton amplitude led to a number of conjectured relations between modular functions of a given$w$, but different numbers of loops$\le w-1$. In this paper we shall prove the simplest of these conjectured relations, namely the one that arises at weight$w=4$and expresses the three-loop modular function$D_4$in terms of modular functions with one and two loops. As a byproduct, we prove three intriguing new holomorphic modular identities. 02/09/2015 PDF More Graviton Physics P/15/05 Voir plus > Voir moins > The interactions of gravitons with spin-1 matter are calculated in parallel with the well known photon case. It is shown that graviton scattering amplitudes can be factorized into a product of familiar electromagnetic forms, and cross sections for various reactions are straightforwardly evaluated using helicity methods. Universality relations are identified. Extrapolation to zero mass yields scattering amplitudes for photon-graviton and graviton-graviton scattering. 02/09/2015 PDF Fractal Tube Formulas and a Minkowski Measurability Criterion for Compact Subsets of Euclidean Spaces M/15/17 Voir plus > Voir moins > We establish fractal tube formulas valid for a large class of compact subsets (and more generally, relative fractal drums, RFDs) in Euclidean spaces of any dimension. These formulas express the volume of the tubular neighborhoods of the fractal under consideration in terms of the residues of the associated fractal zeta functions and their poles (i.e., the complex dimensions). Under suitable assumptions, we also show that a compact subset of R^N is Minkowski measurable if and only if its only complex dimension of maximum real part is the Minkowski (or box) dimension D of the fractal (or RFD), and D is simple. These results extend to arbitrary dimensions N greater than 1 the corresponding ones obtained by Lapidus and van Frankenhuijsen for fractal strings (i.e., when N =1). We illustrate them by means of several examples. 27/07/2015 PDF Fractal Zeta Functions and Complex Dimensions of Relative Fractal Drums M/15/14 15/07/2015 PDF Distance and Tube Zeta Functions of Fractals and Arbitrary Compact Sets M/15/15 15/07/2015 PDF Fractal Zeta Functions and Complex Dimensions: A General Higher-Dimensional Theory M/15/16 15/07/2015 PDF Deformation approach to quantisation of field models M/15/13 Voir plus > Voir moins > Associativity-preserving deformation quantisation via Kontsevich's summation over weighted graphs is lifted from the algebras of functions on finite-dimensional Poisson manifolds to the algebras of local functionals within the variational Poisson geometry of gauge fields over the space-time. 14/07/2015 PDF The Sound of Fractal Strings and the Riemann Hypothesis M/15/11 07/07/2015 PDF Towards Quantized Number Theory: Spectral Operators and an Asymmetric Criterion for the Riemann Hypothesis M/15/12 Voir plus > Voir moins > An asymmetric criterion for the Riemann hypothesis is provided, expressed in terms of the invertibility of the spectral operator. This criterion is asymmetric, in the sense that it is valid for all fractal dimensions c in (0,1/2) if and only if the Riemann hypothesis is true, but that (unconditionally) it fails to be true for any value of c in (1/2, 1), a fact which is closely connected to the universality of the Riemann zeta function. 07/07/2015 PDF Energetics and phasing of nonprecessing spinning coalescing black hole binaries P/15/19 28/06/2015 PDF Deformations of complex structures on Riemann surfaces and integrable structures of Whitham type hierarchies M/15/10 Voir plus > Voir moins > We obtain variational formulas for holomorphic objects on Riemann surfaces with respect to arbitrary local coordinates on the moduli space of complex structures. These formulas are written in terms of a canonical object on the moduli space which corresponds to the pairing between the space of quadratic differentials and the tangent space to the moduli space. This canonical object satisfies certain commutation relations which appear to be the same as the ones that emerged in the integrability theory of Whitham type hierarchies. Driven by this observation, we develop the theory of Whitham type hierarchies integrable by hydrodynamic reductions as a theory of certain differential-geometric objects. As an application we prove that the universal Whitham hierarchy is integrable by hydrodynamic reductions. 12/06/2015 PDF On the large-scale geometry of the$L^p$-metric on the symplectomorphism group of the two-sphere M/15/09 Voir plus > Voir moins > We prove that the vector space$R^d$of any finite dimension$d$with the standard metric embeds in a bi-Lipschitz way into the group of area-preserving diffeomorphisms of the two-sphere endowed with the$L^p$-metric for$p>2$. Along the way we show that the$L^p$-metric on this group is unbounded for$p>2$by elementary methods. 10/06/2015 PDF The Equivalence Principle in a Quantum World P/15/06 20/05/2015 PDF Lectures on Regular and Irregular Holonomic D-modules M/15/08 Voir plus > Voir moins > This is a survey paper based on lectures given by the authors at Ihes, February/March 2015. In a first part, we recall the main results on the tempered holomorphic solutions of D-modules in the language of indsheaves and, as an application, the Riemann-Hilbert correspondence for regular holonomic modules. In a second part, we present the enhanced version of the first part, treating along the same lines the irregular holonomic case. 18/05/2015 PDF Fourth post-Newtonian effective one-body dynamics P/15/18 30/04/2015 PDF Analytic determination of high-order post-Newtonian self-force contributions to gravitational spin precession P/15/17 04/03/2015 PDF Detweiler's gauge-invariant redshift variable: analytic determination of the nine and nine-and-a-half post-Newtonian self-force contributions P/15/16 09/02/2015 PDF Differential symmetry breaking operators I. General theory and F-method M/15/02 Voir plus > Voir moins > We prove a one-to-one correspondence between differential symmetry breaking operators for equivariant vector bundles over two homogeneous spaces and certain homomorphisms for representations of two Lie algebras, in connection with branching problems of the restriction of representations. We develop a new method (F-method) based on the algebraic Fourier transform for generalized Verma modules, which characterizes differential symmetry breaking operators by means of certain systems of partial differential equations. In contrast to the setting of real flag varieties, continuous symmetry breaking operators of Hermitian symmetric spaces are proved to be differential operators in the holomorphic setting. In this case symmetry breaking operators are characterized by differential equations of second order via the F-method. 11/01/2015 PDF Differential symmetry breaking operators. II. Rankin--Cohen operators for symmetric pairs M/15/03 Voir plus > Voir moins > Rankin--Cohen brackets are symmetry breaking operators for the tensor product of two holomorphic discrete series representations of SL(2,R). We address a general problem to find explicit formulae, for such intertwining operators in the setting of multiplicity-free branching laws for reductive symmetric pairs. For this purpose we use a new method (F-method) developed in the first part of the series and based on the algebraic Fourier transform for generalized Verma modules. The method characterizes symmetry breaking operators by means of certain systems of partial differential equations of second order. We discover explicit formulae, of new differential symmetry breaking operators for all the six different complex geometries arising from semisimple symmetric pairs of split rank one, and reveal an intrinsic reason why the coefficients of orthogonal polynomials appear in these operators (Rankin--Cohen type) in the three geometries and why normal derivatives are symmetry breaking operators in the other three cases. Further, we analyze a new phenomenon that the multiplicities in the branching laws of Verma modules may jump up at singular parameters. 11/01/2015 PDF Global uniqueness of small representations M/15/01 Voir plus > Voir moins > We prove that automorphic representations whose local components are certain small representations have multiplicity one. The proof is based on the multiplicity-one theorem for certain functionals of small representations, also proved in this paper. 04/01/2015 PDF On the integral law of thermal radiation P/14/43 Voir plus > Voir moins > The integral law of thermal radiation by finite size emitters is studied. Two geometrical characteristics of a radiating body or a cavity, its volume and its boundary area, define two terms in its radiance. The term defined by the volume corresponds to the Stefan-Boltzmann law. The term defined by the boundary area is proportional to the third power of temperature and inversely proportional to the emitter's effective size, which is defined as the ratio of its volume to its boundary area. This generalized law is valid for arbitrary temperature and effective size. It is shown that the cubic temperature contribution is observed in experiments. This term explains the intrinsic uncertainty of the NPL experiment on radiometric determination of the Stefan-Boltzmann constant. It is also quantitatively confirmed by data from the NIST calibration of cryogenic blackbodies. Its relevance to the size of source effect in optical radiometry is proposed and supported by the experiments on thermal emission from nano-heaters. 12/12/2014 PDF On weight modules of algebras of twisted differential operators on the projective space M/14/42 Voir plus > Voir moins > We classify blocks of categories of weight and generalized weight modules of algebras of twisted differential operators on P^n. Necessary and sufficient conditions for these blocks to be tame and proofs that some of the blocks are Koszul are provided. We also establish equivalences of categories between these blocks and categories of bounded and generalized bounded weight sl(n+1)-modules in the cases of nonintegral and singular central character. 11/12/2014 PDF Topological invariants in magnetohydrodynamics and DNA supercoiling M/14/41 Voir plus > Voir moins > We discuss the structure of topological invariants in two different media. The first example relates to the problem of reconnection in magnetohydrodynamics and the second one to the supercoiling of DNA. Despite the apparently different systems, the behavior of magnetic spread lines and supercoiling process in DNA display some common features based on the existence of topological invariants of Hopf's type. 10/12/2014 PDF$r_\infty$-Matrices, triangular$L_\infty$-bialgebras, and quantum$_\infty$groups M/14/40 Voir plus > Voir moins > A homotopy analogue of the notion of a triangular Lie bialgebra is proposed with a goal of extending the basic notions of theory of quantum groups to the context of homotopy algebras and, in particular, introducing a homotopical generalization of the notion of a quantum group, or quantum$_\infty$-group. 09/12/2014 PDF The calculus of multivectors on noncommutative jet spaces M/14/39 Voir plus > Voir moins > The Leibniz rule for derivations is invariant under cyclic permutations of the co-multiples within the derivations' arguments. We now explore the implications of this fundamental principle, developing the calculus of variations on the infinite jet spaces for maps from sheaves of free associative algebras over commutative manifolds to the quotients of free associative algebras over the linear relation of equivalence under cyclic shifts. In the frames of such variational noncommutative symplectic geometry, we prove the main properties of the Batalin-Vilkovisky Laplacian and variational Schouten bracket. As a by-product of this intrinsically regularised picture, we show that the structures that arise in the classical variational Poisson geometry of infinite-dimensional integrable systems - such as the KdV, NLS, KP, or 2D Toda - do actually not refer to the graded commutativity assumption. 07/12/2014 PDF Smooth approximation of plurisubharmonic functions on almost complex manifolds M/14/38 Voir plus > Voir moins > This note establishes smooth approximation from above for J-plurisubharmonic functions on an almost complex manifold (X,J). The following theorem is proved. Suppose X is J-pseudoconvex, i.e., X admits a smooth strictly J-plurisubharmonic exhaustion function. Let u be an (upper semi-continuous) J-plurisubharmonic function on X. Then there exists a sequence {u_j} of smooth strictly J-plurisubharmonic functions point-wise decreasing down to u. In any almost complex manifold (X,J) each point has a fundamental neighborhood system of J-pseudoconvex domains, and so the theorem above establishes local smooth approximation on X. This result was proved in complex dimension 2 by the third author, who also showed that the result would hold in general dimensions if a parallel result for continuous approximation were known. This paper establishes the required step by solving the obstacle problem. 27/11/2014 PDF Une interprétation modulaire de la variété trianguline M/14/37 Voir plus > Voir moins > En utilisant le système de Taylor-Wiles-Kisin construit dans un travail récent de Caraiani, Emerton, Gee, Geraghty, Paškūnas et Shin, nous construisons un analogue de la variété de Hecke. Nous montrons que cette variété coïncide avec une union de composantes irréductibles de l'espace des représentations galoisiennes triangulines. Nous précisons les relations de cette construction avec les conjectures de modularité dans le cas cristallin ainsi qu'avec une conjecture de Breuil sur le socle des vecteurs localement analytiques de la cohomologie complétée. Nous donnons également une preuve d'une conjecture de Bellaïche et Chenevier sur l'anneau local complété en certains points des variétés de Hecke. 26/11/2014 PDF The BV formalism for L$_\infty$-algebras M/14/36 Voir plus > Voir moins > The notions of a BV$_\infty$-morphism and a category of BV$_\infty$-algebras are investigated. The category of L$_\infty$-algebras with L$_\infty$-morphisms is characterized as a certain subcategory of the category of BV$_\infty$-algebras. This provides a Fourier-dual, BV alternative to the standard characterization of the category of L$_\infty$-algebras as a subcategory of the category of dg cocommutative coalgebras or formal pointed dg manifolds. The functor assigning to a BV$_\infty$-algebra the L$_\infty$-algebra given by higher derived brackets is also shown to have a left adjoint. 23/10/2014 PDF Graviton-Photon Scattering P/14/32 Voir plus > Voir moins > We use that the gravitational Compton scattering factorizes on the Abelian QED amplitudes to evaluate various gravitational Comp- ton processes. We examine both the QED and gravitational Compton scattering from a massive spin-1 system by the use of helicity am- plitudes. In the case of gravitational Compton scattering we show how the massless limit can be used to evaluate the cross-section for graviton-photon scattering and discuss the difference between photon interactions and the zero mass spin-1 limit. We show that the forward scattering cross-section for graviton photo-production has a very pecu- liar behaviour, differing from the standard Thomson and Rutherford cross-sections for a Coulomb-like potential. 16/10/2014 PDF Three dimensional Sklyanin algebras and Groebner bases M/14/35 Voir plus > Voir moins > We consider a Sklyanin algebra S with 3 generators, which is the quadratic algebra over a field K with 3 generators x,y,z given by 3 relations pxy+qyx+rzz=0, pyz+qzy+rxx=0 and pzx+qxz+ryy=0. This class of algebras enjoyed much of attention, in particular, using tools from algebraic geometry, Feigin, Odesskii, and Artin, Tate, Van den Berg, showed that if at least two of the parameters p, q and r are non-zero and at least two of three cubes of p, q and r are distinct, then S is Koszul and has the same Hilbert series as the algebra of commutative polynomials in 3 variables. It became commonly accepted, that it is impossible to achieve the same objective by purely algebraic and combinatorial means, like the Groebner basis technique. The main purpose of this paper is to trace the combinatorial meaning of the properties of Sklyanin algebras, such as Koszulity, PBW, PHS, Calabi-Yau, and to give a new constructive proof of the above facts due to Artin, Tate and Van den Bergh. 14/10/2014 PDF Geometry of Morphogenesis M/14/34 01/10/2014 PDF On Koszulity for operads of Conformal Field Theory M/14/31 Voir plus > Voir moins > We study two closely related operads: the Gelfand-Dorfman operad GD and the Conformal Lie Operad CLie. The latter is the operad governing the Lie conformal algebra structure. We prove Koszulity of the Conformal Lie operad using the Gr ̈bner bases theory for operads and an operadic analogue of the Priddy criterion. An example of deformation of an operad coming from the Hom structures is considered. In particular we study possible deformations of the Associative operad from the point of view of the confluence property. Only one deformation, the operad which governs the identity (α(ab))c = a(α(bc)) turns out to be confluent. We introduce a new Hom structure, namely Hom–Gelfand-Dorfman algebras and study their basic properties. 24/09/2014 PDF The proof of the Kontsevich periodicity conjecture on noncommutative birational transformations M/14/30 Voir plus > Voir moins > For an arbitrary associative unital ring R, let J1 and J2 be the following noncommutative birational partly defined involutions on the set M3 (R) of 3 × 3 matrices over R: J1 (M ) = M −1 (the usual matrix inverse) and J2 (M )jk = (Mkj )−1 (the transpose of the Hadamard inverse). We prove the following surprising conjecture by Kontsevich (1996) saying that (J2 ◦ J1 )3 −1 is the identity map modulo the DiagL × DiagR action (D1 , D2 )(M ) = D1 M D2 of pairs of invertible diagonal matrices. That is, we show that for each M in the domain where (J2 ◦J1 )3 is defined, there are invertible −1 diagonal 3 × 3 matrices D1 = D1 (M ) and D2 = D2 (M ) such that (J2 ◦ J1 )3 (M ) = D1 M D2. 24/09/2014 PDF Topos-theoretic background M/14/27 Voir plus > Voir moins > This text, which will form the first chapter of my book in preparation "Lattices of theories", is a self-contained introduction to topos theory, geometric logic and the 'bridge' technique. 23/09/2014 PDF Lattice-ordered abelian groups and perfect MV-algebras: a topos-theoretic perspective M/14/28 Voir plus > Voir moins > We establish, generalizing Di Nola-Lettieri's categorical equivalence, a Morita-equivalence between the theory of lattice-ordered abelian groups and that of perfect MV-algebras. Further, after observing that the two theories are not bi-interpretable in the classical sense, we identify, by considering appropriate topos-theoretic invariants on their common classifying topos, three levels of bi-interpretability holding for particular classes of formulas: irreducible formulas, geometric sentences and imaginaries. Lastly, by investigating the classifying topos of the theory of perfect MV-algebras, we obtain various results on its syntax and semantics also in relation to the cartesian theory of the variety generated by Chang's MV-algebra, including a concrete representation for the finitely presentable models of the latter theory as finite products of finitely presentable perfect MV-algebras. Among the results established on the way, we mention a Morita-equivalence between the theory of lattice-ordered abelian groups and that of cancellative lattice-ordered abelian monoids with bottom element. 23/09/2014 PDF Quasi-exact-solvability of the$A_2$elliptic model: Algebraic form,$sl(3)$hidden algebra, polynomial eigenfunctions P/14/29 23/09/2014 PDF Dimensional exactness of self-measures for random countable iterated function systems with overlaps. M/14/26 Voir plus > Voir moins > We study projection measures for random countable (finite or infinite) conformal iterated function systems with arbitrary overlaps. In this setting we extend Feng's and Hu's result from [6] about deterministic finite alphabet iterated function systems. We prove, under a mild assumption of finite entropy, the dimensional exactness of the projections of invariant measures from the shift space, and we give a formula for their dimension, in the context of random infinite conformal iterated function systems with overlaps. There exist numerous differences between our case and the finite deterministic case. We give then applications and concrete estimates for pointwise dimensions of measures, with respect to various classes of random countable IFS with overlaps. Namely, we study several types of randomized systems related to Kahane-Salem sets; also, a random system related to a statistical problem of Sinai; and randomized infinite IFS in the plane for which the number of overlaps is uniformly bounded from above. 09/09/2014 PDF Principe de fonctorialité et transformations de Fourier non linéaires : proposition de définitions et esquisse d'une possible (?) démonstration M/14/25 Voir plus > Voir moins > Ce texte rassemble les notes écrites d'une série de quatre exposés donnés à l'IHES les 19 juin, 26 juin, 3 juillet et 8 juillet 2014. Il introduit une nouvelle approche pour une éventuelle démonstration - encore à vérifier - du transfert automorphe de Langlands sur les corps globaux. En attendant donc de vérifier soigneusement si cela marche ou bien non, d'abord dans le cas de GL(2) et des représentations de puissances symétriques de son dual. Le point le plus essentiel, sur lequel tout est fondé, est la propriété de stabilité du tore maximal par convolution (définie comme la transformée de Fourier de la multiplication point par point des fonctions) apparue dans la dernière partie de la partie III. 21/08/2014 PDF Extensions of flat functors and theories of presheaf type M/14/23 Voir plus > Voir moins > We develop a general theory of extensions of flat functors along geometric morphisms of toposes, and apply it to the study of the class of theories whose classifying topos is equivalent to a presheaf topos. As a result, we obtain a characterization theorem providing necessary and sufficient semantic conditions for a theory to be of presheaf type. This theorem subsumes all the previous partial results obtained on the subject and has several corollaries which can be used in practice for testing whether a given theory is of presheaf type as well as for generating new examples of theories belonging to this class. Along the way, we establish a number of other results of independent interest, including developments about colimits in the context of indexed categories, expansions of geometric theories and methods for constructing theories classified by a given presheaf topos. 26/06/2014 PDF Cyclic theories M/14/22 Voir plus > Voir moins > We describe a geometric theory classified by Connes-Consani's epicylic topos and two related theories respectively classified by the cyclic topos and by the topos$[{\mathbb N}^{\ast}, \Set]$. 26/06/2014 PDF A Feynman integral via higher normal functions P/14/06 Voir plus > Voir moins > We study the Feynman integral for the three-banana graph defined as the scalar two-point self-energy at three-loop order. The Feynman integral is evaluated for all identical internal masses in two space-time dimensions. Two calculations are given for the Feynman integral; one based on an interpretation of the integral as an inhomogeneous solution of a classical Picard-Fuchs differential equation, and the other using arithmetic algebraic geometry, motivic cohomology, and Eisenstein series. Both methods use the rather special fact that the Feynman integral is a family of regulator periods associated to a family of$K3$surfaces. We show that the integral is given by a sum of elliptic trilogarithms evaluated at sixth roots of unity. This elliptic trilogarithm value is related to the regulator of a class in the motivic cohomology of the$K3$family. We prove a conjecture by David Broadhurst that at a special kinematical point the Feynman integral is given by a critical value of the Hasse-Weil$L$-function of the$K3$surface. This result is shown to be a particular case of Deligne's conjectures relating values of$L$-functions inside the critical strip to periods. 10/06/2014 PDF Beltrami-Courant Differentials and$G_{\infty}$-algebras M/14/19 Voir plus > Voir moins > Using the symmetry properties of two-dimensional sigma models, we introduce a notion of the Beltrami-Courant differential, so that there is a natural homotopy Gerstenhaber algebra related to it. We conjecture that the generalized Maurer-Cartan equation for the corresponding$L_{\infty}$subalgebra gives solutions to the Einstein equations. 06/05/2014 PDF 2-CY-tilted algebras that are not Jacobian M/14/17 Voir plus > Voir moins > Over any field of positive characteristic we construct 2-CY-tilted algebras that are not Jacobian algebras of quivers with potentials. As a remedy, we propose an extension of the notion of a potential, called hyperpotential, that allows to prove that certain algebras defined over fields of positive characteristic are 2-CY-tilted even if they do not arise from potentials. In another direction, we compute the fractionally Calabi-Yau dimensions of certain orbit categories of fractionally CY triangulated categories. As an application, we construct a cluster category of type G2. 29/04/2014 PDF Algebras of quasi-quaternion type M/14/18 Voir plus > Voir moins > We define algebras of quasi-quaternion type, which are symmetric algebras of tame representation type whose stable module category has certain structure similar to that of the algebras of quaternion type introduced by Erdmann. We observe that symmetric tame algebras that are also 2-CY-tilted are of quasi quaternion type. We present a combinatorial construction of such algebras by introducing the notion of triangulation quivers. The class of algebras that we get contains Erdmann's algebras of quaternion type on the one hand and the Jacobian algebras of the quivers with potentials associated by Labardini to triangulations of closed surfaces with punctures on the other hand, hence it serves as a bridge between modular representation theory of finite groups and cluster algebras. 29/04/2014 PDF SL(2,Z)-invariance and D-instanton contributions to the$D^6 R^4$interaction P/14/07 Voir plus > Voir moins > The modular invariant coefficient of the$D^6R^4$interaction in the low energy expansion of type~IIB string theory has been conjectured to be a solution of an inhomogeneous Laplace eigenvalue equation, obtained by considering the toroidal compactification of two-loop Feynman diagrams of eleven-dimensional supergravity. In this paper we determine its exact$SL(2,\Z)$-invariant solution$f(condition as $y\to \infty$ (the weak coupling limit). The solution is presented as a Fourier series with modes $\widehat{f}_n(y) e^{2\pi i n x}$, where the mode coefficients, $\widehat{f}_n(y)$ are bilinear in $K$-Bessel functions. Invariance under $SL(2,\Z)$ requires these modes to satisfy the nontrivial boundary condition $\widehat{f}_n(y) =O(y^{-2})$ for small $y$, which uniquely determines the solution. The large-$y$ expansion of $f(ower-behaved) terms, together with precisely-determined exponentially decreasing contributions that have the form expected of D-instantons, anti-D-instantons and D-instanton/anti-D-instanton pairs. 09/04/2014 PDF Algebraic rational cells, equivariant intersection theory, and Poincaré duality M/14/15 Voir plus > Voir moins > We provide a notion of algebraic rational cell with applications to intersection theory on singular varieties with torus action. Based on this notion, we study the algebraic analogue of Q-fi ltrable varieties: algebraic varieties where a torus acts with isolated fixed points, such that the associated Bialynicki-Birula decomposition consists of algebraic rational cells. We show that the rational equivariant Chow group of any Q-filtrable variety is freely generated by the cell closures. We apply this result to group embeddings, and more general spherical varieties. In view of the localization theorem for equivariant operational Chow rings, we get some conditions for Pointcarré duality in this setting. 09/04/2014 PDF Motivic Cohomology Spectral Sequence and Steenrod Algebra M/14/16 Voir plus > Voir moins > For an odd prime number$p$, it is shown that differentials$d_n$in the motivic cohomology spectral sequence with$p$-local coefficients vanish unless$p-1$divides$n-1$. We obtain an explicit formula for the first non-trivial differential$d_p$, expressing it in terms of motivic Steenrod$p$-power operations and Bockstein homomorphisms. Finally, we construct examples of varieties, having non-trivial differentials$d_p$in their motivic spectral sequences. 09/04/2014 PDF Boundedness of non-homogeneous square functions and$L^q$type testing conditions with$q \in (1,2)$M/14/14 01/04/2014 PDF Calculabilité de la cohomologie étale modulo l M/14/13 20/03/2014 PDF Scattering Equations and String Theory Amplitudes P/14/11 Voir plus > Voir moins > Scattering equations for tree-level amplitudes are viewed in the context of string theory. As a result of the comparison we are led to define a new dual model which coincides with string theory in both the small and large$\alpha'$limit, and whose solution is found algebraically on the surface of solutions to the scattering equations. Because it has support only on the scattering equations, it can be solved exactly, yielding a simple resummed model for$\alpha'$-corrections to all orders. We use the same idea to generalize scattering equations to amplitudes with fermions and any mixture of scalars, gluons and fermions. In all cases checked we find exact agreement with known results. 19/03/2014 PDF Localization in equivariant operational K-theory and the Chang-Skjelbred property M/14/12 Voir plus > Voir moins > We establish a localization theorem of Borel-Atiyah-Segal type for the equivariant operational K-theory of Anderson and Payne. Inspired by the work of Chang-Skjelbred and Goresky-Kottwitz-MacPherson, we establish a general form of GKM theory in this setting, applicable to singular schemes with torus action. Our results are deduced from those in the smooth case via Gillet-Kimura's technique of cohomological descent for equivariant envelopes. As an application, we extend Uma's description of the equivariant K-theory of smooth compactifi cations of reductive groups to the equivariant operational K-theory of all, possibly singular, projective group embeddings. 18/03/2014 PDF The physics of quantum gravity P/14/08 Voir plus > Voir moins > Quantum gravity is still very mysterious and far from being well under- stood. In this text we review the motivations for the quantification of gravity, and some expected physical consequences. We discuss the remarkable rela- tions between scattering processes in quantum gravity and in Yang-Mills theory, and the role of string theory as an unifying theory. 17/03/2014 PDF Polylogarithms and multizeta values in massless Feynman amplitudes P/14/10 Voir plus > Voir moins > The last two decades have seen a remarkable development of analytic methods in the study of Feynman amplitudes in perturbative quantum field theory. The present lecture offers a physicists' oriented survey of Francis Brown's work on singlevalued multiple polylogarithms, the associated multizeta periods and their application to Schnetz's graphical functions and to$x$-space renormalization. To keep the discussion concrete we restrict attention to explicit examples of primitively divergent graphs in a massless scalar QFT. 19/02/2014 PDF Particle in a field of two centers in prolate spheroidal coordinates: integrability and solvability P/14/09 17/02/2014 PDF The physics and the mixed Hodge structure of Feynman integrals P/14/04 Voir plus > Voir moins > This expository text is an invitation to the relation between quantum field theory Feynman integrals and periods. We first describe the relation between the Feynman parametrization of loop amplitudes and world-line methods, by explaining that the first Symanzik polynomial is the determinant of the period matrix of the graph, and the second Symanzik polynomial is expressed in terms of world-line Green s functions. We then review the relation between Feynman graphs and variations of mixed Hodge structures. Finally, we provide an algorithm for generating the Picard-Fuchs equation satisfied by the all equal mass banana graphs in a two-dimensional space-time to all loop orders. 14/02/2014 PDF Du transfert automorphe de Langlands aux formules de Poisson non linéaires M/14/05 17/01/2014 PDF Non-Abelian Lie algebroids over jet spaces M/14/03 Voir plus > Voir moins > We associate Hamiltonian homological evolutionary vector fields -- which are the non-Abelian variational Lie algebroids' differentials -- with Lie algebra-valued zero-curvature representations for partial differential equations. 07/01/2014 PDF Une loi de réciprocité explicite pour le polylogarithme elliptique M/14/01 Voir plus > Voir moins > On démontre une compatibilité entre la réalisation p-adique et la réalisation de de Rham des sections de torsion du profaisceau polylogarithme elliptique. La preuve utilise une variante pour H1 de la loi de réciprocité explicite de Kato pour le H2 des courbes modulaires. 06/01/2014 PDF Le système d'Euler de Kato (II) M/14/02 Voir plus > Voir moins > Ce texte est le deuxième article d’une série de trois articles sur une généralisation de système d’Euler de Kato. Il est consacré e d’Euler de Kato raffiné associé systèmes d’Euler de Kato. 06/01/2014 PDF Studying Quantum Field Theory P/13/38 13/12/2013 PDF Topological pressure and measure-theoretic degrees for non-expanding transformations M/13/39 11/12/2013 PDF Density of potentially crystalline representations of fixed weight M/13/37 Voir plus > Voir moins > Let K be a finite extension of Qp. We fix a continuous absolutely irreducible representation of the absolute Galois group of K over a finite dimensional vector space with coefficient in a finite field of characteristic p and consider its universal deformation ring R. If we fix a regular set of Hodge-Tate weights k, we prove, under some hypothesis, that the closed points of Spec(R[1/p]) corresponding to potentially crystalline representations of fixed Hodge-Tate weights k are dense in Spec(R[1/p]) for the Zariski topology. The main hypothesis we need is the existence of a potentially diagonalizable lift, so that in the two-dimensional case, the result is unconditional. 15/11/2013 PDF Eigenvalues of Laplacian and multi-way isoperimetric constants on weighted Riemannian manifolds M/13/36 Voir plus > Voir moins > We investigate the distribution of eigenvalues of the weighted Laplacian on closed weighted Riemannian manifolds of nonnegative Bakry-s. These inequalities are quantitative versions of the previous theorem by the author with Shioya. We also study some geometric quantity, called multi-way isoperimetric constants, on such manifolds and obtain similar universal inequalities among them. Multi-way isoperimetric constants are generalizations of the Cheeger constant. Extending and following the heat semigroup argument by Ledoux and E. Milman, we extend the Buser-Ledoux result to the k-th eigenvalue and the -way isoperimetric constant. As a consequence the k-th eigenvalue of the weighted Laplacian and the k-way isoperimetric constant are equivalent up to polynomials of k on closed weighted manifolds of nonnegative Bakry- 21/10/2013 PDF Allure of Quotations and Enchantment of Ideas M/13/35 02/10/2013 PDF Ergostructures, Ergologic and the Universal Learning Problem: Chapters 1, 2, 3 M/13/34 01/10/2013 PDF Poisson varieties from Riemann surfaces M/13/33 30/09/2013 PDF The elliptic dilogarithm for the sunset graph P/13/24 24/09/2013 PDF The geometry of variations in Batalin-Vilkovisky formalism M/13/32 Voir plus > Voir moins > We explain why no sources of divergence are built into the Batalin-Vilkovisky (BV) Laplacian, whence there is no need to postulate any ad hoc conventions such as "delta(0)=0" and "log delta(0)=0" within BV-approach to quantisation of gauge systems. Remarkably, the geometry of iterated variations does not refer at all to the construction of Dirac's delta-function as a limit of smooth kernels. We illustrate the reasoning by re-deriving -but not just "formally postulating"- the standard properties of BV-Laplacian and Schouten bracket and by verifying their basic inter-relations (e.g., cohomology preservation by gauge symmetries of the quantum master-equation). 15/09/2013 PDF On-shell Techniques and Universal Results in Quantum Gravity P/13/23 04/09/2013 PDF Quantum supersymmetric cosmology and its hidden Kac--Moody structure P/13/29 04/09/2013 PDF Analytical determination of the two-body gravitational interaction potential at the 4th post-Newtonian approximation P/13/30 04/09/2013 PDF Merger states and final states of black hole coalescences: a numerical-relativity-assisted effective-one-body approach P/13/31 04/09/2013 PDF The tensor hierarchy algebra P/13/27 02/09/2013 PDF The tensor hierarchy simplified P/13/28 02/09/2013 PDF The Nielsen and the Reidemeister Zeta Functions of maps on infra-solvmanifolds of type (R) M/13/26 Voir plus > Voir moins > We prove the rationality, the functional equations and calculate the radii of convergence of the Nielsen and the Reidemeister zeta functions of continuous maps on infra-solvmanifolds of type (R). We find a connection between the Reidemeister and Nielsen zeta functions and the Reidemeister torsions of the corresponding mapping tori. We show that if the Reidemeister zeta function is defined for a homeomorphism on an infra-solvmanifold of type (R), then this manifold is an infra-nilmanifold. We also prove that a map on an infra-solvmanifold of type (R) induced by an affine map minimizes the topological entropy in its homotopy class and it has a rational Artin-Mazur zeta function. Our main technical tool is the averaging formulas for the Lefschetz, the Nielsen and the Reidemeister numbers on infra-solvmanifolds of type (R). 30/08/2013 PDF Anyon wave functions and probability distributions P/13/25 Voir plus > Voir moins > The problem of determining the ground state energy for a quantum gas of anyons in two dimensions is considered. A recent approach to this problem by means of lower bounds is here refined to bring out the dependence on the n-particle probability distributions encoded in the wave functions. Furthermore, a class of states which has been proposed in the context of upper bounds for a related many-anyon problem, is here considered from the point of view of these refined lower bounds. A numerical approach to determining their corresponding probability distributions is employed for a limited number of particles. 06/08/2013 PDF Ramification and nearby cycles for l-adic sheaves on relative curves M/13/22 Voir plus > Voir moins > Deligne and Kato proved a formula computing the dimension of the nearby cycles complex of an l-adic sheaf on a relative curve over an excellent strictly henselian trait. In this article, we reprove this formula using Abbes-Saito’s ramification theory. 09/07/2013 PDF Conformal and Einstein gravity in twistor space P/13/13 18/06/2013 PDF Distributional Geometry of Squashed Cones P/13/21 Voir plus > Voir moins > A regularization procedure developed in \cite{Fursaev:1995ef} for integral curvature invariants on manifolds with conical singularities is generalized to the case of squashed cones. In general, the squashed conical singularities do not have rotational$O(2)$symmetry in a subspace orthogonal to a singular surface$\Sigma$so that the surface is allowed to have extrinsic curvatures. A new feature of squashed conical singularities is that the surface terms in the integral invariants, in the limit of small angle deficit, now depend also on the extrinsic curvatures of$\Sigma$. A case of invariants which are quadratic polynomials of the Riemann curvature is elaborated in different dimensions and applied to several problems related to entanglement entropy. The results are in complete agreement with computations of the logarithmic terms in entanglement entropy of 4D conformal theories \cite{Solodukhin:2008dh}. Among other applications of the suggested method are logarithmic terms in entanglement entropy of non-conformal theories and a holographic formula of entanglement entropy for theories with gravity duals. 17/06/2013 PDF Nonholonomic deformation of coupled and supersymmetric KdV equation and Euler-Poincaré-Suslov method M/13/15 Voir plus > Voir moins > Recently Kupershmidt \cite{Kup} presented a Lie algebraic derivation of a new sixth-order wave equation, which was proposed by Karasu-Kalkani et al \cite{KKK}. In this paper we demonstrate that Kupershmidt's method can be interpreted as an infinite-dimensional analogue of the Euler-Poincarnsional construction to construct nonholonomic deformation of a wide class of coupled KdV equations, all these equations follow from the Euler-PoincarS^1) \bo C^{\infty}(S^1)}}$, where $Diff(S^1)$ is the group of orientation preserving diffeomorphisms on a circle. We generalize our construction to two component Camassa-Holm equation. We also give a derivation of a nonholonomic deformation of the $N=1$ supersymmetric KdV equation, dubbed as sKdV6 equation and this method can be interpreted as an infinite-dimensional supersymmetric analogue of the Euler-Poincar

14/06/2013 PDF

Application of Jacobi's last multiplier for construction of singular Hamiltonian of the activator-inhibitor model and conformal Hamiltonian dynamics M/13/16

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The relationship between Jacobi's last multiplier and the Lagrangian of a second-order ordinary differential equation is quite well known. In this article we demonstrate the significance of the last multiplier in Hamiltonian theory by explicitly constructing the Hamiltonians of certain well known first-order systems of differential equations arising in the activator and inhibitor model and these are connected to conformal Hamiltonian structure.

14/06/2013 PDF

The role of the Jacobi Last Multiplier in Nonholonomic Systems and Almost Symplectic Structure M/13/17

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The relationship between Jacobi's last multiplier (JLM) and nonholonomic systems endowed with the almost symplectic structure is elucidated in this paper. In particular, we present an algorithmic way to describe how the two form and almost Poisson structure associated to nonholonomic system, studied by L. Bates and his coworkers, can be mapped to symplectic form and canonical Poisson structure using JLM. We demonstrate how JLM can be used to map an integrable nonholonomic system to a Liouville integrable system. We map the toral fibration defined by the common level sets of the integrals of a Liouville integrable Hamiltonian system with a toral fibration coming from a completely integrable nonholonomic system.

14/06/2013 PDF

Contiguity relations for linearisable systems of Gambier type P/13/18

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We introduce the Schlesinger transformations for the Gambier, linearisable, equation and by combining the former construct the contiguity relations of the solutions of the latter. We extend the approach to the discrete domain obtaining thus the Schlesinger transformations and the contiguity relations of the solutions of the Gambier mapping. In all cases the resulting contiguity relation is a linearisable equation, involving free functions, and which can be related to the generic Gambier mapping.

14/06/2013 PDF

Quantum aspects of the Liénard II equation and Jacobi's Last Multiplier-II P/13/19

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This is a continuation of the paper [J. Phys. A: Math. Theor. 46 (2013) 165202], in which we mapped the Lis ordering technique. In this paper we present further results on the construction of three sets of exactly solvable potentials giving rise to bound-state solutions of the Schr

14/06/2013 PDF

Chemotherapy in heterogeneous cultures of cancer cells with interconversion M/13/20

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Recently, it has been observed the interconversion between differentiated and stem-like cancer cells. Here, we model the \textit{in vitro} growth of heterogeneous cell cultures in the presence of interconversion from differentiated cancer cells to cancer stem cells, showing that, targeting only cancer stem cells with cytotoxic agents, it is not always possible to eradicate cancer. We have determined the kinetic conditions under which cytotoxic agents in \textit{in vitro} heterogeneous cultures of cancer cells eradicate cancer. In particular, we have shown that the chemotherapeutic elimination of \textit{in vitro} cultures of heterogeneous cancer cells is effective if it targets all the cancer cell types, and if the induced death rates for the different subpopulations of cancer cell types are large enough.

14/06/2013 PDF

Nouveaux développements sur les valeurs des caractères des groupes symétriques; méthodes combinatoires M/13/14

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L’étude asymptotique des diagrammes de Young de grande taille a été entreprise cu dans le but de contrôler les algèbres d’opérateurs liées aux groupes libres ; elles ont servi ensuite tude des zéros de la fonction zeta de Riemann. Plus récemment , elles ont été appliquées par Biane aux propriétés asymptotiques des permutations. Nous insisterons surtout sur les formules exactes qui sous-tendent ces formules asymptotiques obtenues par les collaborateurs de Biane (Sniady, Féray), et qui développent de nouveaux domaines de la combinatoire (principalement cartes planaires).

05/06/2013 PDF

Truncated Infinitesimal Shifts, Spectral Operators and Quantized Universality of the Riemann Zeta Function M/13/12

15/05/2013 PDF

mRNA diffusion as a mechanism of morphogenesis in Drosophila early development M/13/11

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In Drosophila early development, bicoid mRNA of maternal origin is deposited in one of the poles of the egg, determining the anterior tip of the embryo and the position of the head of larvae. The deposition of mRNAs is done during oogenesis by the mother ovary cells and is transported into the oocyte along microtubules. Initially, the oocyte has only one nucleus, but after fertilization and deposition of the egg, nuclear duplication by mitosis is initiated without the formation of cellular membranes. During the first 14 nuclear divisions of the developing embryo, bicoid mRNA of maternal origin is translated into protein in the ribosomes and accumulates near the external nuclear walls of the recently formed nuclei. Here, we show that mRNA diffusion is the main morphogenesis mechanism explaining consistently the establishment of Bicoid protein gradients. Moreover, we show that if diffusion for both bicoid mRNA and Bicoid protein were assumed, a steady distribution of Bicoid protein would result, with a constant concentration along the embryo, contradicting observations.

29/04/2013 PDF

Equivariant operational Chow rings of spherical varieties and T-linear varieties. M/13/10

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We stablish localization theorems for the equivariant operational Chow rings (or equivariant Chow cohomology) of singular spherical varieties and T-linear varieties. Our main results provide a GKM description of these rings in the case of spherical varieties admitting a BB-decomposition into algebraic rational cells. Our description extends certain topological results to intersection theory on singular varieties.

12/04/2013 PDF

Dynamic trajectory control of gliders M/13/09

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A new dynamic control algorithm in order to direct the trajectory of a glider to a pre-assigned target point is proposed. The algorithm runs iteratively and the approach to the target point is self-correcting. The algorithm is applicable to any non-powered lift-enabled vehicle (glider) travelling in planetary atmospheres. As a proof of concept, we have applied the new algorithm to the command and control of the trajectory of the Space Shuttle during the Terminal Area Energy Management (TAEM) phase.

04/04/2013 PDF

Chemotaxis with directional sensing during Dictyostelium aggregation M/13/07

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With an in silico analysis, we show that the chemotactic movements of colonies of the starving amoeba Dictyostelium discoideum are driven by a force that depends on both the direction of propagation (directional sensing) of reaction-diffusion chemotactic waves and on the gradient of the concentration of the chemoattractant. It is shown that the directional sensing of amoebae is due to the sensitivity of the cells to the time variation of the concentration of the chemoattractant combined with its gradient. It is also shown that chemotaxis exclusively driven by local concentration gradients leads to unstable local motion, preventing cells from aggregation. These facts show that the formation of mounds, which initiate multicellularity in Dictyostelium discoideum, is caused by the sensitivity of the amoebae to three factors, namely, to the direction of propagation of the chemoattractant, to its gradient, and to the spiral spatial topology of the propagating chemoattractant.

24/03/2013 PDF

The regulation of gene expression in eukaryotes: bistability and oscillations in repressilator models M/13/08

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To model the regulation of gene expression in eukaryotes by transcriptional activators and repressors, we introduce delays in conjugation with the mass action law. Delays are associated with the time gap between the mRNA transcription in the nucleoplasm and the protein synthesis in the cytoplasm. After re-parameterisation of the m-repressilator model with the Hill cooperative parameter n, for n=1, the m-repressilator is deductible from the mass action law and, in the limit $n \to \infty$, it is a Boolean type model. With this embedding and with delays, if m is odd and n>1, we show that there is always a choice of parameters for which the m-repressilator model has sustained oscillations (limit cycles), implying that the 1-repressilator is the simplest genetic mechanism leading to sustained oscillations in eukaryotes. If m is even and n>1, there is always a choice of parameters for which the m-repressilator model has bistability.

24/03/2013 PDF

Noyaux du transfert automorphe de Langlands et formules de Poisson non linéaires: Notes de cours M/13/06

18/02/2013 PDF

On weakly group-theoretical non-degenerate braided fusion categories M/13/05

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We show that the Witt class of a weakly group-theoretical non-degenerate braided fusion category belongs to the subgroup generated by classes of non-degenerate pointed braided fusion categories and Ising braided categories. This applies in particular to solvable non-degenerate braided fusion categories. We also give some sufficient conditions for a braided fusion category to be weakly group-theoretical or solvable in terms of the factorization of its Frobenius-Perron dimension and the Frobenius-Perron dimensions of its simple objects. As an application, we prove that every non-degenerate braided fusion category whose Frobenius-Perron dimension is a natural number less than 1800, or an odd natural number less than 33075, is weakly group-theoretical.

01/02/2013 PDF

Special Functions in Minimal Representations M/13/04

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Minimal representations of a real reductive group G are the smallest' irreducible unitary representations of G. We discuss special functions that arise in the analysis of L^2-model of minimal representations.

22/01/2013 PDF

Rankin-Cohen Operators for Symmetric Pairs M/13/03

10/01/2013 PDF

Sur la correspondance de Simpson p-adique. II : aspects globaux M/13/02

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We develop a new approach for the p-adic Simpson correspondence, closely related to the original approach of Faltings, but also inspired by the work of Ogus and Vologodsky on an analogue in characteristic p>0. This second article is devoted to the global aspects of the theory.

09/01/2013 PDF

Sur la correspondance de Simpson p-adique. 0 : une vue d'ensemble M/13/01

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We develop a new approach for the p-adic Simpson correspondence, closely related to the original approach of Faltings, but also inspired by the work of Ogus and Vologodsky on an analogue in characteristic p>0. The aim of this article is to give an extensive overview of the theory that has been developped in two articles, the first one (arXiv:1102.5466) devoted to the local aspects and the second one (arXiv:1301.0904) to the global aspects.

09/01/2013 PDF

F-method for constructing equivariant differential operators M/12/36

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Using an algebraic Fourier transform of operators, we develop a method (F-method) to obtain explicit highest weight vectors in the branching laws by differential equations. This article gives a brief explanation of the F-method and its applications to a concrete construction of some natural equivariant operators that arise in parabolic geometry and in automorphic forms.

31/12/2012 PDF

Gravitational self-force and the effective-one-body formalism between the innermost stable circular orbit and the light ring P/12/35

10/12/2012 PDF

Gravitational radiation reaction along general orbits in the effective one-body formalism P/12/34

10/12/2012 PDF

Minkowski Measurability Results for Self-Similar Tilings and Fractals with Monophase Generators M/12/33

04/12/2012 PDF

Dirac Operators and Geodesic Metric on the Harmonic Sierpinski Gasket and Other Fractal Sets M/12/32

02/12/2012 PDF

The Decimation Method for Laplacians on Fractals: Spectra and Complex Dynamics M/12/31

27/11/2012 PDF

Towards an axiomatic geometry of fundamental interactions in noncommutative space-time at Planck scale P/12/30

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We outline an axiomatic quantum picture unifying the four fundamental interactions; this is done by exploring a possible physical meaning of the notions, structures, and logic in a class of noncommutative geometries which has been introduced in [IHES/M-12-13]. We try to recognise a mathematical formalisation of such phenomena of Nature as the oriented space-time, gravity (here, dark matter and vacuum energy), Hubble's law, inflation, formation and structure of sub-atomic particles, antimatter, annihilation, CP-symmetry violation, mass and mass endowment mechanism, three lepton-neutrino matchings, spin, helicity and chirality, electric charge and electromagnetism, as well as the weak and strong interaction between particles, admissible transition mechanisms (e.g., muon to muon neutrino, electron, and electron antineutrino), and decays (e.g., neutron to proton, electron, and electron antineutrino). Our approach is based on the understanding of Physics as text which is written in the language of affine Lie algebras and associated homeo-class noncommutative structures over the space-time.

12/11/2012 PDF

Temperedness of Reductive Homogeneous Spaces M/12/29

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Let G be a semisimple algebraic Lie group and H a reductive subgroup. We compute geometrically the best even integer p for which the representation of G in L^2(G/H) is almost L^p. As an application, we give a criterion which detects whether this representation is tempered.

05/11/2012 PDF

Noyaux du transfert automorphe de Langlands et formules de Poisson non lin\'eaires M/12/28

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On montre qu'un certain type de formules de Poisson non lin'eaires explicites, qui est impliqu'e par le principe de fonctorialit'e de Langlands, permet de construire des noyaux'' du transfert automorphe. Il y a donc 'equivalence entre le principe de fonctorialit'e et ces formules de Poisson non lin'eaires.

29/10/2012 PDF

The Current State of Fractal Billiards M/12/27

16/10/2012 PDF

Varna Lecture on $L^2$-Analysis of Minimal Representations M/12/26

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Minimal representations of a real reductive group G are the 'smallest' irreducible unitary representations of G. The author suggests a program of global analysis built on minimal representations from the philosophy: 'small' representation of a group = 'large' symmetries in a representation space. This viewpoint serves as a driving force to interact algebraic representation theory with geometric analysis of minimal representations, yielding a rapid progress on the program. We give a brief guidance to recent works with emphasis on the Schroedinger model.

12/10/2012 PDF

Fractal Complex Dimensions, Riemann Hypothesis and Invertibility of the Spectral Operator M/12/25

30/09/2012 PDF

Discrete Spectrum for non-Riemannian Locally Symmetric Spaces --- I. Construction and Stability M/12/24

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We study the discrete spectrum of the Laplacian on certain pseudo-Riemannian manifolds M which are quotients of reductive symmetric spaces X by discrete groups of isometries acting properly discontinuously. Assuming that X admits a maximal compact subsymmetric space of full rank, we construct L^2-eigenfunctions on M for an infinite set of eigenvalues. In contrast to the classical setting where the nonzero discrete spectrum varies on the Teichmüller space of a compact Riemann surface, we prove that this infinite set of eigenvalues is stable under any small deformation of discrete groups, for a large class of groups. We actually construct joint L2-eigenfunctions for the whole commutative algebra of invariant differential operators on M.

19/09/2012 PDF

Minkowski Measurability and Exact Fractal Tube Formulas for p-Adic Self-Similar Strings M/12/23

14/08/2012 PDF

One-Loop four-graviton amplitudes in N=4 supergravity models P/12/21

13/08/2012 PDF

From Global to Local M/12/18

27/07/2012 PDF

The zero locus of the infinitesimal invariant M/12/20

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Let $\nu$ be a normal function on a complex manifold $X$. The infinitesimal invariant of $\nu$ has a well-defined zero locus inside the tangent bundle $TX$. When $X$ is quasi-projective, and $\nu$ is admissible, we show that this zero locus is constructible in the Zariski topology.

27/07/2012 PDF

Box-Counting Fractal Strings, Zeta Functions, and Equivalent Forms of Minkowski Dimension M/12/22

27/07/2012 PDF

Multifractal Analysis via Scaling Zeta Functions and Recursive Structure of Lattice Strings M/12/19

26/07/2012 PDF

On dimension growth of groups M/12/17

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The (asymptotic) dimension growth functions of groups were introduced by Gromov in 1999. In this paper, we show connections between dimension growth and expansion properties of graphs, Ramsey theory and the Kolmogorov-Ostrand dimension of groups and prove that all solvable subgroups of the R.Thompson group F have polynomial dimension growth. We introduce controlled dimension growth function and prove that the exponentially controlled dimension growth is exponential for the Thompson group F and some solvable of class 3 groups. The paper contains many open questions.

24/07/2012 PDF

Sequences of Compatible Periodic Hybrid Orbits of Prefractal Koch Snowflake Billiards M/12/16

18/07/2012 PDF

Partition Zeta Functions, Multifractal Spectra, and Tapestries of Complex Dimensions M/12/15

11/07/2012 PDF

The twelve lectures in the (non)commutative geometry of differential equations M/12/13

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These notes follow the twelve-lecture course in the geometry of nonlinear partial differential equations of mathematical physics. Briefly yet systematically, we outline the geometric and algebraic structures associated with such equations and study the properties of these structures and their inter-relations. The lectures cover the standard material about the infinite jet bundles, systems of differential equations (e.g., Lagrangian or Hamiltonian), their symmetries and conservation laws (together with the First and Second Noether Theorems), and the construction of the nonlocalities. Besides, in the lectures we introduce the calculus of variational multivectors --- in terms of the Schouten bracket, or the antibracket --- on the (non)commutative jet spaces and proceed with its applications to the variational Poisson formalism and the BRST- or BV-approach to the gauge systems. The course differs from other texts on the subject by its greater emphasis on the physics that motivates the model geometries. Simultaneously, the course attests to the applicability of the algebraic techniques in the analysis of the geometry of fundamental interactions. These lectures could be a precursor to the study of the (quantum) field and string theory.

02/07/2012 PDF

Hyperfunctions and Spectral Zeta Functions of Laplacians on Self-Similar Fractals M/12/14

02/07/2012 PDF

Effective action approach to higher-order relativistic tidal interactions in binary systems and their effective one body description P/12/10

29/06/2012 PDF

Theoretical aspects of the equivalence principle P/12/11

29/06/2012 PDF

Measurability of the tidal polarizability of neutron stars in late-inspiral gravitational-wave signals P/12/12

29/06/2012 PDF

Riemann Zeroes and Phase Transitions via the Spectral Operator on Fractal Strings M/12/09

09/06/2012 PDF

Gravity, strings, modular and quasimodular forms P/12/08

09/05/2012 PDF

On the Tutte-Krushkal-Renardy polynomial for cell complexes M/12/07

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Recently V.Krushkal and D.Renardy generalized the Tutte polynomial from graphs to cell complexes. We show that evaluating this polynomial at the origin gives the number of cellular spanning trees in the sense of A.~Duval, C.~Klivans, and J.~Martin. Moreover, after a slight modification, the Tutte-Krushkal-Renardy polynomial evaluated at the origin gives a weighted count of cellular spanning trees, and therefore its free term can be calculated by the cellular matrix-tree theorem of Duval et al. In the case of cell decomposition of a sphere, this modified polynomial satisfies the same duality identity as before. We find that evaluating the Tutte-Krushkal-Renardy along a certain line is the Bott polynomial. Finally we prove skein relations for the Tutte-Krushkal-Renardy polynomial.

16/04/2012 PDF

Hopf Galois (Co)Extensions In Noncommutative Geometry M/12/06

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We introduce an alternative proof, with the use of tools and notions for Hopf algebras, to show that Hopf Galois coextensions of coalgebras are the sources of stable anti Yetter-Drinfeld modules. Furthermore we show that two natural cohomology theories related to a Hopf Galois coextension are isomorphic.

10/04/2012 PDF

Location of the Lee-Yang zeros and absence of phase transitions in some Ising spin systems P/12/05

03/04/2012 PDF

On Generalizations of Connes-Moscovici Characteristic Map M/12/04

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In this paper we generalize the Connes-Moscovici characteristic map for cyclic cohomology of extended version of Hopf algebras called x-Hopf algebras. To do this, we define a pairing for cyclic cohomology of module algebras and module coalgebras under the symmetry of a x-Hopf algebra. We introduce more examples of similar generalized characteristic maps for quantum algebraic torus and enveloping algebras.

12/03/2012 PDF

Algebras for Amplitudes P/12/03

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Tree-level amplitudes of gauge theories are expressed in a basis of auxiliary amplitudes with only cubic vertices. The vertices in this formalism are explicitly factorized in color and kinematics, clarifying the color-kinematics duality in gauge theory amplitudes. The basis is constructed making use of the KK and BCJ relations, thereby showing precisely how these relations underlie the color-kinematics duality. We express gravity amplitudes in terms of a related basis of color-dressed gauge theory amplitudes, with basis coefficients which are permutation symmetric.

05/03/2012 PDF

A $R^4$ non-renormalisation theorem in ${\mathcal N} = 4$ supergravity P/12/02

17/02/2012 PDF

Quantization is a mystery P/12/01

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Expository notes which combine a historical survey of the development of quantum physics with a review of selected mathematical topics in quantization theory (addressed to students who have had a first course in quantum mechanics). After recalling in the introduction the early stages of the quantum revolution, and recapitulating in Sect. 2.1 some basic notions of symplectic geometry, we survey in Sect. 2.2 the so called {\it prequantization} thus preparing the ground for an outline of {\it geometric quantization} (Sect. 2.3). In Sect. 3 we apply the general theory to the study of basic examples of {\it quantization of K"ahler manifolds}. In Sect. 4 we review the Weyl and Wigner maps and the work of Groenewold and Moyal that laid the foundations of {\it quantum mechanics in phase space}, ending with a brief survey of the modern development of {\it deformation quantization}. Sect. 5 provides a review of {\it second quantization} and its mathematical interpretation. We point out that the treatment of (nonrelativistic) bound states requires going beyond the neat mathematical formalization of the concept of second quantization. An appendix is devoted to Pascual Jordan, the least known among the creators of quantum mechanics and the chief architect of the theory of quantized matter waves''.

31/01/2012 PDF

Spatial variation of fundamental couplings and Lunar Laser Ranging P/11/30

22/12/2011 PDF

Energy versus Angular Momentum in Black Hole Binaries P/11/31

22/12/2011 PDF

History of Mathematics from a working mathematician's view M/11/29

12/12/2011 PDF

SMALL REPRESENTATIONS, STRING INSTANTONS, AND FOURIER MODES OF EISENSTEIN SERIES P/11/25

21/11/2011 PDF

A Juzvinski\u{i} Addition Theorem for Finitely Generated Free Groups Actions M/11/28

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The classical Juzvinski\u{i} Addition Theorem states that the entropy of an automorphism of a compact group decomposes along invariant subgroups. Thomas generalized the theorem to a skew-product setting. Using L. Bowen's f-invariant we prove the addition theorem for actions of finitely generated free groups on skew-products with compact totally disconnected groups or finitely dimensional Lie Groups and discuss examples.

17/11/2011 PDF

Homological evolutionary vector fields in Korteweg-de Vries, Liouville, Maxwell, and several other models M/11/26

16/11/2011 PDF

Symplectic structures on moduli spaces of framed sheaves on surfaces M/11/27

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We provide generalizations of the notions of Atiyah class and the Kodaira-Spencer map to the case of framed sheaves. Moreover, we construct closed two-forms on the moduli spaces of framed sheaves on surfaces. As an application, we de ne a symplectic structure on the moduli spaces of framed sheaves on the second Hirzebruch surface. This generalizes a result of Bottacin for the locally free case.

15/11/2011 PDF

N=4 SYM Regge Amplitudes and Minimal Surfaces in AdS/CFT Correspondence P/11/24

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The high-energy behavior of N=4 SYM elastic amplitudes at strong coupling is studied by means of the AdS/CFT correspondence. For massless gluon-gluon scattering, we consider the amplitude found by Alday and Maldacena using a minimal surface in $AdS_5$ momentum space. For elastic scattering of massive quarks, we reconsider the eikonal method proposed by Janik and one of the authors of this paper, where the relevant minimal surface is a "generalized helicoid" in hyperbolic space ("Euclidean AdS_5"), from which the physical amplitude is obtained after an appropriate analytic continuation. Exploiting a conformal transformation, we are able to show that the quark-quark amplitude is dominated by the same cusp contribution already found in the gluon-gluon case. Both amplitudes are shown to be of Regge type at high energy, with the same logarithmic Regge trajectory, in agreement with the expected universality property of Regge amplitudes. The subleading constant term in the trajectory is known for gluon-gluon scattering, but it is regularization-scheme dependent. Hence, the full content of Regge universality remains an open problem.

17/10/2011 PDF

The growth rate of symplectic Floer homology M/11/23

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The main theme of this paper is to study for a symplectomorphism of a compact surface, the asymptotic invariant which is defined to be the growth rate of the sequence of the total dimensions of symplectic Floer homologies of the iterates of the symplectomorphism. We prove that the asymptotic invariant coincides with asymptotic Nielsen number and with asymptotic absolute Lefschetz number. We also show that the asymptotic invariant coincides with the largest dilatation of the pseudo-Anosov components of the symplectomorphism and its logarithm coincides with the topological entropy. This implies that symplectic zeta function has a positive radius of convergence.

14/09/2011 PDF

Collapsing of Abelian Fibred Calabi-Yau Manifolds M/11/22

04/08/2011 PDF

Topos co-évanescents et généralisations M/11/20

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Cet article est consacré à l'étude d'un topos introduit par Faltings pour les besoins de la théorie de Hodge $p$-adique. Nous en présentons une nouvelle approche basée sur une généralisation des topos co-évanescents de Deligne. Chemin faisant, nous corrigeons la définition originelle de Faltings. This article is devoted to studying a topos introduced by Faltings for the purpose of $p$-adic Hodge theory. We present a new approach based on a generalisation of Deligne's co-vanishing topos. Along the way, we correct Faltings' original definition.

13/07/2011 PDF

On the double zeta values M/11/21

13/07/2011 PDF

Surjectivity and equidistribution of the word $x^ay^b$ on $PSL(2,q)$ and $SL(2,q)$ M/11/19

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We determine the positive integers a,b and the prime powers q for which the word map w(x,y)=x^ay^b is surjective on the group PSL(2,q) (and SL(2,q)). We moreover show that this map is almost equidistributed for the family of groups PSL(2,q) (and SL(2,q)). Our proof is based on the investigation of the trace map of positive words.

29/06/2011 PDF

From Traditional Set Theory -- that of Cantor, Hilbert, G\"odel, Cohen -- to Its Necessary Quantum Extension M/11/18

17/06/2011 PDF

The vanishing volume of D=4 superspace P/11/14

07/06/2011 PDF

Darboux coordinates, Yang-Yang functional, and gauge theory P/11/16

07/06/2011 PDF

Tempered automorphic representations of the unitary group M/11/17

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Following Arthur's study of the representations of the orthogonal and symplectic groups, we prove many cases of both the local and global Arthur conjectures for tempered representations of the unitary group. This completes the proof of Arthur's description of the discrete series representations of the quasi-split $p$-adic unitary group, and Arthur's description of the tempered discrete automorphic representations of the unitary group, satisfying certain technical conditions.

04/06/2011 PDF

How to take advantage of the blur between the finite and the infinite M/11/15

04/05/2011 PDF

Derived equivalences for cluster-tilted algebras of Dynkin type D M/11/11

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We provide a far reaching derived equivalence classification of cluster-tilted algebras of Dynkin type D. We introduce another notion of equivalence called good mutation equivalence which is slightly stronger than derived equivalence but is algorithmically more tractable, and give a complete classification together with normal forms. We also suggest normal forms for the derived equivalence classes, but some subtle questions in the derived equivalence classification remain open.

28/04/2011 PDF

Mutation classes of certain quivers with potentials as derived equivalence classes M/11/12

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We characterize the marked bordered unpunctured oriented surfaces with the property that all the Jacobian algebras of the quivers with potentials arising from their triangulations are derived equivalent. These are either surfaces of genus g with b boundary components and one marked point on each component, or the disc with 4 or 5 points on its boundary. We show that for each such marked surface, all the quivers in the mutation class have the same number of arrows, and the corresponding Jacobian algebras constitute a complete derived equivalence class of finite-dimensional algebras whose members are connected by sequences of Brenner-Butler tilts. In addition, we provide explicit quivers for each of these classes. We consider also 10 of the 11 exceptional finite mutation classes of quivers not arising from triangulations of marked surfaces excluding the one of the quiver X_7, and show that all the finite-dimensional Jacobian algebras in such class (for suitable choice of potentials) are derived equivalent only for the classes of the quivers E_6^(1,1) and X_6.

28/04/2011 PDF

Which mutation classes of quivers have constant number of arrows? M/11/13

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We classify the connected quivers with the property that all the quivers in their mutation class have the same number of arrows. These are the ones having at most two vertices, or the ones arising from triangulations of marked bordered oriented surfaces of two kinds: either surfaces with non-empty boundary having exactly one marked point on each boundary component and no punctures, or surfaces without boundary having exactly one puncture. This combinatorial property has also a representation-theoretic counterpart: to each such quiver there is a naturally associated potential such that the Jacobian algebras of all the QP in its mutation class are derived equivalent.

28/04/2011 PDF

Multiscale analysis of biological functions: the example of biofilms P/11/10

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Biological functions involve processes at different scales. This statement is obviously true for organismic processes like development. It is already relevant for a bacterial colony, the example on which we shall more specifically focus here. Understanding biological functions thus requires to integrate knowledge and data of different natures, available at different levels, and described within different frameworks, from quantum mechanics (for elementary intracellular processes e.g. light transduction) to stochastic kinetics to deterministic rate equations and continuous medium theory (e.g. elasticity theory or hydrodynamics). Beyond the epistemic issue of capturing a real process in descriptions and measurements prescribed by our own abilities and limitations, biological functions and their regulation present a greater challenge: they are intrinsically and irreducibly multiscale processes. Indeed regulation of a biological function has to bridge the overall state of the cells as well as some surroundings features with the basic ingredients and mechanisms at the atomic or molecular scale, in an adaptive and interrelated way. A bacterial cell itself has to perform a multiscale integration. Accordingly our analysis and modeling should follow the same line. For these two main reasons, multiscale approaches play an essential role in the way towards the integrated understanding of biological functions, and all the more of biological systems.

17/04/2011 PDF

Configuration Space Renormalization of Massless QFT as an Extension Problem for Associate Homogeneous Distributions P/11/07

12/04/2011 PDF

Quantum Einstein-Dirac Bianchi Universes P/11/08

12/04/2011 PDF

Accurate numerical simulations of inspiralling binary neutron stars and their comparison with effective-one-body analytical models P/11/09

12/04/2011 PDF

Absolute algebra III-the saturated spectrum M/11/06

15/03/2011 PDF

Sur la correspondance de Simpson p-adique. I : étude locale M/11/05

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Nous développons une nouvelle approche pour la correspondance de Simpson p-adique, intimement liée à l'approche originelle de Faltings, mais aussi inspirée du travail d'Ogus et Vologodsky sur un analogue en caractéristique p>0. Ce premier article est consacré aux aspects locaux de la théorie. \\ We develop a new approach for the p-adic Simpson correspondence, closely related to the original approach of Faltings, but also inspired by the work of Ogus and Vologodsky on an analogue in characteristic p>0. This first article is devoted to the local aspects of the theory.

01/03/2011 PDF

On Vassiliev invariants of braid groups of the sphere M/11/03

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We construct a universal Vassiliev invariant for braid groups of the sphere and the mapping class groups of the sphere with $n$ punctures. The case of a sphere is different from the classical braid groups or braids of oriented surfaces of genus strictly greater than zero, since Vassiliev invariants in a group without 2-torsion do not distinguish elements of braid group of a sphere.

31/01/2011 PDF

Shannon entropy: a rigorous mathematical notion at the crossroads between probability, information theory, dynamical systems and statistical physics M/11/04

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Statistical entropy was introduced by Shannon as a basic concept in information theory, measuring the average missing information on a random source. Extended into an entropy rate, it gives bounds in coding and compression theorems. I here present how statistical entropy and entropy rate relate to other notions of entropy, relevant either to probability theory (entropy of a discrete probability distribution measuring its unevenness), computer sciences (algorithmic complexity), the ergodic theory of dynamical systems (Kolmogorov-Sinai or metric entropy), or statistical physics (Boltzmann entropy). Their mathematical foundations and correlates (entropy concentration, Sanov, Shannon-McMillan-Breiman, Lempel-Ziv and Pesin theorems) clarify their interpretation and offer a rigorous basis to maximum entropy principles. Although often ignored, these mathematical perspectives give a central position to entropy and relative entropy in statistical laws describing generic collective behaviors. They provide insights into the notions of randomness, typicality and disorder. The relevance of entropy outside the realm of physics, for living systems and ecosystems, is yet to be demonstrated.

31/01/2011 PDF

On finite arithmetic groups M/11/02

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In this paper we study representations of finite groups stable under Galois operation over arithmetic rings in local and global fields.

21/01/2011 PDF

On localization in holomorphic equivariant cohomology M/11/01

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We study a holomorphic equivariant cohomology built out of the Atiyah algebroid of an equivariant holomorphic vector bundle and prove a related localization formula. localization formula.

18/01/2011 PDF

Statistical Properties of Cosmological Billiards P/10/16

16/12/2010 PDF

On Effective Action of Multiple M5-branes and ABJM Action P/10/45

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We calculate the fluctuations from the classical multiple M5-brane solution of ABJM action which we found in the previous paper. We obtain D4-brane-like action but the gauge coupling constant depends on the spacetime coordinate. This is consistent with the expected proporties of M5-brane action, although we will need to take into account the monopole operators in order to fully understand M5-branes. We also see that the Nambu-Poisson bracket is hidden in the solution.

15/12/2010 PDF

Cyclic structures in algebraic (co)homology theories M/10/44

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This note discusses the cyclic cohomology of a left Hopf algebroid ($\times_A$-Hopf algebra) with coefficients in a right module-left comodule, defined using a straightforward generalisation of the original operators given by Connes and Moscovici for Hopf algebras. Lie-Rinehart homology is a special case of this theory. A generalisation of cyclic duality that makes sense for arbitrary para-cyclic objects yields a dual homology theory. The twisted cyclic homology of an associative algebra provides an example of this dual theory that uses coefficients that are not necessarily stable anti Yetter-Drinfel'd~modules.

06/12/2010 PDF

Generalized matrix models and AGT correspondence at all genera P/10/43

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We study generalized matrix models corresponding to n-point Virasoro conformal blocks on Riemann surfaces with arbitrary genus g. Upon AGT correspondence, these describe four dimensional N=2 SU(2)^{n+3g-3} gauge theories with generalized quiver diagrams. We obtain the generalized matrix models from the perturbative evaluation of the Liouville correlation functions and verify the consistency of the description with respect to degenerations of the Riemann surface. Moreover, we derive the Seiberg-Witten curve for the $\CN=2$ gauge theory as the spectral curve of the generalized matrix model, thus providing a check of AGT correspondence at all genera.

24/11/2010 PDF

Incompressibility of generic orthogonal grassmannians M/10/42

23/11/2010 PDF

A uniqueness theorem for meromorphic mappings with two families of hyperplanes M/10/41

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In this paper, we extend the uniqueness theorem for meromorphic mappings to the case where the family of hyperplanes depends on the meromorphic mapping and where the meromorphic mappings may be degenerate.

07/11/2010 PDF

TENSOR STRUCTURE FROM SCALAR FEYNMAN MATROIDS P/10/40

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We show how to interpret the scalar Feynman integrals which appear when reducing tensor integrals as scalar Feynman integrals coming from certain nice matroids.

04/11/2010 PDF

Precession effect of the gravitational self-force in a Schwarzschild spacetime and the effective one-body formalism P/10/37

27/10/2010 PDF

Analytic modelling of tidal effects in the relativistic inspiral of binary neutron stars P/10/38

27/10/2010 PDF

Accuracy and effectualness of closed-form, frequency-domain waveforms for non-spinning black hole binaries P/10/39

27/10/2010 PDF

Open Gromov-Witten invariants and superpotentials for semi-Fano toric surfaces M/10/36

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We compute the open Gromov-Witten invariants for every compact semi-Fano toric surface, i.e. a toric surface $X$ with nef anticanonical bundle. Unlike the Fano case, this involves non-trivial obstructions in the corresponding moduli problem. As an application, an explicit expression of the superpotential $W$ for the mirror of $X$ is obtained, which in turn gives an explicit ring presentation of the small quantum cohomology of $X$. We also give a computational verification of the natural ring isomorphism between the small quantum cohomology of $X$ and the Jacobian ring of $W$.

25/10/2010 PDF

The momentum kernel of gauge and gravity theories P/10/32

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We derive an explicit formula for factorizing an n-point closed string amplitude into open string amplitudes. Our results are phrased in terms of a momentum kernel which in the limit of infinite string tension reduces to the corresponding field theory kernel. The same momentum kernel encodes the monodromy relations which lead to the minimal basis of color-ordered amplitudes in Yang-Mills theory. There are interesting consequences of the momentum kernel pertaining to soft limits of amplitudes. We also comment on surprising links between gravity and certain combinations of kinematic and color factors in gauge theory.

20/10/2010 PDF

Highly Transitive Actions of Out(Fn) M/10/35

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An action of a group on a set is called k-transitive if it is transitive on ordered k-tuples and highly transitive if it is k-transitive for every k. We show that for n>3 the group Out(Fn) = Aut(Fn)/Inn(Fn) admits a faithful highly transitive action on a countable set.

05/10/2010 PDF

Seiberg-Witten curve via generalized matrix model P/10/34

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We study the generalized matrix model which corresponds to the n-point toric Virasoro conformal block. This describes four-dimensional N=2 SU(2)^n gauge theory with circular quiver diagram by the AGT relation. We first verify that it is obtained from the perturbative calculation of the Liouville correlation function. We derive the Seiberg-Witten curve for N=2 gauge theory as a spectral curve of the generalized matrix model.

28/09/2010 PDF

Geometric analysis on small representations of GL(N,R) M/10/33

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The most degenerate unitary principal series representations \pi_{i\lambda,\delta} (\lambda \in R, \delta \in Z/2 Z) of G = GL(N,R) attain the minimum of the Gelfand--Kirillov dimension among all irreducible unitary representations of G. This article gives an explicit formula of the irreducible decomposition of the restriction \pi_{i\lambda,\delta}|_H (\textit{branching law}) with respect to all symmetric pairs (G,H). For N=2n with n \ge 2, the restriction \pi_{i\lambda,\delta}|_H remains irreducible for H=Sp(n,R) if \lambda\ne0 and splits into two irreducible representations if \lambda=0. The branching law of the restriction \pi_{i\lambda,\delta}|_H is purely discrete for H = GL(n,C), consists only of continuous spectrum for H = GL(p,R) \times GL(q,R) (p+q=N), and contains both discrete and continuous spectra for H=O(p,q) (p>q \ge 1). Our emphasis is laid on geometric analysis, which arises from the restriction of small representations' to various subgroups.

27/09/2010 PDF

On the Surjectivity of Engel Words on PSL(2,q) M/10/31

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We investigate the surjectivity of the word map defined by the n-th Engel word on the groups PSL(2,q) and SL(2,q). For SL(2,q), we show that this map is surjective onto the subset SL(2,q)\{-id} provided that q>Q(n) is sufficiently large. Moreover, we give an estimate for Q(n). We also present examples demonstrating that this does not hold for all q. We conclude that the n-th Engel word map is surjective for the groups PSL(2,q) when q>Q(n). By using the computer, we sharpen this result and show that for any n<5, the corresponding map is surjective for all the groups PSL(2,q). This provides evidence for a conjecture of Shalev regarding Engel words in finite simple groups. In addition, we show that the n-th Engel word map is almost measure preserving for the family of groups PSL(2,q), with q odd, answering another question of Shalev. Our techniques are based on the method developed by Bandman, Grunewald and Kunyavskii for verbal dynamical systems in the group SL(2,q).

23/09/2010 PDF

a-Maximization in N=1 Supersymmetric Spin(10) Gauge Theories P/10/30

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A summary is reported on our previous publications about four dimensional N=1 supersymmetric Spin(10) gauge theory with chiral superfields in the spinor and vector representations in the non-Abelian Coulomb phase. Carrying out the method of \amax, we exlpored decoupling operators in the infared and the renormalization flow of the theory. \ We also give a brief review on the non-Abelian Coulomb phase of the theory after recalling the unitarity bound and the a-maximization procedure in four-dimensional conformal field theory. This is a review article invited to International Journal of Modern Physics A.

17/09/2010 PDF

Topological invariants and moduli spaces of Gorenstein quasi-homogeneous surface singularities. M/10/29

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We describe all connected components of the space of hyperbolic Gorenstein quasi-homogeneous surface singularities. We prove that any connected component is homeomorphic to a quotient of Rd by a discrete group.

15/09/2010 PDF

Restrictions of generalized Verma modules to symmetric pairs M/10/28

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We initiate a new line of investigation on branching problems for generalized Verma modules with respect to reductive symmetric pairs (g, g'). In general, Verma modules may not contain any simple module when restricted to a reductive subalgebra. In this article we give a necessary and sufficient condition on the triple (g, g', p) such that the restriction X|_g' always contains simple g'-modules for any g-module X lying in the parabolic BGG category O^p attached to a parabolic subalgebra p of g. Formulas are derived for the Gelfand–Kirillov dimension of any simple module occurring in a simple generalized Verma module. We then prove that the restriction X|_g' is generically multiplicity-free for any p and any X\in O^p if and only if (g, g') is isomorphic to (A_n,A_n−1), (B_n,D_n), or (D_n+1,B_n). Explicit branching laws are also presented.

26/08/2010 PDF

Ramification and cleanliness M/10/27

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This article is devoted to studying the ramification of Galois torsors and of $\ell$-adic sheaves in characteristic $p>0$ (with $\ell\not=p$). Let $k$ be a perfect field of characteristic $p>0$, $X$ be a smooth, separated and quasi-compact $k$-scheme, $D$ be a simple normal crossing divisor on $X$, $U=X-D$, $\Lambda$ be a finite local ${\mathbb Z}_\ell$-algebra, $F$ be a locally constant constructible sheaf of $\Lambda$-modules on $U$. We introduce a boundedness condition on the ramification of $F$ along $D$, and study its main properties, in particular, some specialization properties that lead to the fundamental notion of cleanliness and to the definition of the characteristic cycle of $F$. The cleanliness condition extends the one introduced by Kato for rank one sheaves. Roughly speaking, it means that the ramification of $F$ along $D$ is controlled by its ramification at the generic points of $D$. Under this condition, we propose a conjectural Riemann-Roch type formula for $F$. Some cases of this formula have been previously proved by Kato and by the second author (T.S.).

23/08/2010 PDF

Noncommutative Toda Chains, Hankel quasideterminants and Painlev\'e II equation M/10/25

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We construct solutions of an infinite Toda system and an analogue of Painlev'e II equation over noncommutative differential division rings in terms of quasideterminants of Hankel matrices.

27/07/2010 PDF

Global Stringy Orbifold Cohomology, K-theory and de Rham Theory M/10/26

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There are two approaches to constructing stringy multiplications for global quotients. The first one is given by first pulling back and then pushing forward. This has been used to define a global stringy extension of the functors $K_0,K^{top}, A^*,H^*$. The second one is given by first pushing forward and then pulling back. This has been used in the cyclic case and in particular for singularities with symmetries and for symmetric products. For Abelian quotients Chen and Hudiscussed such a construction in the de Rham setting. We give a rigorous formulation of de Rham theory for any global quotient from both points of view. We also show that the pull--push formalism has a solution by the push--pull equationsin the setting of cyclictwisted sectors. In the general case, we introduce ring extensions that allow us to treat all the stringy multiplications mentioned above. The first extension provides formal sections and a second extension fractional Euler classes. The formal sections allow us to give a pull-push solution while fractional Euler classes give a trivialization of the co--cycles of the pull-push formalism using the presentation of the obstruction bundle of Jarvis--Kaufmann--Kimura This trivialization can be interpreted as defining twist fields. We end with an outlook on applications to singularities with symmetries aka. orbifold Landau--Ginzburg models.

27/07/2010 PDF

Phenomenology of the Equivalence Principle with Light Scalars P/10/23

22/07/2010 PDF

Equivalence Principle Violations and Couplings of a Light Dilaton P/10/24

22/07/2010 PDF

Categorification of the Jones-Wenzl Projectors M/10/22

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The Jones-Wenzl projectors play a central role in quantum topology, underlying the construction of SU(2) topological quantum field theories and quantum spin networks. We construct chain complexes, whose graded Euler characteristic is the classical''projector in the Temperley-Lieb algebra.We show that the projectors are homotopy idempotents and uniquely defined up to homotopy. Our results fit within the general framework of Khovanov's categorification of the Jones polynomial. Consequences of our construction include families of knot invariants corresponding to higher representations of quantum su(2), and a categorification of quantum spin networks. We introduce 6j-symbols in this context.

14/07/2010 PDF

FEYNMAN AMPLITUDES AND LANDAU SINGULARITIES FOR 1-LOOP GRAPHS M/10/20

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We use mixed Hodge structures to investigate Feynman amplitudes as functions of external momenta and masses.

02/07/2010 PDF

Algebraic Structures in local QFT P/10/21

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A review of the Hodge and Hopf-algebraic approach to QFT.

02/07/2010 PDF

Instantons on Gravitons P/10/19

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Yang-Mills instantons on ALE gravitational instantons were constructed by Kronheimer and Nakajima in terms of matrices satisfying algebraic equations. These were conveniently organized into a quiver. We construct generic Yang-Mills instantons on ALF gravitational instantons. Our data is formulated in terms of matrix-valued functions of a single variable, that are organized into a bow. We introduce the general notion of a bow, its representation, its associated data and moduli space of solutions.The Nahm transform maps any bow solution to an instanton on an ALF space. We demonstrate that this map respects all complex structures on the moduli spaces, so it is likely to be an isometry, and use this fact to study the asymptotics of the moduli spacesof instantons on ALF spaces.

23/06/2010 PDF

Remarks on some locally Qp-analytic representations of GL2(F) in the crystalline case M/10/18

01/06/2010 PDF

The emerging p-adic Langlands programme M/10/17

20/05/2010 PDF

Classical analog of quantum Schwarzschild black hole: local vs global, and the mystery of log 3 P/10/15

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The model is built in which the main global properties of classical and quasi-classical black holes become local. These are the event horizon, no-hair, temperature and entropy. Our construction is based on the features of a quantum collapse, discovered when studying some quantum black hole models. But our model is purely classical, and this allows to use selfconsistently the Einstein equations and classical (local) thermodynamics and explain in this way the log 3-puzzle.

20/04/2010 PDF

A Non-differentiable Noether's theorem M/10/14

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In the framework of the non-differentiable embedding of Lagrangian systems, defined by Cresson and Greff, we prove a Noether's theorem based on the lifting of one-parameter groups of diffeomorphisms.

14/04/2010 PDF

The critical ultraviolet behaviour of N=8 supergravity amplitudes P/10/13

09/04/2010 PDF

Eisenstein series for higher-rank groups and string theory amplitudes P/10/10

06/04/2010 PDF

Kramers-Wannier Duality for Non-Abelian Lattice Spin Systems and Hecke Surfaces M/10/12

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We discuss two themes: 1.Duality transformation for generalized Potts models and Hecke surfaces and $K$-regular graphs

01/04/2010 PDF

Pseudogroupes de Lie et théorie de Galois différentielle M/10/11

26/03/2010 PDF

Intersecting D4-branes Model of Holographic QCD and Tachyon Condensation P/10/09

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We consider the intersecting D4-brane and anti-D4-brane model of holographic QCD, motivated by the model that has recently been suggested by Van Raamsdonk and Whyte. We analyze such D4-branes by the use of the tachyonic Dirac-Born-Infeld action, so that we find the classical solutions describing the intersecting D4-branes and the U-shaped D4-branes. We show that the bi-fundamental tachyon'' field in the bulk theory provides a current quark mass and a quark condensate to the dual gauge theory and thatthe lowest modes of mesons obtain mass via tachyon condensation. Then evaluating the properties of a pion, one can reproduce Gell-Mann-Oakes-Renner relation.

15/03/2010 PDF

Monodromy and Jacobi-like Relations for Color-Ordered Amplitudes P/10/08

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We discuss monodromy relations between different color-ordered amplitudes in gauge theories. We show that Jacobi-like relations of Bern, Carrasco and Johansson can be introduced in a manner that is compatible with these monodromy relations. The Jacobi-like relations are not the most general set of equations that satisfy this criterion. Applications to supergravity amplitudes follow straightforwardly through the KLT-relations. We explicitly show how the tree-level relations give rise to non-trivialidentities at loop level.

12/03/2010 PDF

Monodromy and Kawai-Lewellen-Tye Relations for Gravity Amplitudes P/10/07

08/03/2010 PDF

A matrix model for the topological string I: deriving the matrix model P/10/06

04/03/2010 PDF

String theory dualities and supergravity divergences P/10/05

22/02/2010 PDF

Single-Lifting Macaulay-Type Formulae of Generalized Unmixed Toric Resultants M/10/04

28/01/2010 PDF

Minimal representations and reductive dual pairs in conformal field theory P/10/03

15/01/2010 PDF

Automorphic properties of low energy string amplitudes in various dimensions P/10/01

14/01/2010 PDF

On the ultraviolet behaviour of N=8 supergravity amplitudes P/10/02

14/01/2010 PDF

Sugawara-type constraints in hyperbolic coset models P/09/47

22/12/2009 PDF

$E_{7(7)}$ invariant Lagrangian of $d=4$ ${\mathcal N} = 8$ supergravity P/09/52

22/12/2009 PDF

Crystal melting on toric surfaces P/09/53

02/12/2009 PDF

Almost etale resolution of foliations M/09/51

25/11/2009 PDF

La cosmologie: un laboratoire pour la théorie des cordes P/09/50

17/11/2009 PDF

Gravitational Self Force in a Schwarzschild Background and the Effective One Body Formalism P/09/49

03/11/2009 PDF

Improved resummation of post-Newtonian multipolar waveforms from circularized compact binaries P/09/48

03/11/2009 PDF

Sur un problème de compatibilité local-global modulo p pour GL2 M/09/46

05/10/2009 PDF

Three \'Etudes in QFT P/09/45

23/09/2009 PDF

Construire un noyau de la fonctorialité~? \\ Le cas de l'induction automorphe \\ sans ramification de ${\rm GL}_1$ à ${\rm GL}_2$ M/09/42

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Le but de cet article (à paraître aux Annales de l'Institut Fourier) est de présenter une nouvelle méthode purement adélique pour réaliser le principe de fonctorialité de Langlands dans le cas de l'induction automorphe sans ramification de GL(1) à GL(2) sur les corps de fonctions. On construit sur le produit des groupes adéliques GL(1) et GL(2) un noyau de la fonctorialité. C’est une version « en famille » et locale de la construction par les modèles de Whittaker globaux, utilisée classiquement dans les « théorèmes réciproques » de Weil et Piatetski-Shapiro. La plus grande partie de la construction et des vérifications nécessaires est locale, c’est-à-dire se fait place par place. Il s’agit de prouver que deux fonctions, dont chacune est définie localement, deviennent égales après sommation sur les éléments rationnels de certains groupes. Cela résulte de la formule de Poisson, sur le modèle de la thèse de Tate, dès lors que l’on comprend comment nos deux fonctions se déduisent localement l’une de l’autre par une certaine transformation de Fourier.

18/09/2009 PDF

Construire des noyaux de la fonctorialité~? \\ Définition générale, \\ cas de l'identité de ${\rm GL}_2$ \\ et construction générale conjecturale \\ de leurs coefficients de Fourier M/09/43

18/09/2009 PDF

We are all your students, Mr. Cartan'' M/09/44

18/09/2009 PDF

Notes sur l'histoire et la philosophie des mathématiques V : le problème de l'espace M/09/41

16/09/2009 PDF

Noncommutative $\mathbf{K}$-correspondence categories, simplicial sets and pro $C^*$-algebras M/09/40

28/08/2009 PDF

A representation-valued relative Riemann-Hurwitz theorem and the Hurwitz-Hodge bundle M/09/39

27/08/2009 PDF

Non-renormalization conditions for four-gluon scattering in supersymmetric string and field theory P/09/35

20/08/2009 PDF

Living in a contradictory world: categories vs sets? M/09/37

20/08/2009 PDF

RENORMALIZATION AND RESOLUTION OF SINGULARITIES P/09/36

07/08/2009 PDF

Two-Dimensional Topological Strings Revisited P/09/17

03/08/2009 PDF

Minimal Basis for Gauge Theory Amplitudes P/09/33

15/07/2009 PDF

Vinberg Algebras and Combinatorics M/09/34

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Vinberg algebras are usually called pre-Lie algebras and were introduced long ago by Gerstenhaber. We propose to follow a diﬀerent route by motivating these algebras by problems coming from diﬀerential geometry, and ﬁrst studied in depth by Vinberg. We shall recall how the Lie bracket of vector ﬁelds can be obtained by skewsymmetriz- ing a more fundamental product. We shall then develop a combinatorial method for the higher order diﬀerential operators, quite similar to the procedure used in study- ing Runge–Kutta methods. We shall then move to nilpotent (or pronilpotent) Lie groups. In the last part of these lectures, I shall apply the previous methods in the renormalization theory of quantum ﬁelds (à la Connes–Kreimer).

26/06/2009 PDF

On the gravitational polarizability of black holes P/09/28

17/06/2009 PDF

Fermionic Kac-Moody Billiards and Supergravity P/09/31

17/06/2009 PDF

The Equivalence Principle and the Constants of Nature P/09/32

17/06/2009 PDF

Supersymmetric Vacua and Bethe Ansatz P/09/09

16/06/2009 PDF

The Effective One Body description of the Two-Body problem P/09/27

16/06/2009 PDF

Relativistic tidal properties of neutron stars P/09/29

16/06/2009 PDF

An improved analytical description of inspiralling and coalescing black-hole binaries P/09/30

16/06/2009 PDF

Simplicity of Amplitudes in Gravity and Yang-Mills Theories P/09/26

15/06/2009 PDF

Algebras for quantum fields P/09/24

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We give an account of the current state of the approach to quantum field theory via Hopf algebras and Hochschild cohomology. We emphasize the versatility and mathematical foundation of this algebraic structure, and collect algebraic structures here inone place which are either scattered over the literature, or only implicit in previous writings. In particular we point out mathematical structures which can be helpful to farther develop our mathematical understanding of quantum fields. 1.1.

09/06/2009 PDF

The QCD $\beta$-function from global solutions to Dyson-Schwinger equations P/09/25

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We study quantum chromodynamics from the viewpoint of untruncated Dyson–Schwinger equations turned to an ordinary differential equation for the gluon anomalous dimension. This nonlinear equation is parameterized by a function P(x) which is unknown beyond perturbation theory. Still, very mild assumptions on P(x) lead to stringent restrictions for possible solutions to Dyson– Schwinger equations. We establish that the theory must have asymptotic freedom beyond perturbation theory and alsoinvestigatethe low energy regime and the possibility for a mass gap in the asymptotically free theory.

09/06/2009 PDF

The Big Bang as the Ultimate Traffic Jam P/09/23

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We present a novel solution to the nature and formation of the initial state of the Universe. It derives from the physics of a generally covariant extension of Matrix theory. We focus on the dynamical state space of this background independent quantum theory of gravity and matter, an infinite dimensional, complex non-linear Grassmannian. When this space is endowed with a Fubini–Study-like metric, the associated geodesic distance between any two of its points is zero. This striking mathematical result translates into a physical description of a hot, zero entropy Big Bang. The latter is then seen as a far from equilibrium, large fluctuation driven, metastable ordered transition, a “freezing by heating” jamming transition. Moreover, the subsequent unjamming transition could provide a mechanism for inflation while rejamming may model a Big Crunch, the final state of gravitational collapse.

18/05/2009 PDF

Surprising simplicity of N=8 supergravity P/09/22

15/05/2009 PDF

An infinite family of solvable and integrable quantum systems on a plane P/09/21

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An infinite family of exactly-solvable and integrable potentials on a plane is introduced. It is shown that all already known rational potentials with the above properties allowing separation of variables in polar coordinates are particular cases of thisfamily. The underlying algebraic structure of the new potentials is revealed as well as its hidden algebra. We conjecture that all members of the family are also superintegrable and demonstrate this for the first few cases. A quasi-exactly-solvableand integrable generalization of the family is found.

13/05/2009 PDF

Quantum Groups and Braid Group Statistics in Conformal Current Algebra Models P/09/18

24/04/2009 PDF

Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence M/09/20

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We discuss derived categories of the first kind for DG-modules, DG-comodules, and DG-contramodules, derived categories of the second kind for CDG-modules, CDG-comodules, and CDG-contramodules. For the latter two, the comodule-contramodule correspondence is constructed. Nonhomogeneous Koszul duality (equivalence of exotic derived categories) is obtained in conilpotent and nonconilpotent versions. A_\infty structures are also considered.

24/04/2009 PDF

Generalised Kostka-Foulkes polynomials and cohomology of line bundles on homogeneous vector bundles M/09/19

23/04/2009 PDF

Sutherland-type Trigonometric Models, Trigonometric Invariants and Multivariable Polynomials. II. $E_7$ case P/09/15

08/04/2009 PDF

Discrete Minimal Surface Algebras M/09/14

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We consider Discrete Minimal Surface Algebras (DMSA) as noncommutative analogues of minimal surfaces in higher dimensional spheres. These algebras appear naturally in the context of Membrane Theory, where sequences of their representations are used as a regularization of the theory. After showing that the defining relations of the algebra are consistent, and that one can compute a basis of the universal enveloping algebra, we give several explicit examples of DMSAs in terms of subsets of sl(n) (any semi-simple Lie algebra providing a trivial example by itself). A special class of DMSAs are Yang-Mills algebras. The representation graph is introduced to study representations of DMSAs of dimension d<=4, and properties of representations are related to properties of graphs. The representation graph of a tensor product is (generically) the Cartesian product of the corresponding graphs. We provide explicit examples of irreducible representations and, for coinciding eigenvalues, classify all the unitary representations of the corresponding algebras.

27/03/2009 PDF

Higher-loop amplitudes in the non-minimal pure spinor formalism P/09/13

23/03/2009 PDF

Recursive relations in the core Hopf algebra P/09/12

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We study co-ideals in the core Hopf algebra underlying a quantum field theory.

17/03/2009 PDF

Integrable Systems in Noncommutative Spaces P/09/11

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We discuss extension of soliton theories and integrable systems into non-commutative (NC) spaces. In the framework of NC integrable hierarchy, we give infinite conserved quantities and exact soliton solutions for many NC integrable equations,which are represented in terms of Strachan's products and quasi-determinants, respectively. We also present a relation to an NC Anti-Self-Dual Yang-Mills equation, and make comments on how integrability'' should be considered in noncommutative spaces.

13/03/2009 PDF

Scattering Amplitudes and BCFW Recursion in Twistor Space P/09/10

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A number of recent advances in our understanding of scattering amplitudes have been inspired by ideas from twistor theory. While there has been much work studying the twistor space support of scattering amplitudes, this has largely been done by examiningthe amplitudes in momentum space. In this paper, we construct the actual twistor scattering amplitudes themselves. The main reasons for doing so are to seek a formulation of scattering amplitudes in N=4 super Yang-Mills in which superconformal symmetry ismanifest, and to use the progress in on-shell methods in momentum space to build our understanding of how to construct quantum field theory in twistor space. We show that the recursion relations of Britto, Cachazo, Feng and Witten have a natural twistorformulation that, together with the three-point seed amplitudes, allow us in principle to recursively construct general tree amplitudes in twistor space. The twistor space BCFW recursion is tractable, and we obtain explicit formulae for n-particle MHV and NMHV amplitudes, together with their CPT conjugates (whose representations are distinct in our chiral framework). The amplitudes are a set of purely geometric, superconformally invariant delta functions, dressed by certain sign operators. These sign operators subtly violate conformal invariance, even for tree-level amplitudes in N=4 super Yang-Mills, and we trace their origin to a topological property of split signature spacetime. Our work is related via a twistor transform to the ambidextrous twistor diagram approach of Hodges and of Arkani-Hamed, Cachazo, Cheung and Kaplan.

12/03/2009 PDF

An accurate few-parameter ground state wave function for the Lithium atom P/09/08

25/02/2009 PDF

On two-dimensional quantum gravity and quasiclassical integrable hierarchies P/09/06

24/02/2009 PDF

Essential hyperbolic Coxeter polytopes M/09/07

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We introduce a notion of essential hyperbolic Coxeter polytope as a polytope which fits some minimality conditions. The problem of classification of hyperbolic reflection groups can be easily reduced to classification of essential Coxeter polytopes. We determine a potentially large combinatorial class of polytopes containing, in particular, all the compact hyperbolic Coxeter polytopes of dimension at least $6$ which are known to be essential, and prove that this class contains finitely many polytopes only. We also construct an effective algorithm of classifying polytopes from this class, and realize it in four-dimensional case.

24/02/2009 PDF

The core Hopf algebra P/09/05

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We study the core Hopf algebra underlying the renormalization Hopf algebra.

13/02/2009 PDF

On a class of hamiltonian fiber bundles M/09/04

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We study an interesting class of hamiltonian fiber bundles whose fibers are compact homogeneous symplectic manifolds. Applications to the cohomology of their symplectomorphism group are given.

04/02/2009 PDF

Exploiting N=2 in consistent coset reductions of type IIA P/09/03

27/01/2009 PDF

Generalized E(7(7)) coset dynamics and D=11 supergravity P/09/02

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The hidden on-shell E(7(7)) symmetry of maximal supergravity is usually discussed in a truncation from D=11 to four dimensions. In this article, we reverse the logic and start from a theory with manifest off-shell E(7(7)) symmetry inspired by West's coset construction. Following de Wit's and Nicolai's idea that a 4+56 dimensional exceptional geometry'' underlies maximal supergravity, we construct the corresponding Lagrangian and the supersymmetry variations for the 56 dimensional subsector. We prove that both the dynamics and the supersymmetry coincide with D=11 supergravity in a truncation to d=7 in the expected way.

11/01/2009 PDF

Un pays dont on ne connaîtrait que le nom (Grothendieck et les 'motifs') M/09/01

08/01/2009 PDF

Topological String on ${\mathcal S}^2$ Revisited P/08/18

31/12/2008 PDF

Théories de Galois géométriques M/08/62

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Nous présentons de manière conceptuelle les idées de Riemann sur les singularités et la monodromie, et nous l'illustrons par l'étude des fonctions algébriques, et des solutions des équations différentielles. Cela fournit une approche unifiée aux diverses théories de Galois.

29/12/2008 PDF

Yet Another Poincaré's Polyhedron Theorem M/08/64

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This work contains a new version of Poincare's Polyhedron Theorem that also suits geometries of nonconstant curvature lacking concepts of convexity. Most conditions of the theorem, being as local as possible, are easy to verify in practice.

22/12/2008 PDF

Notion de spectre M/08/61

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La notion de spectre est au départ une notion physique. Elle a pris progressivement une signification de plus en plus large en mathématique, sa signification mathématique la plus importante lui ayant été donnée par Grothendieck dans sa théorie des schémas. Nous nous promènerons donc de la physique à la géométrie algébrique.

19/12/2008 PDF

Entropy estimation of symbolic sequences: How short is a short sequence? P/08/63

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While entropy per unit time is a meaningful index to quantify the dynamic features of experimental time series, its estimation is often hampered by the finite length of the data. We here investigate the performance of entropy estimation procedures, relying either on block entropies or Lempel-Ziv complexity, when only {\it very short symbolic sequences} are available. Heuristic analytical arguments point at the influence of temporal correlations on the bias and statistical fluctuations, and put forward a reduced effective sequence length suitable for error estimation. Numerical studies are conducted using, as benchmarks, the wealth of different dynamic regimes generated by the family of logistic maps and stochastic evolutions generated by a Markov chain of tunable correlation time. Practical guidelines and validity criteria are proposed, based on the result that the quality of entropy estimation is sensitive to the sequence temporal correlation hence self-consistently depends on the entropy value itself.

19/12/2008 PDF

Quantum Integrability and Supersymmetric Vacua P/08/59

02/12/2008 PDF

Phase space polarization and the topological string: a case study P/08/60

02/12/2008 PDF

Instanton Partition Functions and M-Theory P/08/16

27/11/2008 PDF

24/11/2008 PDF

Simplicity in the Structure of QED and Gravity Amplitudes P/08/54

21/11/2008 PDF

Dimensional Regularization of the Gravitational Interaction of Point Masses in the ADM Formalism P/08/55

21/11/2008 PDF

What is Missing from Minkowski s Raum und Zeit Lecture P/08/56

21/11/2008 PDF

Improved Resummation of Post-Newtonian Multipolar Waveforms from Circularized Compact Binaries P/08/57

21/11/2008 PDF

Two Dimensional Topological Strings and Gauge Theory P/08/17

18/11/2008 PDF

Multi-valued hyperelliptic continued fractions of generalized Halphen type M/08/53

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We introduce and study higher genera generalizations of the Halphen theory of continued fractions. The basic notion we start with is hyperel- liptic Halphen (HH) element p X2g+2 ¡ p Y2g+2 x ¡ y ; depending on parameter y, where X2g+2 is a polynomial of degree 2g + 2 and Y2g+2 = X2g+2(y). We study regular and irregular HH elements, their continued fraction developments and some basic properties of such developments such as: even and odd symmetry and periodicity. There is a 2 $g + 1 dynamics which lies in the basis of the developed continued fractions theory. We give two geometric realizations of this dynamics. The ¯rst one deals with nets of polynomials and with polygons circumscribed about a conic K. The dynamics is realized as a path ofpolygons of g + 1 sides inscribed in a curve B of degree 2g and circumscribed about the conic K obtained by successive moves, so called { °ips along edges. The second geometric realization leads to a new interpretation of generalized Jacobians of hyperelliptic curves. 1 18/11/2008 PDF Partition Functions of Matrix Models as the First Special Functions of String Theory II. Kontsevich Model P/08/52 07/11/2008 PDF Diagrammes de Diamond et$(\varphi,\Gamma)$-modules M/08/51 23/09/2008 PDF Piecewise principal comodule algebras M/08/50 Voir plus > Voir moins > A comodule algebra P over a Hopf algebra H with bijective antipode is called principal if the coaction of H is Galois and P is H-equivariantly projective (faithfully flat) over the coaction-invariant subalgebra B. We prove that principality is a piecewise property: given N comodule-algebra surjections P -> P_i whose kernels intersect to zero, P is principal if and only if all P_i's are principal. Furthermore, assuming the principality of P, we show that the lattice these kernels generate is distributive if and only if so is the lattice obtained by intersection with B. Finally, assuming the above distributivity property, we obtain a flabby sheaf of principal comodule algebras over a certain space that is universal for all such N-families of surjections P -> P_i and such that the comodule algebra of global sections is P. 26/08/2008 PDF Tamagawa defect of Euler systems M/08/48 Voir plus > Voir moins > As remarked by Mazur an Rubin (2004, {\em Mem. Amer. Math. Soc.\/}, 168(799)) one does not expect the Kolyvagin system obtained from an Euler system for a$p$-adic Galois representation$T$to be \emph{primitive} (in the sense of \emph{loc. cit.}) if$p$divides a Tamagawa number at a prime$\ell\neq p$; thus fails to compute the correct size of the relevant Selmer module. In this paper we obtain a lower bound for the size of the cokernel of the Euler system to Kolyvagin system map in terms of the local Tamagawa numbers of$T$, refining a result of \emph{loc. cit.}. We show how this partially accounts for the missing Tamagawa factors in Kato's calculations with his Euler system. 19/08/2008 PDF Stickelberger elements and Kolyvagin systems M/08/49 Voir plus > Voir moins > In this paper we construct (many) Kolyvagin systems out of Stickelberger elements, utilizing ideas borrowed from our previous work on Kolyvagin systems of Stark elements. We show how to apply this construction to prove results on the \emph{odd} parts ofthe ideal class groups of CM fields which are abelian over a totally real field, and deduce the main conjecture of Iwasawa theory for totally real fields (for totally odd characters). Although the main results of this paper have already been established by Wiles, our approach provides another example (which slightly differs from the case of Stark elements) on how to study \emph{Kolyvagin systems of core rank$r>1$} (in the sense of Mazur and Rubin). The analogous (and in some sense complementary) results for \emph{even} parts of the ideal class groups and main conjectures for totally even characters of totally real number fields have been previously obtained by the author using similar ideas. 19/08/2008 PDF Analytic subvarieties with many rational points M/08/46 30/07/2008 PDF Dyson's Theorem for curves M/08/47 30/07/2008 PDF Vertex algebroids over Veronese rings M/08/45 Voir plus > Voir moins > We find a canonical quantization of Courant algebroids over Veronese rings. Part of our approach allows a semi-infinite cohomology interpretation, and the latter can be used to define sheaves of chiral differential operators on some homogeneous spaces including the space of pure spinors punctured at a point. 29/07/2008 PDF LOCAL STABILITY OF A QUASI-LINEAR AGE-SIZE STRUCTURED POPULATION DYNAMICS MODEL M/08/44 Voir plus > Voir moins > The local stability of a quasi-linear age-size structured population model studied in Tchuenche (2007) is analysed. If a certain threshold parameter known as the basic reproductive rate is less than unity, then the trivial steady state is locally asymptotically stable. Also, it is shown that if the only real root of the equation${\cal R}(m')=1$is negative, then, the non trivial steady state is locally exponentially asymptotically stable. 03/07/2008 PDF Pure Spinor Partition Function and the Massive Superstring Spectrum P/08/31 03/07/2008 PDF One-loop$\beta$functions of a translation-invariant renormalizable noncommutative scalar model P/08/43 Voir plus > Voir moins > Recently, a new type of renormalizable$\phi^{\star 4}_{4}$scalar model on the Moyal space was proved to be perturbatively renormalizable. It is translation-invariant and introduces in the action a$a/(\theta^2p^2)$term. We calculate here the$\beta$and$\gamma$functions at one-loop level for this model. The coupling constant$\beta_\lambda$function is proved to have the same behaviour as the one of the$\phi^4$model on the commutative$\mathbb{R}^4$. The$\beta_a$function of the new parameter$a$is also calculated. Some interpretation of these results are done. 24/06/2008 PDF SELF-SIMILAR P-ADIC FRACTAL STRINGS AND THEIR COMPLEX DIMENSIONS M/08/42 Voir plus > Voir moins > We develop a geometric theory of self-similar p-adic fractal strings and their complex dimensions. We obtain a closed-form formula for the geo- metric zeta functions and show that these zeta functions are rational functions in an appropriate variable. We also prove that every self-similar p-adic fractal string is lattice. Finally, we deﬁne the notion of a nonarchimedean self-similar set and discuss its relationship with that of a self-similar p-adic fractal string. We illustrate the general theory by two simple examples, the nonarchimedean Cantor and Fibonacci strings. 11/06/2008 PDF Regularization, renormalization, and renormalization groups: relationships and epistemological aspects P/08/41 Voir plus > Voir moins > This paper confronts renormalization used in quantum field theory and that used in critical phenomena studies in statistical mechanics or dynamical systems theory. Regularization that cures spurious divergences is distinguished from renormalization transformations allowing to compute actual physical divergences. The former generates a group, and is also encountered in singular perturbation analyses in nonlinear physics. The latter generates a semi-group, and is implemented as a flow in a space of models; its analysis, focusing on fixed points and their neighborhood, allows to determine asymptotic scaling behavior, to delineate universality classes and to assess model structural stability (or instability, i.e. crossovers). The renormalization group can be seen as a symmetry group and a general covariant formulation is proposed. Aspects presented here show that renormalization theory has emulated a shift of focus from the investigation of outcomes of a given model to the analysis of models themselves, by relating models of the same system at different scales or grouping models of different systems exhibiting the same large-scale behavior. So doing, not only (subjective and partial) models are distinguished from underlying physical systems, but also intrinsic physical features can be derived from model comparison and classification. 10/06/2008 PDF Open-Closed Moduli Spaces and Related Algebraic Structures M/08/40 Voir plus > Voir moins > We set up a Batalin-Vilkovisky Quantum Master Equation (QME) for open-closed string theory and show that the corresponding moduli spaces give rise to a solution, a generating function for their fundamental chains. The equation encodes the topological structure of the compactification of the moduli space of bordered Riemann surfaces. The moduli spaces of bordered$J$-holomorphic curves are expected to satisfy the same equation, and from this viewpoint, our paper treats the case of the target space equal to a point. We also introduce the notion of a symmetric Open-Closed Topological Conformal Field Theory (OC TCFT) and study the L_infty and A_infty algebraic structures associated to it. 06/06/2008 PDF A trace on fractal graphs and the Ihara zeta function M/08/36 Voir plus > Voir moins > Starting with Ihara's work in 1968, there has been a growing interest in the study of zeta functions of finite graphs, by Sunada, Hashimoto, Bass, Stark and Terras, Mizuno and Sato, to name just a few authors. Then, Clair and Mokhtari-Sharghi have studied zeta functions for infinite graphs acted upon by a discrete group of automorphisms. The main formula in all these treatments establishes a connection between the zeta function, originally defined as an infinite product, and the Laplacian of the graph. In this article, we consider a different class of infinite graphs. They are fractal graphs, i.e. they enjoy a self-similarity property. We define a zeta function for these graphs and, using the machinery of operator algebras, we prove a determinant formula, which relates the zeta function with the Laplacian of the graph. We also prove functional equations, and a formula which allows approximation of the zeta function by the zeta functions of finite subgraphs. 29/05/2008 PDF Ihara's zeta function for periodic graphs and its approximation in the amenable case M/08/37 Voir plus > Voir moins > In this paper, we give a more direct proof of the results by Clair and Mokhtari-Sharghi on the zeta functions of periodic graphs. In particular, using appropriate operator-algebraic techniques, we establish a determinant formula in this context and examine its consequences for the Ihara zeta function. Moreover, we answer in the affirmative one of the questions raised in \cite{GrZu} by Grigorchuk and$\dot{\text{Z}}$uk. Accordingly, we show that the zeta function of a periodic graph with an amenable group action is the limit of the zeta functions of a suitable sequence of finite subgraphs. 29/05/2008 PDF Bartholdi Zeta Functions for Periodic Simple Graphs M/08/38 Voir plus > Voir moins > The definition of the Bartholdi zeta function is extended to the case of infinite periodic graphs. By means of the analytic determinant for semifinite von~Neumann algebras studied by the authors in \cite{GILa03}, a determinant formula and functional equations are obtained for this zeta function. 29/05/2008 PDF Ihara zeta functions for periodic simple graphs M/08/39 Voir plus > Voir moins > The definition and main properties of the Ihara zeta function for graphs are reviewed, focusing mainly on the case of periodic simple graphs. Moreover, we give a new proof of the associated determinant formula, based on the treatment developed by Stark and Terras for finite graphs. 29/05/2008 PDF Toward zeta functions and Complex Dimensions of Multifractals M/08/34 Voir plus > Voir moins > Multifractals are inhomogeneous measures (or functions) which are typically described by a full spectrum of real dimensions, as opposed to a single real dimension. Results from the study of fractal strings in the analysis of their geometry, spectra and dynamics via certain zeta functions and their poles (the complex dimensions) are used in this text as a spring board to define similaar tools for the study of multifractals such as the binomial measure. The goal of this work is to shine light on new ideas and perspectives rather than to summarize a coherent theory. Progress has been made which connects these new perspectives to and expands upon classical results, leading to a healthy variety of natural and interesting questions for further investigation and elaboration. 28/05/2008 PDF Turbulence and Holography P/08/35 28/05/2008 PDF Sutherland-type trigonometric models, trigonometric invariants, and multivariate polynomials P/08/32 16/05/2008 PDF The QED$\beta$-function from global solutions to Dyson-Schwinger equations P/08/33 Voir plus > Voir moins > We discuss the structure of beta functions as determined by the recursive nature of Dyson-Schwinger equations turned into an analysis of ordinary differential equations, with particular emphasis given to quantum electrodynamics. In particular we determine when a separatrix for solutions to such ODEs exists and clarify the existence of Landau poles beyond perturbation theory. Both are determined in terms of explicit conditions on the asymptotics for the growth of skeleton graphs. 15/05/2008 PDF Tube Formulas and complex Dimensions of Self-Similar tilings M/08/27 Voir plus > Voir moins > We use the self-similar tilings constructed by Erin Pearse to define a generating function for the geometry of a self-similar set in Euclidean space. This geometric zeta function encodes scaling and curvature properties related to the complement of the fractal set, and the associated system of mappings. This allows one to obtain the complex dimensions of the self-similar tiling as the poles of the geometric zeta function and hence develop a tube formula for self-similar tilings in Rd. The resulting power series in epsilon is a fractal extension of Steiner's classical tube formula for convex bodies K of Rd. Our sum has coefficients related to the curvatures of the tiling, and contains terms for each integer i=0,1,...,d-1, just as Steiner's does. However, our formula also contains a term for each complex dimension. This provides further justification for the term complex dimension''. It also extends several aspects of the theory of fractal strings to higher dimensions and sheds new light on the tube formula for fractals strings obtained by Michel Lapidus and Machiel van Frankenhuisjen. 02/05/2008 PDF Tube Formulas for Self-Similar Fractals M/08/28 Voir plus > Voir moins > Tube formulas (by which we mean an explicit formula for the volume of an (inner) epsilon-neighbourhood of a subset of a suitable metric space) have been used in many situations to study properties of the subset. For smooth submanifolds of Euclidean space, this includes Weyl's celebrated results on spectral asymptotics, and the subsequent relation between curvature and spectrum. Additionally, a tube formula contains information about the dimension and measurability of rough sets. In convex geometry, thetube formula of a convex subset of Euclidean space allows for the definition of certain curvature measures. These measures describe the curvature of sets which may be too irregular to support derivatives. In this survey paper, we describe some recent advances in the development of tube formulas for self-similar fractals, and their applications and connections to the other topics mentioned here. 02/05/2008 PDF Nonarchimedean Cantor Set and String M/08/29 Voir plus > Voir moins > We construct a nonarchimedean (or p-adic) analogue of the classical ternary set C. In particular, we show that this nonarchimedean Cantor set C3 is self-similar. Furthermore, we characterize C3 as the subset of 3-adic integers whose elements contain only 0's and 2's in their 3-adic expansions and prove that C3 is naturally homeomorphic to C. Finally, from the point of view of fractal strings and their complex dimensions (see the books by Lapidus and van Frankhuijsen), the corresponding non archimedean Cantor string resembles the standard archimedean (or real) Cantor string perfectly. 02/05/2008 PDF MIXED HODGE STRUCTURES AND RENORMALIZATION IN PHYSICS M/08/30 Voir plus > Voir moins > We relate renormalization in perturbative quantum field theory to the theory of limiting mixed Hodge structures using parametric representations of Feynman graphs. 02/05/2008 PDF Recycling the Independent Field Approximation argument in the far field P/08/26 Voir plus > Voir moins > The Independent Field Approximation for the entropy production of Laplacian mild diﬀusional ﬁelds is rigorously introduced and discussed. Some new results due to super-convergent algorithms are presented and the meaning of the active zone concept is enlightened. 23/04/2008 PDF On the origin of time and the Universe P/08/25 22/04/2008 PDF Fractal structure of the block-complexity function M/08/24 Voir plus > Voir moins > We demonstrate that the block-complexity function for words from 3-letter and 4-letter alphabets exhibits a fractal structure. The resulting fractals have dimensions approximately equal to 1.892 and 1.953 respectively. We visualize approximations of the corresponding fractals using sequences of length 6 and 5 respectively. We note that a similar fractal structure has been established recently for the block-complexity function for words from a 2-letter alphabet, using a diﬀerent terminology. In this case, the resulting fractal has dimension approximately equal to 1.584. 18/04/2008 PDF Infinite Dimensional Lie Algebras in 4D Conformal Field Theory P/08/23 09/04/2008 PDF On a class of holonomic D-modules on symmetric matrices attached to the general linear group M/08/22 Voir plus > Voir moins > We give a classification of regular holonomic D-modules on complex symmetric matrices whose characteristic variety is the union of conormal bundles to the orbits of the general linear group 02/04/2008 PDF Accurate Effective-One-Body waveforms of inspiralling and coalescing black-hole binaries P/08/21 01/04/2008 PDF SQCD: A Geometric Apercu P/08/04 Voir plus > Voir moins > We take new algebraic and geometric perspectives on the old subject of SQCD. We count chiral gauge invariant operators using generating functions, or Hilbert series, derived from the plethystic programme and the Molien-Weyl formula. Using the character expansion technique, we also see how the global symmetries are encoded in the generating functions. Equipped with these methods and techniques of algorithmic algebraic geometry, we obtain the character expansions for theories with arbitrary numbers of colours and flavours. Moreover, computational algebraic geometry allows us to systematically study the classical vacuum moduli space of SQCD and investigate such structures as its irreducible components, degree and syzygies. We find the vacuum manifolds of SQCD to be affine Calabi-Yau cones over weighted projective varieties. 01/04/2008 PDF Groupoides de Lie et leurs alg\'ebroides M/08/20 Voir plus > Voir moins > Résumé : Ce texte d’un exposé prochain au Séminaire Bourbaki est une revue des notions de géométrie différentielle liées aux variétés symplectiques et de Poisson, aux groupoides de Lie et aux algébroides de Lie. Ces notions ont des liens multiples,et la correspondance groupoide de Lie$rightleftarrows$algébroide de Lie généralise la correspondance bien connue groupe de Lie$rightleftarrows$algèbre de Lie. Les groupoides de Lie jouent un rôle analogue aux champs algébriques et permettent, grâce à l’équivalence de Morita entre groupoides de Lie, de définir des variétés-quotients généralisées (par exemple, espace des feuilles d’un feuilletage). L’exposé s’achève par une présentation élémentaire de la théorie de Galois des équations différentielles (Picard-Vessiot-Ritt-Kolchin). Abstract: The purpose of this coming lecture at the Bourbaki Seminar is a review of a set of important notions in differential geometry: symplectic and Poisson varieties, Lie groupoids and Lie algebroids. All these notions are strongly connected and the classical correspondence Lie groups$rightleftarrows$Lie algebras extends to a correspondence Lie groupoids$rightleftarrows$Lie algebroids. One can considers Lie groupoids up to an equivalencemodelled after the Morita equivalence in algebra. They form the ob jects of a category, closely related to stacks in algebraic geometry, and representing generalized varieties like the space of leaves of a foliation. We end up with a new presentation of the differential Galois theory. 25/03/2008 PDF Instantons beyond topological theory II P/08/15 25/03/2008 PDF On the conjecture of Kevin Walker M/08/19 Voir plus > Voir moins > In 1985 Kevin Walker in his study of topology of polygon spaces raised an interesting conjecture in the spirit of the well-known question "Can you hear the shape of a drum?" of Marc Kac. Roughly, Walker's conjecture asks if one can recover relativelengths of the bars of a linkage from intrinsic algebraic properties of the cohomology algebra of its configuration space. In this paper we prove that the conjecture is true for polygon spaces in the 3-dimensional space. We also prove that for planar polygon spaces the conjecture holds is several modified forms: (a) if one takes into account the action of a natural involution on cohomology, (b) if the cohomology algebra of the involution's orbit space is known, or (c) if the length vector is normal. Some of our results allow the length vector to be non-generic, the corresponding polygon spaces have singularities. Our main tool is the study of the natural involution and its action on cohomology. A crucial role in our proof plays the solution of the isomorphism problem for monoidal rings due to J. Gubeladze. 21/03/2008 PDF Effective one body approach to the dynamics of two spinning black holes with next-to-leading order spin-orbit coupling P/08/14 11/03/2008 PDF String theory, gravity and experiment P/08/08 04/03/2008 PDF Introductory lectures on the Effective One Body formalism P/08/09 04/03/2008 PDF Faithful Effective One Body waveforms of equal-mass coalescing black-hole binaries P/08/10 04/03/2008 PDF Constraints on the variability of quark masses from nuclear binding P/08/11 04/03/2008 PDF Comparing Effective One Body gravitational waveforms to accurate numerical data P/08/12 04/03/2008 PDF Hamiltonian of two spinning compact bodies with next-to-leading order gravitational spin-orbit coupling P/08/13 04/03/2008 PDF On the Brownian gas: a field theory with a Poissonian ground state P/08/05 Voir plus > Voir moins > As a first step towards a successful field theory of Brownian particles in interaction, we study exactly the non-interacting case, its combinatorics and its nonlinear time-reversal symmetry. Even though the particles do not interact, the field theory contains an interaction term: the vertex is the hallmark of the original particle nature of the gas and it enforces the constraint of a strictly positive density field, as opposed to a Gaussian free field. We compute exactly all the n-point density correlation functions, determine non-perturbatively the Poissonian nature of the ground state and emphasize the futility of any coarse-graining assumption for the derivation of the field theory. We finally verify explicitly, on the n-point functions, the fluctuation-dissipation theorem implied by the time-reversal symmetry of the action. 27/02/2008 PDF NOT SO NON-RENORMALIZABLE GRAVITY P/08/06 Voir plus > Voir moins > We review recent ideas [1] how gravity might turn out to be a renormalizable theory after all. 27/02/2008 PDF Quantum Groups and Braid Group Statistics in Conformal Field Theory P/08/07 Voir plus > Voir moins > A quantum universal enveloping algebra$U_q$and the braid group on$n$strands${\mathcal B}_n$mutually commute when acting on the$n$-fold tensor product of a$U_q$-module. Their combined action is applied to low dimensional systems -- the only ones that admit a nontrivial monodromy and hence a braid group (rather than a permutation group) statistics. The lectures introduce the notions of braid group and Hopf algebra and apply them to examples of 2-dimensional (rational) conformal field theory. The case of the$su(2)$current algebra model, for which the deformation parameter$q$is an even root of unity, is considered in some detail. In particular, the solution to the Schwarz problem for the$su(2)$Knizhnik-Zamolodchikov equation is reviewed. 27/02/2008 PDF Classical and quantum integrability M/08/03 15/02/2008 PDF Twistor Strings, Gauge Theory and Gravity P/08/02 06/02/2008 PDF Cofiniteness conditions, projective covers and the logarithmic tensor product theory M/08/01 Voir plus > Voir moins > We construct projective covers of irreducible V-modules in the category of grading-restricted generalized V-modules when V is a vertex operator algebra satisfying the following conditions: 1. V is C_{1}-cofinite in the sense of Li. 2. There existsa positive integer N such that the differences between the real parts of the lowest conformal weights of irreducible V-modules are bounded by N and such that the associative algebra A_{N}(V) is finite dimensional. This result shows that the category of grading-restricted generalized$V$-modules is a finite abelian category over C. Using the existence of projective covers, we prove that if such a vertex operator algebra V satisfies in addition Condition 3, that irreducible V-modules are$\R$-graded and C_{1}-cofinite in the sense of the author, then the category of grading-restricted generalized V-modules is closed under the P(z)-tensor product operation for$z\in C^{\times}. We also prove that other conditions for applying the logarithmic tensor product theory developed by Lepowsky, Zhang and the author hold. Consequently, for such V, this category has a natural structure of braided tensor category. In particular, when V is of positive energy and C_{2}-cofinite, Conditions 1--3 are satisfied and thus all the conclusions hold.

09/01/2008 PDF

On the Projective Hull of Certain Curves in $C^2$ M/07/39

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The projective hull X^ of a compact set X in projective n-space P^n is an analogue of the classical polynomial hull of a set in C^n. In the special case that X lies in an affine chart C^n in P^n, the part of X^ lying in C^n can be defined as the set of points x in C^n for which there exists a constant M(x) so that |p(x)| < M(x)^d sup_X|p| for all polynomials p of degree less than or equal to d, and any d > 0. Let X^(M) denote the set of points x where M(x) < M. Using an argument of E. Bishop, we show that if g is a compact real analytic curve in C^2 (not necessarily connected), then for any linear projection p:C^2 --> C^1, that part of g^(M) which lies above z is finite for almost all z in C^1. It is then shown that for anycompact stable real-analytic curve g in P^n, the set g^ - g is a 1-dimensional complex analytic subvariety of P^n - g.

14/12/2007 PDF

Operator-Valued Involutive Distributions of Evolutionary Vector Fields and their Affine Geometry M/07/38

04/12/2007 PDF

Describing general cosmological singularities in Iwasawa variables P/07/36

31/10/2007 PDF

When steric hindrance facilitates processivity: polymerase activity within chromatin P/07/37

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During eukaryotic transcription, polymerase activity generates torsional stress in DNA, having a negative impact in polymerase processivity. Using our previous studies of the chromatin fiber structure and conformational transitions, we suggest that this torsional stress can be alleviated thanks to a balance between fiber twist and a nucleosome conformational transition into a reversome state. Our model enlightens the origin of polymerase pauses, and leads to the counter-intuitive conclusion that chromatin organized compaction might facilitate polymerase processivity. Indeed, in a compact and well-structured chromatin loop, steric hindrance between nucleosomes enforce sequential transitions, thus ensuring that the polymerase always meets a permissive nucleosomal state.

31/10/2007 PDF

Quelques remarques sur le principe de fonctorialit\'e M/07/31

28/10/2007 PDF

Dirichlet Duality and the Nonlinear Dirichlet Problem M/07/34

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We study the Dirichlet problem for fully nonlinear, degenerate elliptic equations of the form f(Hess u)=0 on a smoothly bounded domain D in R^n. In our approach the equation is replaced by a subset F of the space of symmetric nxn-matrices with bdy(F) contined in the set {f=0}. We establish the existence and uniqueness of continuous solutions under an explicit geometric F-convexity'' assumption on the boundary bdy(F). The topological structure of F-convex domains is also studied and a theorem of Andreotti-Frankel type is proved for them. Two key ingredients in the analysis are the use of subaffine functions and Dirichlet duality, both introduced here. Associated to F is a Dirichlet dual set F* which gives a dual Dirichlet problem.This pairing is a true duality in that the dual of F* is F and in the analysis the roles of F and F* are interchangeable. The duality also clarifies many features of the problem including the appropriate conditions on the boundary. Many interesting examples are covered by these results including: All branches of the homogeneous Monge-Ampere equation over R, C and H; equations appearing naturally in calibrated geometry, Lagrangian geometry and p-convex riemannian geometry, and all branches of theSpecial Lagrangian potential equation.

23/10/2007 PDF

Scalar curvature on lightlike hypersurfaces. M/07/35

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In a recent paper by K. Duggal, the concept of induced scalar curvature of lightlike hypersurfaces is introduced, restricting on a specific class of the latter. This paper removes some of these constraints and construct this scalar quantity by an approach that is consistent with the well-known nondegenerate theory. Basic calculations supported by examples are provided.

23/10/2007 PDF

Deforming, revolving and resolving - New paths in the string theory landscape P/07/33

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In this paper we investigate the properties of series of vacua in the string theory landscape. In particular, we study minima to the flux potential in type IIB compactifications on the mirror quintic. Using geometric transitions, we embed its one dimensional complex structure moduli space in that of another Calabi-Yau with h^{1,1}=86 and h^{2,1}=2. We then show how to construct infinite series of continuously connected minima to the mirror quintic potential by moving into this larger moduli space, applying its monodromies, and moving back. We provide an example of such series, and discuss their implications for the string theory landscape.

04/10/2007 PDF

L'universalisme math\'ematique M/07/29

20/09/2007 PDF

The diffeomorphism group of a $K3$ surface and Nielsen realization M/07/32

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We use moduli spaces of various geometric structures on a manifold M to probe the cohomology of the diffeomorphism group and mapping class group of M. The general principle is that existence of a moduli problem for which the Teichmuller space resembles apoint implies that the homomorphism from the diffeomorphism group (or the mapping class group) to an appropriate discrete group resembles a retraction after applying the classifying space functor. Our main application of this idea is for K^4 a K3 surface;here the maps BDiff(K) --> BAut(H^2(K;Z)) and B\pi_0 Diff(K) --> BAut(H^2(K;Z)) are injective on real cohomology in degrees * < 10. The work of Borel and Matsushima determines the real cohomology of BAut(H^2(K;Z)) in these degrees. Using the above injections, the Borel classes provide cohomological obstructions to a generalized Nielsen realization problem which asks when subgroups of the mapping class group can be lifted to the diffeomorphism group. We conclude that the homomorphism Diff(M) --> \pi_0 Diff(M) does not admit a section if M contains a K3 surface as a connected summand.

20/09/2007 PDF

Constraints and the $E_{10}$ Coset Model P/07/30

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We continue the study of the one-dimensional $E_{10}$ coset model (massless spinning particlemotion on $E_{10}/K(E_{10}))$ whose dynamics at low levels is known to coincide with the equations of motion of maximal supergravity theories in appropriate truncations. We show that the coset dynamics (truncated at levels $\ell \leq 3$) can be consistently restricted by requiring the vanishing of a set of constraints which are in one-to-one correspondence with the canonical constraints of supergravity. Hence,the resulting constrained $\sigma$-model dynamics captures the full (constrained) supergravity dynamics in this truncation. Remarkably, the bosonic constraints are found to be expressible in a Sugawara-like (current $\times$ current) form in terms of the conserved $E_{10}$ Noether current, and transform covariantly under an upper parabolic subgroup $E_{10}^+ \subset E_{10}$. We discuss the possible implications of this result, and in particular exhibit a tantalising link with the usual affine Sugawara construction in the truncation of $E_{10}$ to its affine subgroup $E_9$.

18/09/2007 PDF

Restricted set addition: The exceptional case of the Erdos-Heilbronn conjecture M/07/28

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Let A,B be different nonempty subsets of the group of integers modulo a prime p. If p is not smaller than |A|+|B|-2, then at least this many residue classes can be represented as a+b, where a and b are different elements of A and B, respectively. This result complements the solution of a problem of Erdos and Heilbronn obtained by Alon, Nathanson, and Ruzsa.

08/08/2007 PDF

Tubular Neighborhoods of Nodal Sets and Diophantine Approximation M/07/27

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We give upper and lower bounds on the volume of a tubular neighborhood of the nodal set of an eigenfunction of the Laplacian on a real analytic closed Riemannian manifold M. As an application we consider the question of approximating points on M by nodal sets, and explore analogy with approximation by rational numbers.

31/07/2007 PDF

Hochschild cohomology, the characteristic morphism and derived deformations M/07/26

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A notion of Hochschild cohomology of an abelian category was defined by Lowen and Van den Bergh (2005) and they showed the existence of a characteristic morphism from the Hochschild cohomology into the graded centre of the (bounded) derived category. An element in the second Hochschild cohomology corresponds to a first order deformation of the abelian category (Lowen and Van den Bergh, 2006). The problem of deforming single objects of the bounded derived category was treated by Lowen (2005). In this paper we show that the image of the Hochschild cohomology element under the characteristic morphism encodes precisely the obstructions to deforming single objects of the bounded derived category to the bounded derived category of the deformed abelian category. Hence this paper provides a missing link between the above works. Finally we discuss some implications of these facts in the direction of a derived deformation theory''.

24/07/2007 PDF

Towards a modulo $p$ Langlands correspondence for GL$_2$ M/07/25

04/07/2007 PDF

Subgroup separability in residually free groups M/07/24

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We prove that the finitely presentable subgroups of residually free groups are separable and that the subgroups of type $FP_\infty$ are virtual retracts. We describe a uniform solution to the membership problem for finitely presentable subgroups of residually free groups.

29/06/2007 PDF

Perverse coherent sheaves and the geometry of special pieces in the unipotent variety M/07/23

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Let X be a scheme of finite type over a Noetherian base scheme S admitting a dualizing complex, and let U be an open subscheme whose complement has codimension at least 2. We extend the theory of perverse coherent sheaves, due to Deligne and disseminated by Bezrukavnikov, by showing that a coherent middle extension functor from perverse sheaves on U to perverse sheaves on X may be defined for a much broader class of perversities that has previously been known. We also introduce a derived category version of the coherent middle extension functor. Under suitable hypotheses, we introduce a construction (called "S_2-extension") in terms of perverse coherent sheaves of algebras on Xthat takes a finite morphism to U and extends it in a canonical way to a finite morphism to X. In particular, this construction gives a canonical "S_2-ification" of appropriate X. The construction also has applications to the "Macaulayfication" problem, and it is particularly well-behaved when X is Gorenstein. Our main goal, however, is to address a conjecture of Lusztig on the geometry of special pieces (certain subvarieties of the unipotent variety of a reductive algebraic group). The conjecture asserts in part that each special piece is the quotient of some variety (previously unknown in the exceptional groups and in positive characteristic) by the action of a certain finite group. We use S_2-extension to give a uniform construction of the desired variety.

28/06/2007 PDF

On the derived category of 1-motives, I M/07/22

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We consider the category of Deligne 1-motives over a perfect field k of exponential characteristic p and its derived category for a suitable exact structure after inverting p. As a first result, we provide a fully faithful embedding into an 'etale version of Voevodsky's triangulated category of geometric motives. Our second main result is that this full embedding almost" has a left adjoint, that we call LAlb. Applied to the motive of a variety we thus get a bounded complex of 1-motives, that we compute fully for smooth varieties and partly for singular varieties. As an application we give motivic proofs of Ro\v\i tman type theorems (in characteristic 0).

14/06/2007 PDF

A remark on quantum gravity P/07/20

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We discuss the structure of Dyson--Schwinger equations in quantum gravity and conclude in particular that all relevant skeletons are of first order in the loop number. There is an accompanying sub Hopf algebra on gravity amplitudes equivalent to identities between n-graviton scattering amplitudes which generalize the Slavnov Taylor identities. These identities map the infinite number of charges and finite numbers of skeletons in gravity to an infinite number of skeletons and a finite number of charges needing renormalization. Our analysis suggests that gravity, regarded as a probability conserving but perturbatively non-renormalizable theory, is renormalizable after all, thanks to the structure of its Dyson--Schwinger equations.

29/05/2007 PDF

Wormholes as Black Hole Foils P/07/19

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We study to what extent wormholes can mimic the observational features of black holes. It is surprisingly found that many features that could be thought of as 'characteristic' of a black hole (endowed with an event horizon) can be closely mimicked by aglobally static wormhole, having no event horizon. This is the case for: the apparently irreversible accretion of matter down a hole, no-hair properties, quasi-normal-mode ringing, and even the dissipative properties of black hole horizons, such as a finite surface resistivity equal to 377 Ohms. The only way to distinguish the two geometries on an observationally reasonable time scale would be through the detection of Hawking's radiation, which is, however, too weak to be of practical relevance for astrophysical black holes. We point out the existence of an interesting spectrum of quantum microstates trapped in the throat of a wormhole which could be relevant for storing the information 'lost' during a gravitational collapse.

04/05/2007 PDF

Cosmological Singularities and a Conjectured Gravity/Coset Correspondence P/07/15

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We review the recently discovered connection between the Belinsky-Khalatnikov-Lifshitz-like "chaotic" structure of generic cosmological singularities in eleven-dimensional supergravity and the "last" hyperbolic Kac-Moody algebra $E_{10}$. This intriguing connection suggests the existence of a hidden "correspondence" between supergravity (or even $M$-theory) and null geodesic motion on the infinite-dimensional coset space $E_{10}/K(E_{10})$. If true, this gravity/coset correspondence would offer a new view of the (quantum) fate of space (and matter) at cosmological singularities.

13/04/2007 PDF

Binary Systems as Test-beds of Gravity Theories P/07/16

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We review the general relativistic theory of the motion, and of the timing, of binary systems containing compact objects (neutron stars or black holes). Then we indicate the various ways one can use binary pulsar data to test the strong-field and/or radiative aspects of General Relativity, and of general classes of alternative theories of relativistic gravity.

13/04/2007 PDF

General Relativity Today P/07/17

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After recalling the conceptual foundations and the basic structure of general relativity, we review some of its main modern developments (apart from cosmology): (i) the post-Newtonian limit and weak-field tests in the solar system, (ii) strong-gravitational fields and black holes, (iii) strong-field and radiative tests in binary pulsar observations, (iv) gravitational waves, (v) general relativity and quantum theory.

13/04/2007 PDF

Chaos and Symmetry in String Cosmology P/07/18

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We review the recently discovered interplay between chaos and symmetry in the general inhomogeneous solution of many string-related Einstein-matter systems in the vicinity of a cosmological singularity. The Belinsky-Khalatnikov-Lifshitz-type chaotic behaviour is found, for many Einstein-matter models (notably those related to the low-energy limit of superstring theory and $M$-theory), to be connected with certain (infinite-dimensional) hyperbolic Kac-Moody algebras. In particular, the billiard chambers describing the asymptotic cosmological behaviour of pure Einstein gravity in spacetime dimension $d+1$, or the metric-three-form system of $11$-dimensional supergravity, are found to be identical to the Weyl chambers of the Lorentzian Kac-Moody algebras $AE_d$, or $E_{10}$, respectively. This suggests that these Kac-Moody algebras are hidden symmetries of the corresponding models. There even exists some evidence of a hidden equivalence between the general solution of the Einstein-three-form system and a null geodesic in the infinite dimensional coset space $E_{10} / K (E_{10})$, where $K (E_{10})$ is the maximal compact subgroup of $E_{10}$.

13/04/2007 PDF

Local Asymmetry and the Inner Radius of Nodal Domains M/07/14

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Let M be a closed Riemannian manifold of dimension n. Let f be an eigenfunction of the Laplace-Beltrami operator corresponding to an eigenvalue lambda. We show that the volume of {f>0} inside any ball B whose center lies on {f=0} is > C|B|/lambda^n. Weapply this result to prove that each nodal domain contains a ball of radius > C/lambda^n.

12/04/2007 PDF

B$_{\rm cris}^{\varphi = 1}$-représentations et $(\varphi , \Gamma)$-modules M/07/13

12/04/2007 PDF

The cyclomatic number of connected graphs without solvable orbits M/07/12

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A graph is without solvable orbits if its group of automorphisms acts on each of its orbits through a non-solvable quotient. We prove that there is a connected graph without solvable orbits of cyclomatic number c if and only if c is equal to 6, 8, 10, 11, 15, 16, 19, 20, 21, 22, or is at least 24, and briefly discuss the geometric consequences.

05/04/2007 PDF

The Geometry of Small Causal Diamonds P/07/11

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The geometry of causal diamonds or Alexandrov open sets whose initial and final events $p$ and $q$ respectively have a proper-time separation $au$ small compared with the curvature scale is a universal. The corrections from flat space are given as a power series in $au$ whose coefficients involve the curvature at the centre of the diamond. We give formulae for the total 4-volume $V$ of the diamond, the area $A$ of the intersection the future light cone of $p$ with the past light coneof$q$ and the 3-volume of the hyper-surface of largest 3-volume bounded by this intersection valid to ${cal O } ( au ^4)$. The formula for the 4-volume agrees with a previous result of Myrheim. Remarkably, the iso-perimetric ratio ${3V_3 øver 4 pi }/({ A øver 4 pi } ) ^{3 øver 2}$ depends only on the energy density at the centre and is bigger than unity if the energy density is positive. These results are also shown to hold in all spacetime dimensions. Formulae are also given, valid to nextnon-trvial order, for causal domains in two spacetime dimensions. We suggest a number of applications, for instance, the directional dependence of the volume allows one to regard the volumes of causal diamonds as an observable providing a measurement of the Ricci tensor.

14/03/2007 PDF

An $A_{\infty}$-structure for lines in a plane M/07/10

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As an explicit example of an $A_{\infty}$-structure associated to geometry, we construct an $A_{\infty}$-structure for a Fukaya category of finitely many lines (Lagrangians) in $R^2$, ie., we define also non-transversal $A_{\infty}$-products. This construction is motivated by homological mirror symmetry of (two-)tori, where $R^2$ is the covering space of a two-torus. The strategy is based on an algebraic reformulation of Morse homotopy theory through homological perturbation theory (HPT) as discussed by Kontsevich and Soibelman in cite{KoSo}, where we introduce a special DG category which is a key idea of our construction.

06/03/2007 PDF

A vanishing theorem in positive characteristic and tilting equivalences M/07/09

02/03/2007 PDF

Notes on instantons in topological field theory and beyond P/07/08

21/02/2007 PDF

Renormalisation of non-commutative field theories P/07/07

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The first renormalisable quantum field theories on non-commutative space have been found recently. We review this rapidly growing subject.

13/02/2007 PDF

Familles de représentations de De Rham et monodromie p-adique M/07/06

08/02/2007 PDF

One-loop Beta Functions for the Orientable Non-commutative Gross-Neveu Model P/07/05

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We compute at the one-loop order the beta-functions for a renormalisable non-commutative analog of the Gross-Neveu model defined on the Moyal plane. The calculation is performed within the so called x-space formalism. We find that this non-commutative field theory exhibits asymptotic freedom for any number of colors. The beta-function for the non-commutative counterpart of the Thirring model is found to be non vanishing.

02/02/2007 PDF

A counterexample to Premet's and Joseph's conjectures M/07/01

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Let $g$ be a finite-dimensional reductive Lie algebra of rank $l$ over an algebraically closed field of characteristic zero. Given an element $x$ of $g$, we denote by $g_x$ the centraliser of $x$ in $g$. It was conjectured by Premet, that the algebra $S(g_x)^{g_x}$ of the symmetrci $mathfrak g_x$-invariants is a graded polynomial algebra in $l$ variables. In this note, we show that this conjecture does not hold for the minimal nilpotent orbit in the simple Lie algebra of type $E_8$. As a consequence, a conjecture of Joseph on the semi-invariants of (bi)parabolics is not true either.

26/01/2007 PDF

Lifetime of a massive particle in a de Sitter universe P/07/02

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We study particle decay in de Sitter space-time as given by first order perturbation theory in an interacting quantum field theory. We show that for fields with masses above a critical mass $m_c$ there is no such thing as particle stability, so thatdecays forbidden in flat space-time do occur there. The lifetime of such a particle also turns out to be independent of its velocity when that lifetime is comparable with de Sitter radius. Particles with lower mass are even stranger: The masses of their decay products must obey quantification rules, and their lifetime is zero.

26/01/2007 PDF

Semi-classical open string corrections and symmetric Wilson loops P/07/03

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In the AdS/CFT correspondence, an AdS_2 x S^2 D3-brane with electric flux in AdS_5 x S^5 spacetime corresponds to a circular Wilson loop in the symmetric representation or a multiply wound one in N=4 super Yang-Mills theory. In order to distinguish the symmetric loop and the multiply wound loop, one should see an exponentially small correction in large 't Hooft coupling. We study semi-classically the disk open string attached to the D3-brane. We obtain the exponent of the term and it agrees with the result of the matrix model calculation of the symmetric Wilson loop.

26/01/2007 PDF

Geometry of differential equations M/07/04

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This is a review of classical and modern methods of geometric-algebraic approach to (overdeteremined) systems of partial differential equations.

26/01/2007 PDF

Recursion and growth estimates in renormalizable quantum field theory P/06/62

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In this paper we show that there is a Lipatov bound for the radius of convergence for superficially divergent one-particle irreducible Green functions in a renormalizable quantum field theory if there is such a bound for the superficially convergent ones. The radius of convergence turns out to be ${ m min}{ ho,1/b_1}$, where $ho$ is the bound on the convergent ones, the instanton radius, and $b_1$ the first coefficient of the $eta$-function.

19/12/2006 PDF

Comparaison des théories de Deitmar et de Zhu M/06/63

19/12/2006 PDF

The next-to-ladder approximation for Dyson-Schwinger equations P/06/64

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We solve the linear Dyson Schwinger equation for a massless vertex in Yukawa theory, iterating the first two primitive graphs.

19/12/2006 PDF

Extended Seiberg-Witten Theory and Integrable Hierarchy P/06/43

08/12/2006 PDF

Sur un corps de caractéristique 1' (d'après Zhu) M/06/61

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Nous exposons la théorie de Zhu concernant un analogue formel du corps ${mathbf F}_{p}$, pour $p=1$'.

06/12/2006 PDF

Homotopy graph-complex for configuration and knot spaces M/06/58

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In the paper we prove that the primitive part of the Sinha homology spectral sequence E2-term for the space of long knots is rationally isomorphic to the homotopy E2-term. We also define natural graph-complexes computing the rational homotopy of configuration and of knot spaces.

05/12/2006 PDF

Modified Hochschild and Periodic Cyclic Homology M/06/59

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The Hochschild and (periodic) cyclic homology of the algebra of continuous functions on a smooth manifold are trivial. In this paper we create an analogue of the Hochschild and periodic cyclic homology which gives the right result when applied onto the algebra of continuous functions on smooth manifolds ($Z_{2}$-graded de Rham co-homology of the manifold). This is realized by replacing the Connes periodic bi-complex (b, B) by the bi-complex $( ilde{b}, d),$ where the operator $ilde{b}$ is obtained by blending the Hochschild boundary $b$ with the Alexander-Spanier boundary $d$; the operator $ilde{b}$ anti-commutes with the operator $d$. In order to reach this objective, as in the classical case, one has to consider the Alexander-Spanier complex of germs. As the notion of germ is a locality notion, our procedure will apply to topological algebras. The problem of producing a tool able to extract the correct homology from the algebra of continuous functions was addressed before byM. Puschnigg.

05/12/2006 PDF

A generalization of residual finiteness M/06/60

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The concept of residual finiteness with respect to automorphic equivalence, a property generalizing residual finiteness and conjugacy separability is introduced. A sufficient condition for a group G to be residually finite with respect to automorphic equivalence is proven (Theorem). It is then used to give some examples of automorphic equivalent residually finite groups.

05/12/2006 PDF

Noncommutative Geometry Approach to Principal and Associated Bundles M/06/65

01/12/2006 PDF

The map from the cyclohedron to the associahedron is left cofinal M/06/66

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Two natural projections from the cyclohedron to the associahedron are defined. We show that the preimages of any point via these projections might not be homeomorphic to (a cell decomposition of) a disc, but are still contractible. We briefly explain anapplication of this result to the study of knot spaces from the point of view of the Goodwillie-Weiss embedding calculus.

01/12/2006 PDF

Injectivity Radius of Lorentzian Manifolds M/06/67

01/12/2006 PDF

Reduction Theorems for characteristic functors on finite $p$-groups and applications to $p$-nilpotence criteria M/06/55

22/11/2006 PDF

Strongly homotopy Lie bialgebras and Lie quasi-bialgebras M/06/57

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Structures of Lie algebras, Lie coalgebras, Lie bialgebras and Lie quasibialgebras are presented as solutions of Maurer-Cartan equations on corresponding governing differential graded Lie algebras. Cohomology theories of all these structures are described in a concise way using the big bracket construction of Kosmann-Schwarzbach. This approach provides a definition of an $L_infty$-(quasi)bialgebra (strong homotopy Lie (quasi)bialgebra). We recover an $L_infty$-algebra structure as a particularcase of our construction. The formal geometry interpretation leads to a definition of an $L_infty$ (quasi)bialgebra structure on $V$ as a differential operator $Q$ on $V,$ self-commuting with respect to the Poisson bracket. Finally, we establish an $L_infty$-version of a Manin (quasi) triple and get a correspondence theorem with $L_infty$-(quasi) bialgebras.

21/11/2006 PDF

Cardy condition for Open-closed field algebras M/06/56

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Let $V$ be a vertex operator algebra satisfying certain reductivity and finiteness conditions such that $mathcal{C}_V$, the category of $V$-modules, is a modular tensor category. We study open-closed field algebras over $V$ equipped with nondegenerateinvariant bilinear forms for both open and closed sectors. We show that they give algebras over certain $C$-extension of the Swiss-cheese partial dioperad, and we obtain Ishibashi states easily in such algebras. We formulate Cardy condition algebraically in terms of the action of the modular transformation $S: au mapsto -frac{1}{ au}$ on the space of intertwining operators. We then derive a graphical representation of $S$ in the modular tensor category $mathcal{C}_V$. This result enablesus to give a categorical formulation of Cardy condition and modular invariant conformal full field algebra over $Vøtimes V$. Then we incorporate the modular invariance condition for genus-one closed theory, Cardy condition and the axioms for open-closed field algebra over $V$ equipped with nondegenerate invariant bilinear forms into a tensor-categorical notion called Cardy $mathcal{C}_V|mathcal{C}_{Vøtimes V}$-algebra. We also give a categorical construction of Cardy $mathcal{C}_V|mathcal{C}_{Vøtimes V}$-algebra in Cardy case.

15/11/2006 PDF

Field rotation parameters and limit cycle bifurcations M/06/54

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In this paper, the global qualitative analysis of planar polynomial dynamical systems is established and a new geometric approach to solving Hilbert's Sixteenth Problem on the maximum number and relative position of their limit cycles in two special cases of such systems is suggested. First, using geometric properties of four field rotation parameters of a new canonical system which is constructed in this paper, we present a proof of our earlier conjecture that the maximum number of limit cycles ina quadratic system is equal to four and the only possible their distribution is (3:1). Then, by means of the same geometric approach, we solve the Problem for Li'{e}nard's polynomial system (in this special case, it is considered as Smale's Thirteenth Problem). Besides, generalizing the obtained results, we present a solution of Hilbert's Sixteenth Problem on the maximum number of limit cycles surrounding a singular point for an arbitrary polynomial system and, applying the Wintner-Perko terminationprinciple for multiple limit cycles, we develop an alternative approach to solving the Problem. By means of this approach, for example, we give another proof of the main theorem for a quadratic system and complete the global qualitative analysis ofa generalized Li'{e}nard's cubic system with three finite singularities. We discuss also some different approaches to the Problem.

08/11/2006 PDF

Correlator of Fundamental and Anti-symmetric Wilson loops in AdS/CFT Correspondence P/06/53

27/10/2006 PDF

Instantons beyond topological theory I P/06/42

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Many quantum field theories in one, two and four dimensions possess remarkable limits in which the instantons are present, the anti-instantons are absent, and the perturbative corrections are reduced to one-loop. We analyze the corresponding models as full quantum field theories, beyond their topological sector. We show that the correlation functions of all, not only topological (or BPS), observables may be studied explicitly in these models, and the spectrum may be computed exactly. An interesting feature is that the Hamiltonian is not always diagonalizable, but may have Jordan blocks, which leads to the appearance of logarithms in the correlation functions. We also find that in the models defined on Kahler manifolds the space of states exhibits holomorphic factorization. We conclude that in dimensions two and four our theories are logarithmic conformal field theories.

17/10/2006 PDF

Open-closed field algebras, operads and tensor categories M/06/51

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We introduce the notions of open-closed field algebra and open-closed field algebra over a vertex operator algebra $V$. In the case that $V$ satisfies certain finiteness and reductivity conditions, we show that an open-closed field algebra over $V$ canonically gives an algebra over a $C$-extension of the Swiss-cheese partial operad. We also give a tensor categorical formulation and categorical constructions of open-closed field algebras over $V$.

10/10/2006 PDF

Classification of low energy sign-changing solutions of an almost critical problem M/06/52

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In this paper we make the analysis of the blow-up of low energy sign-changing solutions of a semi-linear elliptic problem involving nearly critical exponent. Our results allow to classify these solutions according to the concentration speeds of the positive and negative part and, in high dimensions, lead to complete classification of them. Additional qualitative results, such as symmetry or location of the concentration points are obtained when the domain is a ball.

10/10/2006 PDF

Remarks on compact shrinking Ricci solitons of dimension four M/06/50

03/10/2006 PDF

A primer of Hopf algebras M/06/40

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In this paper, we review a number of basic results about so-called Hopf algebras. We begin by giving a historical account of the results obtained in the 1930's and 1940's about the topology of Lie groups and compact symmetric spaces. The climax is provided by the structure theorems due to Hopf, Samelson, Leray and Borel. The main part of this paper is a thorough analysis of the relations between Hopf algebras and Lie groups (or algebraic groups). We emphasize especially the category of unipotent (and prounipotent) algebraic groups, in connection with Milnor-Moore's theorem. These methods are a powerful tool to show that some algebras are free polynomial rings. The last part is an introduction to the combinatorial aspects of polylogarithm functions andthecorresponding multiple zeta values.

26/09/2006 PDF

Rigidity theorem for expanding Gradient Ricci Solitons M/06/49

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In this paper, we study the rigidity problem for expanding gradient Ricci soliton equation on a complete conformally compact Riemannian manifold. We show that under a natural condition on the Ricci curvature and the scalar curvature, the expanding Ricci soliton is Poincare-Einstein.

21/09/2006 PDF

Linear dependence in Mordell-Weil groups M/06/48

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We consider a local to global principle for detecting linear dependence of nontorsion points, by reduction maps, in the Mordell-Weil group of an abelian variety defined over a number field.

19/09/2006 PDF

A Lie theoretic approach to renormalization P/06/46

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Motivated by recent work of Connes and Marcolli, based on the Connes-Kreimer approach to renormalization, we augment the latter by a combinatorial, Lie algebraic point of view. Our results rely both on the properties of the Dynkin idempotent, one of the fundamental Lie idempotents in the theory of free Lie algebras, and on the fine properties of Hopf algebras and their associated descent algebras. Besides leading very directly to proofs of the main combinatorial properties of the renormalization procedures, the new techniques do not depend on the geometry underlying the particular case of dimensional regularization and the Riemann-Hilbert correspondence. This is illustrated with a discussion of the BPHZ renormalization scheme.

13/09/2006 PDF

Le Calcul des Probabilités de Poincaré M/06/47

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Résumé: Le présent texte est extrait d'un ouvrage à paraître sur L'héritage mathématique de Poincaré' (éditions Belin). Il s'agit d'une analyse du livre de Poincaré sur les probabilités, qui reprend l'un de ses cours, et son texte fameux sur le hasard. Le plus intéressant est la manière dont Poincaré traite des problèmes de théorie cinétique des gaz et des cheminements aléatoires. Abstract: We analyse the classical work of Poincaré about probability theory. This report is part of a forthcoming book on the Mathematical Heritage of Poincaré'. We focus on the relevance of the book for the development of statistical physics and random walks.

13/09/2006 PDF

Multiloop Superstring Amplitudes from Non-Minimal Pure Spinor Formalism P/06/41

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Using the non-minimal version of the pure spinor formalism, manifestly super-Poincare covariant superstring scattering amplitudes can be computed as in topological string theory without the need of picture-changing operators. The only subtlety comes fromregularizing the functional integral over the pure spinor ghosts. In this paper, it is shown how to regularize this functional integral in a BRST-invariant manner, allowing the computation of arbitrary multiloop amplitudes. The regularization method simplifies for scattering amplitudes which contribute to ten-dimensional F-terms, i.e. terms in the ten-dimensional superspace action which do not involve integration over the maximum number of $heta$'s.

06/09/2006 PDF

Dessins d'enfants: Solving equations determining Belyi pairs M/06/44

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This paper deals with the Grothendieck dessins d'enfants, that is tamely embedded graphs on surfaces. We investigate combinatorics of systems of equations determining corresponding Belyi pair, that is a rational function with at most 3 critical values onan algebraic curve, such that its preimage is the dessin under consideration. Several properties of extra, or so-called parasitic, solutions of such systems are described.

05/09/2006 PDF

Dyson Schwinger Equations: From Hopf algebras to Number Theory P/06/45

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We consider the structure of renormalizable quantum field theories from the viewpoint of their underlying Hopf algebra structure. We review how to use this Hopf algebra and the ensuing Hochschild cohomology to derive non-perturbative results for the short-distance singular sector of a renormalizable quantum field theory. We focus on the short-distance behaviour and thus discuss renormalized Green functions $G_R(alpha,L)$ which depend on a single scale $L=ln q^2/mu^2$.

05/09/2006 PDF

Séries hypergéométriques multiples et polyzêtas M/06/36

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Nous décrivons un algorithme théorique et effectif permettant de démontrer que des séries et intégrales hypergéométriques multiples relativement générales se décomposent en combinaisons linéaires à coefficients rationnels de polyzêtas.

29/06/2006 PDF

Phénomènes de symétries dans des formes linéaires en polyzêtas M/06/37

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On donne deux généralisations, en profondeur quelconque, du phénomène de symétrie utilisé par Ball-Rivoal pour démontrer qu'une infinité de valeurs de la fonction zeta de Riemann aux entiers impairs sont irrationnelles. Ces généralisations concernent des séries multiples de type hypergéométrique qui s'écrivent comme formes linéaires en certains polyzêtas. La preuve utilise notamment la régularisation des polyzêtas à divergence logarithmique.

29/06/2006 PDF

Tensor gauge fields in arbitrary representations of $GL(D,{\bf R})$: II. Quadratic actions P/06/33

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Quadratic, second-order, non-local actions for tensor gauge fields transforming in arbitrary irreducible representations of the general linear group Quadratic, second-order, non-local actions for tensor gauge fields transforming in arbitrary irreduciblein $D$-dimensional Minkowski space are explicitly written in a compact form by making use of Levi--Civita tensors. The field equations derivedfrom these actions ensure the propagation of the correct massless physical degrees of freedom and are shown to be equivalent to non-Lagrangian local field equations proposed previously.Moreover, these actions allow a frame-like reformulation \a la MacDowell--Mansouri, without any trace constraint in the tangent indices.

29/06/2006 PDF

On matrix differential equations in the Hopf algebra of renormalization P/06/38

29/06/2006 PDF

Bounds for the dimensions of $p$-adic multiple $L$-value spaces M/06/39

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First, we will define $p$-adic multiple $L$-values ($p$-adic MLV's), which are generalizations of Furusho's $p$-adic multiple zeta values ($p$-adic MZV's) in Section 2. Next, we prove bounds for the dimensions of $p$-adic MLV-spaces in Section 3, assuming results in Section 4. The bounds come from the rank of $K$-groups of ring of $S$-integers of cyclotomic fields, and these are $p$-adic analogues of Deligne-Goncharov's bounds for the dimensions of (complex) MLV-spaces. In the case of MLV-spaces, the gap between the dimensions and the bounds is related to spaces of modular forms similarly as the complex case. In Section 4, we define the crystalline realization of mixed Tate motives and show a comparison isomorphism, by uing $p$-adic Hodge theory.

29/06/2006 PDF

Quantum effects in gravitational wave signals from cuspy superstrings P/06/35

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We study the gravitational emission, in Superstring Theory, from fundamental strings exhibiting cusps. The classical computation of the gravitational radiation signal from cuspy strings features strong bursts in the special null directions associated to the cusps. We perform a quantum computation of the gravitational radiation signal from a cuspy string, as measured in a gravitational wave detector using matched filtering and located in the special null direction associated to the cusp. We study the quantum statistics (expectation value and variance) of the measured filtered signal and find that it is very sharply peaked around the classical prediction. Ultimately, this result follows from the fact that the detector is a low-pass filter which is blind tothe violent high-frequency quantum fluctuations of both the string worldsheet, and the incoming gravitational field.

23/06/2006 PDF

Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I M/06/34

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We develop geometric approach to A-infinity algebras and A-infinity categories based on the notion of formal scheme in the category of graded vector spaces. Geometric approach clarifies several questions, e.g. the notion of homological unit or A-infinity structure on A-infinity functors. We discuss Hochschild complexes of A-infinity algebras from geometric point of view. The paper contains homological versions of the notions of properness and smoothness of projective varieties as well as the non-commutative version of Hodge-to-de Rham degeneration conjecture. We also discuss a generalization of Deligne's conjecture which includes both Hochschild chains and cochains. We conclude the paper with the description of an action of the PROP of singular chains of the topological PROP of 2-dimensional surfaces on the Hochschild chain complex of an A-infinity algebra with the scalar product (this action is more or less equivalent to the structure of 2-dimensional Topological Field Theory associated with an abstract'' Calabi-Yau manifold).

21/06/2006 PDF

$K(E_{10})$, Supergravity and Fermions P/06/28

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We study the fermionic extension of the E10/K(E10) coset model and its relation to eleven-dimensional supergravity. Finite-dimensional spinor representations of the compact subgroup K(E10) of E(10,R) are studied and the supergravity equations are rewritten using the resulting algebraic variables. The canonical bosonic and fermionic constraints are also analysed in this way, and the compatibility of supersymmetry with local K(E10) is investigated. We find that all structures involving A9 levels 0,1 and 2 nicely agree with expectations, and provide many non-trivial consistency checks of the existence of a supersymmetric extension of the E10/K(E10) coset model, as well as a new derivation of the bosonic dictionary' between supergravity and coset variables.However, there are also definite discrepancies in some terms involving level 3, which suggest the need for an extension of the model to infinite-dimensional faithful representations of the fermionic degrees of freedom.

15/06/2006 PDF

Coherent algebras and noncommutative projective lines M/06/32

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A well-known conjecture says that every one-relator discrete group is coherent. We state and partly prove an analogous statement for graded associative algebras. In particular, we show every that Gorenstein algebra $A$ of global dimension 2 is graded coherent. This allows us to define a noncommutative analogue of the projective line $PP1$ as a noncommutative scheme based on the coherent noncommutative spectrum $cohp A$ of such an algebra $A$, that is, the category of coherent $A$-modulesmodulo the torsion ones. This category is always abelian Ext-finite hereditary with Serre duality, like the category of coherent sheaves on $PP1$. In this way, we obtain a sequence $PP1_n$ ($nge 2$) of pairwise non-isomorphic noncommutative schemes which generalize the scheme $PP1 = PP1_2$.

13/06/2006 PDF

About the Trimmed and the Poincaré-Dulac normal form of diffeomorphisms M/06/29

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We study two particular continuous prenormal forms as defined by Jean Ecalle and Bruno Vallet for local analytic diffeomorphism: the Trimmed form and the Poincare-Dulac normal form. We first give a self-contain introduction to the mould formalism of Jean Ecalle. We provide a dictionary between moulds and the classical Lie algebraic formalism using non-commutative formal power series. We then give full proofs and details for results announced by J. Ecalle and B. Vallet about the Trimmed form of diffeomorphisms. We then discuss a mould approach to the classical Poincare-Dulac normal form of diffeomorphisms. We discuss the universal character of moulds taking place in normalization problems.

07/06/2006 PDF

Fractional embedding of differential operators and Lagrangian systems M/06/30

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This paper is a contribution to the general program of embedding theories of dynamical systems. Following our previous work on the Stochastic embedding theory developed with S. Darses, we define the fractional embedding of differential operators and ordinary differential equations. We construct an operator combining in a symmetric way the left and right (Riemann-Liouville) fractional derivatives. For Lagrangian systems, our method provide a fractional Euler-Lagrange equation. We prove, developing the corresponding fractional calculus of variations, that such equation can be derived via a fractional least-action principle. We then obtain naturally a fractional Noether theorem and a fractional Hamiltonian formulation of fractional Lagrangian systems. All these constructions are coherents, i.e. that the embedding procedure is compatible with the fractional calculus of variations. We then extend our results to cover the Ostrogradski formalism. Using the fractional embedding and following a previous work of F. Riewe, we obtain a fractional Ostrogradski formalism which allows us to derive non-conservative dynamical systems via a fractional generalized least-action principle. We also discuss the Whittaker equation and obtain a fractional Lagrangian formulation. Last, we discuss the fractional embedding of continuous Lagrangian systems. In particular, we obtain a fractional Lagrangian formulation of the classical fractional wave equation introduced by Schneider and Wyss as well as the fractional diffusion equation.

07/06/2006 PDF

Mode conversion in the cochlea? linear analysis P/06/31

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It is suggested that the frequency selectivity of the ear may be based on the phenomenon of mode conversion rather than critical layer resonance. The distinction is explained and supporting evidence discussed.

07/06/2006 PDF

Stochastic Embedding of Dynamical Systems M/06/27

19/05/2006 PDF

Théorème de Noether stochastique M/06/25

17/05/2006 PDF

Non-differentiable deformations of $R^n$ M/06/26

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Many problems of physics or biology involve very irregular objects like the rugged surface of a malignant cell nucleus or the structure of space-time at the atomic scale. We define and study non-differentiable deformations of the classical Cartesian space $R^n$ which can be viewed as the basic bricks to construct irregular objets. They are obtain by taking the topological product of $n$-graphs of nowhere differentiable real valued functions. Our point of view is to replace the study of a non-differentiable function by the dynamical study of a one-parameter family of smooth regularization of this function. In particular, this allows us to construct a one parameter family of smooth coordinates systems on non-differentiable deformations of $R^n$ whichdepend on the smoothing parameter via an explicit differential equation called a scale-law. Deformations of $R^n$ are examples of a new class of geometrical objects called scale manifolds which are defined in this paper. As an application, we derive rigorously the main results of the scale relativity theory developed by L. Nottale in the framework of a Scale space-time manifold.

17/05/2006 PDF

Etude for linear Dyson-Schwinger Equations P/06/23

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We discuss properties of linear Dyson-Schwinger equations.

10/05/2006 PDF

An Etude in non-linear Dyson-Schwinger equations P/06/24

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We show how to use the Hopf algebra structure of quantum field theory to derive nonperturbative results for the short-distance singular sector of a renormalizable quantum field theory in a simple but generic example. We discuss renormalized Green functions $G_R(alpha,L)$ in such circumstances which depend on a single scale $L=ln q^2/mu^2$ and start from an expansion in the scale $G_R(alpha,L)=1+sum_k gamma_k(alpha)L^k$. We derive recursion relations between the $gamma_k$ which make full use of the renormalization group. We then show how to determine the Green function by the use of a Mellin transform on suitable integral kernels. We exhibit our approach in an example for which we find a functional equation relating weak and strong coupling expansions.

10/05/2006 PDF

Constructing conformal field theory models P/06/05

09/05/2006 PDF

Calcul Moulien M/06/22

09/05/2006 PDF

Curvature corrections and Kac-Moody compatibility conditions P/06/21

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We study possible restrictions on the structure of curvature corrections to gravitational theories in the context of their corresponding Kac--Moody algebras, following the initial work on E10 in Class. Quant. Grav. 22 (2005) 2849. We first emphasize thatthe leading quantum corrections of M-theory can be naturally interpreted in terms of (non-gravity) fundamental weights of E10. We then heuristically explore the extent to which this remark can be generalized to all over-extended algebras by determining which curvature corrections are compatible with their weight structure, and by comparing these curvature terms with known results on the quantum corrections for the corresponding gravitational theories.

26/04/2006 PDF

Evolution in random environment and structural instability M/06/20

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We consider stability and evolution of complex biological systems in particular, genetic networks. We focus our attention on supporting of homeostasis in these systems with respect to fluctuations of an external medium (the problem is posed by M. Gromov, A.Carbone cite{Gr}). Using a measure of stochastic stability we show that a generic system with fixed parameters is unstable, i.e., the probability to support homeostasis converges to zero as time $T o infty$. However, if we consider a population of unstable systems, which are capable to evolve (change their parameters), then such a population can be stable as $T o infty$. This means that the probability to survive may be non-zero as $T o infty$. Evolution algorithms, that provide stability of populations, are not trivial. We show that the mathematical results on evolution algorithms are consistent with experimental data on genetic evolution.

13/04/2006 PDF

L'isomorphisme entre les tours de Lubin-Tate et de Drinfeld au niveau des points M/06/19

11/04/2006 PDF

Wilson Loops of Anti-symmetric Representation and D5-branes P/06/18

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We use a D5-brane with electric flux in AdS_5 x S^5 background to calculate the circular Wilson loop of anti-symmetric representation in N=4 super Yang-Mills theory in 4 dimensions. The result agrees with the Gaussian matrix model calculation.

30/03/2006 PDF

Nonequilibrium statistical mechanics and entropy production in a classical infinite system of rotators. P/06/17

29/03/2006 PDF

Matrix Representation of Renormalization in Perturbative Quantum Field Theory P/06/15

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We formulate the Hopf algebraic approach of Connes and Kreimer to renormalization in perturbative quantum field theory using triangular matrix representation. We give a Rota-Baxter anti-homomorphism from general regularized functionals on the Feynmangraph Hopf algebra to triangular matrices with entries in a Rota--Baxter algebra. For characters mapping to the group of unipotent triangular matrices we derive the algebraic Birkhoff decomposition for matrices using Spitzer's identity. This simple matrix factorization is applied to characterize and calculate perturbative renormalization.

28/03/2006 PDF

L'isomorphisme entre les tours de Lubin-Tate et de Drinfeld : Décomposition cellulaire de la tour de Lubin-Tate M/06/16

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Cet article est le premier d'une série visant à construire un isomorphisme entre les tours p-adiques de Lubin-Tate et de Drinfeld, décrire cet isomorphisme et en donner des applications. Nous-y construisons un modèle entier p-adique équivariant en niveauinfini de la tour de Lubin-Tate. Ce schéma formel p-adique sera comparé plus tard à un autre associé à la tour de Drinfeld.

28/03/2006 PDF

Birkhoff type decompositions and the Baker--Campbell--Hausdorff recursion P/06/14

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We describe a unification of several apparently unrelated factorizations arisen from quantum field theory, vertex operator algebras, combinatorics and numerical methods in differential equations. The unification is given by a Birkhoff type decompositionthat was obtained from the Baker-Campbell-Hausdorff formula in our study of the Hopf algebra approach of Connes and Kreimer to renormalization in perturbative quantum field theory. There we showed that the Birkhoff decomposition of Connes and Kreimer canbe obtained from a certain Baker-Campbell-Hausdorff recursion formula in the presence of a Rota-Baxter operator. We will explain how the same decomposition generalizes the factorization of formal exponentials and uniformization for Lie algebras that arose in vertex operator algebra and conformal field theory, and the even-odd decomposition of combinatorial Hopf algebra characters as well as to the Lie algebra polar decomposition as used in the context of the approximation of matrix exponentials in ordinary differential equations.

23/03/2006 PDF

The structure of double groupoids M/06/13

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We give a general description of the structure of a discrete double groupoid (with an extra, quite natural, filling condition) in terms of groupoid factorizations and groupoid 2-cocycles with coefficients in abelian group bundles. Our description goesas follows: in a first step we prove that every double groupoid is obtained as an extension of its pith groupoid, which is an abelian group bundle, by its frame double groupoid. The frame satisfies that every box is determined by its edges, and thusis called a 'thin' double groupoid. In a second, independent, step we prove that every thin double groupoid with filling condition is completely determined by a factorization of a certain canonically defined 'diagonal' groupoid.

22/03/2006 PDF

First steps towards p-adic Langlands functoriality M/06/12

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By the theory of Colmez and Fontaine, a de Rham representation of the Galois group of a local field roughly corresponds to a representation of the Weil-Deligne group equipped with an admissible filtration on the underlying vector space. Using a modification of the classical local Langlands correspondence, we associate with any pair consisting of a Weil-Deligne group representation and a type of a filtration (admissible or not) a specific locally algebraic representation of a general linear group. We advertise the conjecture that this pair comes from a de Rham representation if and only if the corresponding locally algebraic representation carries an invariant norm. In the crystalline case, the Weil-Deligne group representation is unramified and the associated locally algebraic representation can be studied using the classical Satake isomorphism. By extending the latter to a specific norm completion of the Hecke algebra, we show that the existence of an invariant norm implies that our pair, indeed, comes from a crystalline representation. We also show, by using the formalism of Tannakian categories, that this latter fact is compatible with classical unramified Langlands functoriality and therefore generalizes to arbitrary split reductive groups.

21/03/2006 PDF

Existence of closed $G_2$-structures on 7-manifolds M/06/11

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In this note we propose a new way of constructing compact 7-manifolds with a closed $G_2$-structure. As a result we find a first example of a closed $G_2$-structure on $S^3 imes S^4$. We also prove that any integral closed $G_2$-structure on a compact7-manifold $M^7$ can be obtained by embedding $M^7$ to a universal space $(W ^{3(80 +8 cdot C^3 _8)}, h)$.

08/03/2006 PDF

Existence d'immeubles triangulaires quasi-périodiques M/06/08

03/03/2006 PDF

Ergodic pumping: a mechanism to drive biomolecular conformation changes P/06/09

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We propose that a significant contribution to the power stroke of myosin and similar conformation changes in other biomolecules is the pressure of a single molecule (e.g. a phosphate ion) expanding a trap, a mechanism we call ergodic pumping''. We demonstrate the principle with a toy computer model and discuss the mathematics governing the evolution of slow degrees of freedom in large Hamiltonian systems. We indicate in detail how the mechanism could fit with known features of the myosin cycle. Manyother biomolecular conformation changes could be driven in part by ergodic pumping. We suggest the use of ergodic pumping as a design principle in nanobiotechnology.

03/03/2006 PDF

Interaction of two charges in a uniform magnetic field I: planar problem P/06/10

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The interaction of two charges moving in $R^2$ in a magnetic field ${f B}$ can be formulated as a hamiltonian system with 4 degrees of freedom. Assuming that the magnetic field is uniform and the interaction potential has rotational symmetry we reducethis Hamiltonian system to one with 2 degrees of freedom; for certain values of the conserved quantities and choices of parameters, we obtain an integrable system. Furthermore, when the interaction potential is of Coulomb type, we prove that, for suitableregime of parameters, there are invariant subsets on which this system contains a suspension of a subshift of finite type. This implies non-integrability for this system with a Coulomb-type interaction. Explicit knowledge of the reconstruction map and adynamical analysis of the reduced Hamiltonian systems are the tools we use in order to give a description for the various types fo dynamical behaviours in this system: from periodic ro quasiperiodic and chaotic orbits, from bounded to unbounded motion.

03/03/2006 PDF

Cerbelli and Giona's map is pseudo-Anosov and 9 consequences M/06/06

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It is shown that a piecewise affine area-preserving homeomorphism of the 2-torus studied by Cerbelli and Giona is pseudo-Anosov. This enables one to prove various of their conjectures, quantify the multifractality of its "w-measures", calculate many other quantities for its dynamics, and construct an exact area-preserving tilt map of the cylinder with proved diffusive behaviour.

10/02/2006 PDF

Discommensuration Theory and Shadowing in Frenkel-Kontorova Models P/06/07

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We prove that if the minimum energy advancing discommensuration of mean spacing $p/q$ for a Frenkel-Kontorova chain is unique up to translations and has phonon gap then all minimum energy states with mean spacing $\omega$ just above $p/q$ are approximated exponentially well in $q\omega - p$ by concatenations of advancing $p/q$ discommensuration.

10/02/2006 PDF

Dirichlet forms and Markov semigroups on non-associative vector bundles M/06/04

01/02/2006 PDF

Sur quelques représentations potentiellement cristallines de $GL_2(Q_p)$ M/06/03

24/01/2006 PDF

Jordan structures in harmonic functions and Fourier algebras on homogeneous spaces M/06/02

20/01/2006 PDF

Bubbling Geometries for Half BPS Wilson lines P/06/01

18/01/2006 PDF

Maximal surface group representations in isometry groups of classical Hermitian symmetric spaces M/05/51

21/12/2005 PDF

On the geometry of moduli spaces of holomorphic chains over compact Riemann surfaces M/05/56

21/12/2005 PDF

Universal enveloping algebras and some applications in physics P/05/26

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These notes are intended to provide a self-contained and pedagogical introduction to the universal enveloping algebras and some of their uses in mathematical physics. After reviewing their abstract definitions and properties, the focus is put on their relevance in Weyl calculus, in representation theory and their appearance as higher symmetries of physical systems.

20/12/2005 PDF

Semi-global invariants of piecewise smooth Lagrangian fibrations M/05/55

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We study certain types of piecewise smooth Lagrangian fibrations of smooth symplectic manifolds, which we call stitched Lagrangian fibrations. We extend the classical theory of action-angle coordinates to these fibrations by encoding the information on the non-smoothness into certain invariants consisting, roughly, of a sequence of closed 1-forms on a torus. The main motivation for this work is given by the piecewise smooth Lagrangian fibrations previously constructed by the authors, which topologicallycoincide with the local models used by Gross in "Topological Mirror Symmetry".

20/12/2005 PDF

Hidden symmetries and the fermionic sector of eleven-dimensional supergravity P/05/53

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We study the hidden symmetries of the fermionic sector of D=11 supergravity, and the role of K(E10) as a generalised R symmetry'. We find a consistent model of a massless spinning particle on an E10/K(E10) coset manifold whose dynamics can be mapped onto the fermionic and bosonic dynamics of D=11 supergravity in the near space-like singularity limit. This E10-invariant superparticle dynamics might provide the basis of a new definition of M-theory, and might describe the de-emergence' of space-time near a cosmological singularity.

15/12/2005 PDF

On Killing tensors and cubic vertices in higher-spin gauge theories P/05/54

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The problem of determining all consistent non-Abelian local interactions is reviewed in flat space-time. The antifield-BRST formulation of the free theory is an efficient tool to address this problem. Firstly, it allows to compute all on-shell local Killing tensor fields, which are important because of their deep relationship with higher-spin algebras. Secondly, under the sole assumptions of locality and Poincar?e invariance, all non-trivial consistent deformations of a sum of spin-three quadratic actions deforming the Abelian gauge algebra were determined. They are compared with lower-spin cases.

15/12/2005 PDF

Relative Seiberg-Witten and Ozsvath-Szabo invariants for surfaces in 4-manifolds M/05/57

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We study the invariants of surfaces in 4-manifolds extracted from the Seiberg-Witten and the Ozsvath-Szabo invariants of their fiber sums with auxiliary Lefschetz fibrations. Such invariants involve relative Spin_c structures and can be treated as refinements of the usual Seiberg-Witten and Ozsvath-Szabo invariants. We prove several properties of the relative invariants, for instance the adjunction inequality for membranes, which estimates their genus, and the product formula.

01/12/2005 PDF

Topological strings and two dimensional electrons P/05/34

01/12/2005 PDF

Collapsed 5-manifolds with pinched sectional curvature M/05/52

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Let $M$ be a closed $5$-manifold of pinched curvature $0<\delta\le \text{sec}_M\le 1$. We prove that $M$ is homeomorphic to a spherical space form if $M$ satisfies one of the following conditions: (i) $\delta =1/4$ and the fundamental groupis a non-cyclic group of order $\ge C$, a constant. (ii) The center of the fundamental group has index $\ge w(\delta)$, a constant depending on $\delta$. (iii) The ratio of the volume and the maximal injectivity radius is $<\epsilon(\delta)$. (iv) The volume is less than $\epsilon(\delta)$ and the fundamental group $\pi_1(M)$ has a center of index at least $w$, a universal constant, and $\pi_1(M)$ is either isomorphic to a spherical $5$-space group or has an odd order.

23/11/2005 PDF

A Chiral Perturbation Expansion for Gravity P/05/48

22/11/2005 PDF

Filtration de monodromie et cycles évanescents formels M/05/50

16/11/2005 PDF

Refined Analytic Torsion as an Element of the Determinant Line M/05/49

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We construct a canonical element, called the refined analytic torsion, of the determinant line of the cohomology of a closed oriented odd-dimensional manifold M with coefficients in a flat complex vector bundle E. We compute the Ray-Singer norm of the refined analytic torsion. In particular, if there exists a flat Hermitian metric on $E$, we show that this norm is equal to 1. We prove a duality theorem, establishing a relationship between the refined analytic torsions corresponding to a flat connection and its dual.

08/11/2005 PDF

Lectures on curved beta-gamma systems, pure spinors, and anomalies P/05/35

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The curved beta-gamma system is the chiral sector of a certain infinite radius limit of the non-linear sigma model with complex target space. Naively it only depends on the complex structures on the worldsheet and the target space. It may suffer from theworldsheet and target space diffeomorphism anomalies. We analyze the curved beta-gamma system on the space of pure spinors, aiming to verify the consistency of Berkovits covariant superstring quantization. We demonstrate that under certain conditions bothanomalies can be cancelled for the pure spinor sigma model, in which case one reproduces the old construction of B.Feigin and E.Frenkel.

02/11/2005 PDF

On motives associated to graph polynomials M/05/46

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The appearance of multiple zeta values in anomalous dimensions and $\beta$-functions of renormalizable quantum field theories has given evidence towards a motivic interpretation of these renormalization group functions. In this paper we start to hunt the motive, restricting our attention to a subclass of graphs in four dimensional scalar field theory which give scheme independent contributions to the above functions.

11/10/2005 PDF

On obstructions to asphericity of certain crossed modules M/05/45

11/10/2005 PDF

Noncritical osp($1 vert 2$,R) M-theory matrix model with an arbitrary time dependent cosmological constant P/05/42

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Dimensional reduction of the D=2 minimal super Yang-Mills to the D=1 matrix quantum mechanics is shown to double the number of dynamical supersymmetries, from N=1 to N=2. We analyze the most general supersymmetric deformation of the latter, in order to construct the noncritical 3D M-theory matrix model on generic supersymmetric backgrounds. It amounts to adding a harmonic oscillator potential with an arbitrary time dependent coefficient or cosmological constant,' \Lambda(t). The resulting matrix modelenjoys, irrespective of \Lambda(t), two dynamical supersymmetries which further reveal three hidden so(1,2) symmetries. All together they form the supersymmetry algebra, osp(1|2,R). Each so(1,2) multiplet in the Hilbert space visualizes a dynamics constrained on either Euclidean or Minkowskian dS_{2}/AdS_{2} space, depending on its Casimir. In particular, all the unitary as well as BPS multiplets have the Euclidean dS_{2}/AdS_{2} geometry. We conjecture that the matrix model provides holographic duals tothe 2D superstring theories on various backgrounds having the spacetime signature Minkowskian if \Lambda(t)>0, or Euclidean if \Lambda(t)<0. In particular, we argue that the choice of the negative constant \Lambda corresponds to the N=2 super Liouvilletheory.

11/10/2005 PDF

Anatomy of a gauge theory P/05/40

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Resumé: We exhibit the role of Hochschild cohomology in quantum field theory with particular emphasis on gauge theory and Dyson--Schwinger equations, the quantum equations of motion. These equations emerge from Hopf- and Lie algebra theory and free quantum field theory only. In the course of our analysis we exhibit an intimate relation between the Slavnov-Taylor identities for the couplings and the existence of Hopf sub-algebras defined on the sum of all graphs at a given loop order, surpassing the need to work on single diagrams.

07/10/2005 PDF

Théorie d'Iwasawa des représentations cristallines II M/05/41

07/10/2005 PDF

Représentations modulaires de $mathrm{GL}_2(mathbf{Q}_p)$ et représentations galoisiennes de dimension $2$ M/05/44

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On montre la conjecture de Breuil concernant la r\'eduction modulo $p$ des repr\'esentations triangulines $V$ et des repr\'esentations $\Pi(V)$ de $\mathrm{GL}_2(\mathbf{Q}_p)$ qui leur sont associ\'ees par la correspondance de Langlands $p$-adique. L'ingr\'edient principal de la d\'emonstration est l'\'etude de certaines repr\'esentations lisses irr\'eductibles de $\mathrm{B}(\mathbf{Q}_p)$ via des mod\eles construits en utilisant les $(\varphi,\Gamma)$-modules.

05/10/2005 PDF

Kähler flat manifolds of low dimensions M/05/43

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We give a list of six dimensional flat K\"ahler manifolds. Moreover, we present an example of eight dimensional flat K\"ahler manifold M with finite Out(\pi_1(M)) group.

04/10/2005 PDF

The Hopf algebra structure of renormalizable Quantum Field Theory P/05/39

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Review for the Encyclopedia of Mathematical Physics.

20/09/2005 PDF

Quaternion Landau-Ginsburg models and noncommutative Frobenius manifolds M/05/37

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We extend topological Landau-Ginsburg models with boundaries to Quaternion Landau-Ginsburg models that satisfy the axioms for open-closed topological field theories. Later we prove that moduli spaces of Quaternion Landau-Ginsburg models are non-commutative Frobenius manifolds in means of [J. Geom. Phys, 51 (2003),387-403.].

14/09/2005 PDF

IR Free or Interacting? A Proposed Diagnostic P/05/38

14/09/2005 PDF

The continuous spin limit of higher spin equations of motion P/05/36

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We show that the Wigner equations describing the continuous spin representations can be obtained as a limit of massive higher-spin equations. The limit involves a suitable scaling of the wave function, the mass going to zero and the spin to infinity with their product being fixed. The result allows to transform the Wigner equations to a gauge invariant Fronsdal-like form. We also give the generalisation of the Wigner equations to higher dimensions with fields belonging to arbitrary representations of the massless little group.

13/09/2005 PDF

On non-commutative analytic spaces over non-archimedean fields M/05/33

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We discuss various examples of non-commutative spaces which can be called non-commutative rigid analytic spaces

30/08/2005 PDF

On the torsion of optimal elliptic curves over function fields M/05/32

26/08/2005 PDF

Torsion as a function on the space of representations M/05/30

16/08/2005 PDF

Dynamics, Laplace transform and Spectral geometry M/05/31

16/08/2005 PDF

Jean Dieudonné (1906-1992) mathematician M/05/28

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Jean Dieudonn\'e has been one of the most influential French mathematicians during the 20$^{\rm th}$ century, especially through his association -- even identification -- with the Bourbaki group. An excellent biography has been written by his friend P. Dugac, a historian of science [4]. We shall retrace here his long and distinguished career.

09/08/2005 PDF

An algebraic proof of a cancellation theorem for surfaces M/05/29

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Let $\K$ be an algebraically closed field of arbitrary characteristic. We give a short self-contained algebraic proof of the following statement: If the cylinder over an affine surface (i. e. the product of our surface and an affine line) over $\K$ is (isomorphic to) an affine space then the surface is (isomorphic to) an affine plane.

09/08/2005 PDF

Spin three gauge theory revisited P/05/25

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We study the problem of consistent interactions for spin-$3$ gauge fields in flat spacetime of arbitrary dimension $n > 3$. Under the sole assumptions of Poincar\'e and parity invariance, local and perturbative deformation of the free theory, we determine all nontrivial consistent deformations of the abelian gauge algebra and classify the corresponding deformations of the quadratic action, at first order in the deformation parameter. We prove that all such vertices are cubic, contain a total ofeither three or five derivatives and are uniquely characterized by a rank-three constant tensor (an internal algebra structure constant). The covariant cubic vertex containing three derivatives is the vertex discovered by Berends, Burgers and van Dam, which however leads to inconsistencies at second order in the deformation parameter. In dimensions $n>4$ and for completely antisymmetric structure constant tensor, a new covariant cubic vertex exists, which contains five derivatives and passes the consistency test where the previous vertex failed.

05/08/2005 PDF

A group of diffeomorphisms of the interval with intermediate growth M/05/27

04/08/2005 PDF

Sur la dynamique unidimensionnelle en régularité intermédiaire M/05/24

02/08/2005 PDF

Représentations p-adiques ordinaires de GL2(Qp) et compatibilité local-global M/05/22

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On définit les représentations p-adiques de GL2(Qp) "correspondant" aux représentations potentiellement cristallines réductibles (et éventuellement scindées) de Gal(Qpbar/Qp) de dimension 2 et on montre qu'elles apparaissent naturellement dans la cohomologie étale complétée de la tour en p des courbes modulaires.

21/07/2005 PDF

Représentations semi-stables de $GL_2 ({mathbb Q}_p)$, demi-plan $p$-adique et réduction modulo $p$ M/05/23

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On calcule par voie cohomologique la r\'eduction modulo $p$ de certaines repr\'esentations $p$-adiques semi-stables de ${\rm GL}_2({\mathbb Q}_p)$. Les calculs exploitent la g\'eom\'etrie du demi-plan $p$-adique. Ils permettent de retrouver certaines formules de la r\'eduction modulo $p$ de repr\'esentations $p$-adiques semi-stables de ${\rm Gal}(\overline{\bf Q}_p/{\bf Q}_p)$.

20/07/2005 PDF

Cauchy-Davenport theorem in group extensions M/05/21

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Let A and B be nonempty subsets of a finite group G in which the order of the smallest nontrivial subgroup is not smaller than d=|A|+|B|-1. Then the product set AB has at least d elements. This extends a classical theorem of Cauchy and Davenport to noncommutative groups. We also generalize Vosper's inverse theorem in the same spirit, giving a complete description of the critical pairs. The proofs depend on the structure of group extensions.

13/07/2005 PDF

Sur la conjecture faible de Greenberg dans le cas abélien $p$-décomposé M/05/20

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Let $p$ be an odd prime. For any CM number field $K$ containing a primitive $p^{\rm th}$-root of unity, class field theory and Kummer theory put together yield the well known reflection inequality $\lambda^+ \leq \lambda^-$ between the plus'' andminus'' parts of the $\lambda$-invariant of $K$. Greenberg's classical conjecture predicts the vanishing of $\lambda^+$. We propose a weak form of this conjecture: $\lambda^+ = \lambda^-$ if and only if $\lambda^+ = \lambda^- = 0$, and we proveit when $K^+$ is abelian, $p$ is totally split in $K^+$, and certain (mild) conditions on the cohomology of circular units are satisfied.

13/07/2005 PDF

On the Riemann zeta-function and analytic characteristic functions M/05/18

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Set $f(s) :=1/(\sin(\pi s/4)q(1/2 + s))$ with $q(s) := \pi^{-s/2} \, 2 \Gamma (1 + s/2)(s-1) \zeta (s)$. The Riemann hypothesis, RH, and the simple zeros conjecture, SZC, together with conjectures advanced by the author are used to show that $f(s)$on each vertical strip $V_{4n}$ of $s$ with $4n < {\rm Re} \, (s) < 4(n+1)$ provides an analytic characteristic function, $(-1)^n \cdot f(s) = \int_R (dy) e^{sy} P_{4n} (y)$ with $P_{4n} (y)$ positive. The essential case with $n = 0$ implies RH. A formula is obtained for $P_{4n} (y)$, which for $y$ negative involves the critical zeros. An alternative formula is obtained for $P_{4n} (y)$, without relying on RH, SZC or other unproven conjectures. It does not involve the critical zeros. Analogous resultsfor the cases of the Dirichlet $L$-functions and the Ramanujan tau Dirichlet $L$-function are conjectured.

28/06/2005 PDF

Hopf algebras in renormalization theory: Locality and Dyson-Schwinger equations from Hochschild cohomology P/05/19

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In this review we discuss the relevance of the Hochschild cohomology of renormalization Hopf algebras for local quantum field theories and their equations of motion.

24/06/2005 PDF

Semi-stable extensions on arithmetic surfaces M/05/17

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On a given arithmetic surface, inspired by work of Miyaoka, we consider vector bundles which are extensions of a line bundle by another one. We give sufficient conditions for their restriction to the generic fiber to be semi-stable. We then apply the arithmetic analog of Bogomolov inequality in Arakelov theory, and deduce from it a lower bound for some successive minima in the lattice of extension classes between these line bundles.

07/06/2005 PDF

Estimates from below for the spectral function and for the remainder in local Weyl's law M/05/16

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We obtain asymptotic lower bounds for the spectral function of the Laplacian and for the remainder in local Weyl's law on manifolds. In the negatively curved case, thermodynamic formalism is applied to improve the estimates. Key ingredients of the proof include the wave equation parametrix, a pretrace formula and the Dirichlet box principle.

19/05/2005 PDF

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Given a symplectic manifold $M$, we may define an operad structure on the the spaces $\op^k$ of the Lagrangian submanifolds of $(\bar{M})^k\times M$ via symplectic reduction. If $M$ is also a symplectic groupoid, then its multiplication space is an associative product in this operad. Following this idea, we provide a deformation theory for symplectic groupoids analog to the deformation theory of algebras. It turns out that the semi-classical part of Kontsevich's deformation of $C^\infty(\R^d)$ is adeformation of the trivial symplectic groupoid structure of $T^*\R^d$.

18/05/2005 PDF

Gauge invariants and Killing tensors in higher-spin gauge theories P/05/15

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In free completely symmetric tensor gauge field theories on Minkowski space-time, all gauge invariant functions and Killing tensor fields are computed, both on-shell and off-shell. These problems are addressed in the metric-like formalisms.

18/05/2005 PDF

Partial wave expansion and Wightman positivity in conformal field theory P/05/13

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A new method for computing exact conformal partial wave expansions is developed and applied to approach the problem of Hilbert space (Wightman) positivity in a non-perturbative four-dimensional quantum field theory model. The model is based on the assumption of global conformal invariance on compactified Minkowski space (GCI). Bilocal fields arising in the harmonic decomposition of the operator product expansion (OPE) prove to be a powerful instrument in exploring the field content. In particular,in the theory of a field $\LL$ of dimension 4 which has the properties of a (gauge invariant) Lagrangian, the scalar field contribution to the 6-point function of the twist 2 bilocal field is analyzed with the aim to separate the free field part fromthe nontrivial part.

26/04/2005 PDF

Sur la réduction des représentations cristallines de dimension 2 en poids moyens M/05/12

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On calcule la r\'eduction modulo $p$ des repr\'esentations cristallines de dimension $2$ dont les poids de Hodge-Tate sont $0$ et $k-1$ avec $k \in \{p+2,\cdots,2p-1\}$.

20/04/2005 PDF

On the Hochschild-Kostant-Rosenberg map for graded manifolds M/05/10

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We show that the Hochschild--Kostant--Rosenberg map from the space of multivector fields on a graded manifold $N$ (endowed with a Berezinian volume) to the cohomology of the algebra of multidifferential operators on $N$ (as a subalgebra of the Hochschild complex of $C^\infty(N)$) is an isomorphism of Batalin--Vilkovisky algebras. Moreover, with an example inspired by string topology, we prove that in general the inclusion of multidifferential operators in the Hochschild complex is not a quasi-isomorphism.

15/04/2005 PDF

Rota-Baxter Algebras, Dendriform Algebras and Poincare-Birkhoff-Witt Theorem M/05/11

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Rota-Baxter algebras appeared in both the physics and mathematics literature. It is of great interest to have a simple construction of the free object of this algebraic structure. For example, free commutative Rota-Baxter algebras relate to double shuffle relations for multiple zeta values. The interest in the non-commutative setting arose in connection with the work of Connes and Kreimer on the Birkhoff decomposition in renormalization theory in perturbative quantum field theory. We construct free non-commutative Rota-Baxter algebras and apply the construction to obtain universal enveloping Rota-Baxter algebras of dendriform dialgebras and trialgebras. We also prove an analog of the Poincare-Birkhoff-Witt theorem for universal enveloping algebra in the context of dendriform trialgebras. In particular, every dendriform dialgebra and trialgebra is a subalgebra of a Rota-Baxter algebra. We explicitly show that the free dendriform dialgebras and trialgebras, as represented by planar trees, are canonical subalgebras of free Rota-Baxter algebras.

15/04/2005 PDF

The character of pure spinors P/05/01

14/04/2005 PDF

Dynamics of higher spin fields and tensorial space P/05/06

14/04/2005 PDF

Biorthogonal Laurent polynomials, Töplitz determinants, minimal Toda orbits and isomonodromic tau functions M/05/08

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We consider the class of biorthogonal polynomials that are used to solve the inverse spectral problem associated to elementary co-adjoint orbits of the Borel group of upper triangular matrices; these orbits are the phase space of generalizedintegrable lattices of Toda type. Such polynomials naturally interpolate between the theory of orthogonal polynomials on the line and orthogonal polynomials on the unit circle and tie together the theory of Toda, relativistic Toda, Ablowitz-Ladikand Volterra lattices. We establish corresponding Christoffel-Darboux formul\ae . For all these classes of polynomials a $2\times 2$ system of Differential-Difference-Deformation equations is analyzed in the most general setting of pseudomeasures with arbitrary rational logarithmic derivative. They provide particular classes of isomonodromic deformations of rational connections on the Riemann sphere. The corresponding isomonodromic tau function is explicitly related to the shifted T\"oplitz determinants of the moments of the pseudo-measure. In particular, the results imply that any (shifted) T\"oplitz (H\"ankel) determinant of a symbol (measure) with arbitrary rational logarithmic derivative is an isomonodromic taufunction.

14/04/2005 PDF

On the Distribution of the Wave Function for Systems in Thermal Equilibrium P/05/09

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A density matrix that is not pure can arise, via averaging, from many different distributions of the wave function. This raises the question, which distribution of the wave function, if any, should be regarded as corresponding to systems in thermal equilibrium as represented, for example, by the density matrix $\rho_\beta = (1/Z) \exp(- \beta H)$ of the canonical ensemble. To answer this question, we construct, for any given density matrix $\rho$, a measure on the unit sphere in Hilbert space, denoted GAP($\rho$), using the Gaussian measure on Hilbert space with covariance $\rho$. We argue that GAP($\rho_\beta$) corresponds to the canonical ensemble.

14/04/2005 PDF

An invariant for non simply connected manifolds M/05/07

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For a closed manifold $M$ we introduce the set of co-Euler structures and we define the modified Ray-Singer torsion, a positive real number associated to $M,$ a co-Euler structure and an acyclic representation $\rho$ of the fundamental group of $M$ with $H^\ast(M;\rho)=0.$ If the co-Euler structure is integral we show that the modified Ray--Singer torsion, regarded as a positive (real valued) function on the variety of some complex representations, is the absolute value of a (complex valued) rational function which carries interesting topological information about the manifold. This rational function is the invariant in the title. If the co-Euler structure is arbitrary one obtains a more general object, a holomorphic 1-cocycle. Interesting rational functions in topology appear in this way.The argument of this rational function when defined, is an interesting and apparently unexplored invariant which reminds the Atiyah--Patodi--Singer eta invariant.

02/02/2005 PDF

Uniform uniformisation M/05/03

01/02/2005 PDF

Factorization Conjecture and the Open/Closed String Correspondence P/05/04

01/02/2005 PDF

Analyticity of the susceptibility function for unimodal Markovian maps of the interval P/05/05

01/02/2005 PDF

Semi-stable reduction of fiolations M/05/02

26/01/2005 PDF

Gromov-Witten invariants and pseudo symplectic capacities M/04/60

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We introduce the concept of pseudo symplectic capacities which is a mild generalization of that of symplectic capacities. As a generalization of the Hofer-Zehnder capacity we construct a Hofer-Zehnder type pseudo symplectic capacity and estimate itin terms of Gromov-Witten invariants. The (pseudo) symplectic capacities of Grassmannians and some product symplectic manifolds are computed. As applications we first derive some general nonsqueezing theorems that generalize and unite many previous versions, then prove the Weinstein conjecture for cotangent bundles over a large class of symplectic uniruled manifolds (including the uniruled manifolds in algebraic geometry) and also show that the Hofer-Zehnder capacity is finite on a small neighborhood of a rational connected closed symplectic submanifold of codimension two in a symplectic manifold. Finally, we give two results on symplectic packings in Grassmannians and on Seshadri constants.

24/12/2004 PDF

A LA RECHERCHE DE LA m-THÉORIE PERDUE Z-THEORY: CHASING m/f THEORY P/04/52

23/12/2004 PDF

LECTURES ON NONPERTURBATIVE ASPECTS OF SUPERSYMMETRIC GAUGE THEORIES P/04/53

23/12/2004 PDF

FROM SUPERSTRINGS TO QUANTUM FOAM USING SUPERSYMMETRY P/04/54

23/12/2004 PDF

GROMOV-WITTEN THEORY AND DONALDSON-THOMAS THEORY, II M/04/55

23/12/2004 PDF

Global geometric deformations of current algebras as Krichever-Novikov type algebras M/04/56

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We construct algebraic-geometric families of genus one (i.e. elliptic) current and affine Lie algebras of Krichever-Novikov type. These families deform the classical current, respectively affine Kac-Moody Lie algebras. The construction is induced by the geometric process of degenerating the elliptic curve to singular cubics. If the finite-dimensional Lie algebra defining the infinite dimensional current algebra is simple then, even if restricted to local families, the constructed families are non-equivalent to the trivial family. In particular, we show that the current algebra is geometrically not rigid, despite its formal rigidity. This shows that in the infinite-dimensional Lie algebra case the relations between geometric deformations, formal deformations and Lie algebra two-cohomology are not that close as in the finite-dimensional case. The constructed families are e.g. of relevance in the global operator approach to the Wess-Zumino-Witten-Novikov models appearing in the quantization of Conformal Field Theory.

23/12/2004 PDF

Vertex algebras and the Landau-Ginzburg/Calabi-Yau correspondence M/04/57

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We construct a spectral sequence that converges to the cohomology of the chiral de Rham complex over a Calabi-Yau hypersurface and whose first term is a vertex algebra closely related to the Landau-Ginzburg orbifold. As an application, we prove an explicit orbifold formula for the elliptic genus of Calabi-Yau hypersurfaces.

23/12/2004 PDF

On the Galois cohomology of unipotent algebraic groups over local and global function fields M/04/58

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We discuss some results on the triviality and finiteness for Galois cohomology of connected unipotent groups over local and global function fields, and their relation with the closedness of orbits. As application, we show that a separable additive polynomial over a global field $k$ of characteristic $p>0$ in two variables is universal over $k$ if and only if it is so over all completions $k_v$ of $k$.

05/12/2004 PDF

Nonlinear Higher Spin Theories in Various Dimensions P/04/47

01/12/2004 PDF

Some Rationality Properties of Observable Groups and Related Questions M/04/62

01/12/2004 PDF

Chiral polyhedra in ordinary space, II M/04/61

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A chiral polyhedron has a geometric symmetry group with two orbits on the flags, such that adjacent flags are in distinct orbits. Part I of the paper described the discrete chiral polyhedra in ordinary Euclidean 3-space with finite skew faces and finite skew vertex-figures. Part II completes the enumeration of all discrete chiral polyhedra in 3-space. There exist several families of chiral polyhedra with infinite, helical faces. In particular, there are no discrete chiral polyhedra with finite faces in addition to those described in Part I.

01/12/2004 PDF

EXPLICIT MUMFORD ISOMORPHISM FOR HYPERELLIPTIC CURVES M/04/51

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We give an explicit version of the Mumford isomorphism on the moduli stack of hyperelliptic curves of any given genus.

16/11/2004 PDF

LA MONODROMIE HAMILTONIENNE DES CYCLES ÉVANESCENTS M/04/50

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Nous montrons sous des hypoth\eses pr\'ecis\'ees dans l'\'enonc\'e que le premier groupe d'homologie \'evanescente d'une fibration lagrangienne singuli\ere est librement engendr\'e par les cycles \'evanescents. Nous en d\'eduisons que l'op\'erateur de variation associ\'e est un isomorphisme.

05/11/2004 PDF

Four basic symmetry types in the universal 7-cluster structure of 143 complete bacterial genomic sequences M/04/49

28/10/2004 PDF

Fermions in the harmonic potential and string theory P/04/41

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We explicitly derive collective field theory description for the system of fermions in the harmonic potential. This field theory appears to be a coupled system of free scalar and (modified) Liouville field. This theory should be considered as an exact bosonization of the system of non-relativistic fermions in the harmonic potential. Being surprisingly similar to the world-sheet formulation of c=1 string theory, this theory has quite different physical features and it is conjectured to give space-time description of the string theory, dual to the fermions in the harmonic potential. A vertex operator in this theory is shown to be a field theoretical representation of the local fermion operator, thus describing a D0 brane in the string language. Possible generalization of this result and its derivation for the case of c=1 string theory (fermions in the inverse harmonic potential) is discussed.

28/10/2004 PDF

Creation of Toy Universe P/04/48

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General ideas of gauge/gravity duality allow for the possibility of time dependent solutions that interpolate between a perturbative gauge theory phase and a weakly curved string/gravity phase. Such a scenario applied to cosmology would exhibit a non-geometric phase before the big bang. We investigate a toy model for such a cosmology, whose endpoint is the classical limit of the two-dimensional non-critical string. We discuss the basic dynamics of this model, in particular how it evolves toward the double scaling limit required for stringy dynamics. We further comment on the physics that will determine the fluctuation spectrum of the scalar tachyon. Finally, we discuss various features of this model, and what relevance they might have for a more realistic, higher dimensional scenario.

18/10/2004 PDF

REPRÉSENTATIONS CRISTALLINES IRRÉDUCTIBLES DE ${ M/04/46 Voir plus > Voir moins > Nous démontrons certaines conjectures (non nullit\'e, irr\'eductibilit\'e topologique, admissibilit\'e) dues au second auteur concernant des représentations unitaires de GL2(Qp) sur des espaces de Banach p-adiques associées aux représentations cristallines irréductibles de dimension 2 de Gal(Qpbar/Qp). Pour cela, nous réinterprétons ces espaces de Banach comme espaces de fonctions sur Qp d'un certain type, puis nous utilisons la théorie des (phi,Gamma)-modules associés aux représentations cristallines correspondantes. 05/10/2004 PDF Boundary Liouville theory at$c=1$P/04/42 Voir plus > Voir moins > The c=1 Liouville theory has received some attention recently as the Euclidean version of an exact rolling tachyon background. In an earlier paper it was shown that the bulk theory can be identified with the interacting c=1 limit of unitary minimal models. Here we extend the analysis of the c=1-limit to the boundary problem. Most importantly, we show that the FZZT branes of Liouville theory give rise to a new 1-parameter family of boundary theories at c=1. These models share many features with the boundary Sine-Gordon theory, in particular they possess an open string spectrum with band-gaps of finite width. We propose explicit formulas for the boundary 2-point function and for the bulk-boundary operator product expansion in the c=1 boundary Liouville model. As a by-product of our analysis we also provide a nice geometric interpretation for ZZ branes and their relation with FZZT branes in the c=1 theory. 29/09/2004 PDF On the Landau Background Gauge Fixing and the IR Properties of YM Green Functions P/04/23 Voir plus > Voir moins > We analyse the complete algebraic structure of the background field method for Yang--Mills theory in the Landau gauge and show several structural simplifications within this approach. In particular we present a new way to study the IR behavior of Green functions in the Landau gauge and show that there exists a unique Green function whose IR behaviour controls the IR properties of the gluon and the ghost propagators. 29/09/2004 PDF Towards Integrability of Topological Strings I: Three-forms on Calabi-Yau manifolds P/04/43 Voir plus > Voir moins > The precise relation between Kodaira-Spencer path integral and a particular wave function in seven dimensional quadratic field theory is established. The special properties of three-forms in 6d, as well as Hitchin's action functional, play an important role. The latter defines a quantum field theory similar to Polyakov's formulation of 2d gravity; the curious analogy with world-sheet action of bosonic string is also pointed out. 29/09/2004 PDF THE K-THEORY OF HEEGAARD-TYPE QUANTUM 3-SPHERES M/04/44 Voir plus > Voir moins > We use a Heegaard splitting of the topological 3-sphere as a guiding principle to construct a family of its noncommutative deformations. The main technical point is an identification of the universal C*-algebras defining our quantum 3-spheres with an appropriate fiber product of crossed-product C*-algebras. Then we employ this result to show that the K-groups of our family of noncommutative 3-spheres coincide with their classical counterparts. 29/09/2004 PDF Lattice Geometry P/04/45 29/09/2004 PDF The Structure of the Ladder Insertion-Elimination Lie Algebra M/04/40 Voir plus > Voir moins > We continue our investigation into the insertion-elimination Lie algebra of Feynman graphs in the ladder case, emphasizing the structure of this Lie algebra relevant for future applications in the study of Dyson-Schwinger equations. We work out the relation of this Lie algebra to some classical infinite dimensional Lie algebra and we determine its cohomology. 01/09/2004 PDF Saddle point equations in Seiberg-Witten theory P/04/38 Voir plus > Voir moins > N=2 supersymmetric Yang-Mills theories for all classical gauge groups, that is, for SU(N), SO(N), and Sp(N) is considered. The equations which define the Seiberg-Witten curve are proposed. In some cases they are solved. It is shown that for (almost) all models allowed by the asymptotic freedom the 1-instanton corrections which follows from these equations agree with the direct computations and with known results. 24/08/2004 PDF What is the trouble with Dyson--Schwinger equations? P/04/37 Voir plus > Voir moins > We discuss similarities and differences between Green Functions in Quantum Field Theory and polylogarithms. Both can be obtained as solutions of fixpoint equations which originate from an underlying Hopf algebra structure. Typically, the equation is linear for the polylog, and non-linear for Green Functions. We argue though that the crucial difference lies not in the non-linearity of the latter, but in the appearance of non-trivial representation theory related to transcendental extensions of the number field which governs the linear solution. An example is studied to illuminate this point. 11/08/2004 PDF Spitzer's identity and the algebraic Birkhoff decomposition in pQFT P/04/39 Voir plus > Voir moins > In this article we continue to explore the notion of Rota-Baxter algebras in the context of the Hopf algebraic approach to renormalization theory in perturbative quantum field theory. We show in very simple algebraic terms that the solutions of the recursively defined formulae for the Birkhoff factorization of regularized Hopf algebra characters, i.e. Feynman rules, naturally give a non-commutative generalization of the well-known Spitzer's identity. The underlying abstract algebraic structure is analyzed in terms of complete filtered Rota-Baxter algebras. 11/08/2004 PDF Approximation by analytic operator functions. Factorizations and very badly approximable functions M/04/36 Voir plus > Voir moins > This is a continuation of our earlier paper. We consider here operator-valued functions (or infinite matrix functions) on the unit circle$\T$and study the problem of approximation by bounded analytic operator functions. We discuss thematic and canonical factorizations of operator functions and study badly approximable and very badly approximable operator functions. We obtain algebraic and geometric characterizations of badly approximable and very badly approximable operator functions. Note that there is an important difference between the case of finite matrix functions and the case of operator functions. Our criteria for a function to be very badly approximable in the case of finite matrix functions also guarantee that the zero function is the only superoptimal approximant. However in the case of operator functions this is not true. 11/08/2004 PDF AN EXTENSION OF THE KOPLIENKO-NEIDHARDT TRACE FORMULAE M/04/35 Voir plus > Voir moins > Koplienko found a trace formula for perturbations of self-adjoint operators by operators of Hilbert Schmidt class$\bS_2$. A similar formula in the case of unitary operators was obtained by Neidhardt. In this paper we improve their results and obtain sharp conditions under which the Koplienko--Neidhardt trace formulae hold. 30/07/2004 PDF Bethe Ansatz for Arrangements of Hyperplanes and the Gaudin Model M/04/34 Voir plus > Voir moins > We show that the Shapovalov norm of a Bethe vector in the Gaudin model is equal to the Hessian of the logarithm of the corresponding master function at the corresponding isolated critical point. We show that different Bethe vectors are orthogonal. These facts are corollaries of a general Bethe ansatz type construction, suggested in this paper and associated with an arbitrary arrangement of hyperplanes. 29/07/2004 PDF Differentiating the absolutely continuous invariant measure of an interval map$f$with respect to$f$M/04/32 23/07/2004 PDF Differentiation of SRB states for hyperbolic flows M/04/33 23/07/2004 PDF ABCD of instantons P/04/19 Voir plus > Voir moins > We solve N=2 supersymmetric Yang-Mills theories for arbitrary classical gauge group, i.e. SU(N), SO(N), Sp(N). In particular, we derive the prepotential of the low-energy effective theory, and the corresponding Seiberg-Witten curves. We manage to do thiswithout resolving singularities of the compactified instanton moduli spaces. 21/07/2004 PDF Bernoulli Number Identities from Quantum Field Theory P/04/31 Voir plus > Voir moins > We present a new method for the derivation of convolution identities for finite sums of products of Bernoulli numbers. Our approach is motivated by the role of these identities in quantum field theory and string theory. We first show that the Miki identity and the Faber-Pandharipande-Zagier (FPZ) identity are closely related, and give simple unified proofs which naturally yield a new Bernoulli number convolution identity. We then generalize each of these three identities into new families ofconvolution identities depending on a continuous parameter. We rederive a cubic generalization of Miki's identity due to Gessel and obtain a new similar identity generalizing the FPZ identity. The generalization of the method to the derivation of convolution identities of arbitrary order is outlined. We also describe an extension to identities which relate convolutions of Euler and Bernoulli numbers. 29/06/2004 PDF Holomorphically Covariant Matrix Models P/04/30 Voir plus > Voir moins > We present a method to construct matrix models on arbitrary simply connected oriented real two dimensional Riemannian manifolds. The actions and the path integral measure are invariant under holomorphic transformations of matrix coordinates. 28/06/2004 PDF Distance Function, Linear quasi-Connections and Chern Character. M/04/27 Voir plus > Voir moins > In Sect.4 we show how the Chern character of the tangent bundle of a smooth manifold may be extracted from the geodesic distance function by means of cyclic homology. In Sect.5 we introduce the notion of coarse linear connection in vector bundles, notion which generalizes the notion of linear connection. We show next that the algebraic procedure for constructing the Chern character, discussed in Sect.4, applies also in the case of coarse linear connections. Looking in retrospect, the constructions we present here represent the non commutative counterpart of a geometric construction of the Chern character, see Teleman N.[Tn] and Teleman C.[Tc]. The arguments discussed here may be formulated within the language of groupoids. In a subsequent paper we are going to improve some of the considerations presented here and extend their field of application to more singular situations. 25/06/2004 PDF Optimal Trade-Off for Merkle Tree Traversals M/04/29 Voir plus > Voir moins > We prove tight upper and lower bounds for computing Merkle tree traversals, and display optimal trade-offs between time and space complexity of that problem. 17/06/2004 PDF Gravitational radiation from inspiralling compact binaries completed at the third post-Newtonian order P/04/25 Voir plus > Voir moins > The gravitational radiation from point particle binaries is computed at the third post-Newtonian (3PN) approximation of general relativity. Three previously introduced ambiguity parameters, coming from the Hadamard self-field regularization of the 3PN source-type mass quadrupole moment, are consistently determined by means of dimensional regularization, and proved to have the values xi = -9871/9240, kappa = 0 and zeta = -7/33. These results complete the derivation of the general relativistic prediction for compact binary inspiral up to 3.5PN order, and should be of use for searching and deciphering the signals in the current network of gravitational wave detectors. 15/06/2004 PDF Mathematical models in population dynamics and ecology M/04/26 Voir plus > Voir moins > We introduce the most common quantitative approaches to population dynamics and ecology, emphasizing the different theoretical foundations and assumptions. These populations can be aggregates of cells, simple unicellular organisms, plants or animals. The basic types of biological interactions are analysed: consumer-resource, prey-predation, competition and mutualism. Some of the modern developments associated with the concepts of chaos, quasi-periodicity, and structural stability are discussed. To describe short- and long-range population dispersal, the integral equation approach is derived, and some of its consequences are analysed. We derive the standard McKendrick age-structured density dependent model, and a particular solution of theMcKendrick equation is obtained by elementary methods. The existence of demography growth cycles is discussed, and the differences between mitotic and sexual reproduction types are analysed. 15/06/2004 PDF Approximation Hardness of Short Symmetric Instances of MAX-3SAT M/04/28 Voir plus > Voir moins > We prove approximation hardness of short symmetric instances of MAX-3SAT in which each literal occurs exactly twice, and each clause is exactly of size 3. We display also an explicit approximation lower bound for that problem. The bound two on the number of occurrences of literals in symmetric MAX-3SAT is thus the smallest possible bound for the MAX-3SAT hardness gap property to exist and making the instances hard to approximate. 15/06/2004 PDF Deformation of outer representations of Galois group M/04/24 Voir plus > Voir moins > To a hyperbolic smooth curve defined over a number-field one naturally associates an "anabelian" representation of the absolute Galois group of the base field landing in outer automorphism group of the algebraic fundamental group. In this paper, weintroduce several deformation problems for Lie-algebra versions of the above representation and show that, this way we get a richer structure than those coming from deformations of "abelian" Galois representations induced by the Tate module of associated Jacobian variety. We develop an arithmetic deformation theory of graded Lie algebras with finite dimensional graded components to serve our purpose. 02/06/2004 PDF Phasing of gravitational waves from inspiralling eccentric binaries P/04/22 Voir plus > Voir moins > We provide a method for analytically constructing high-accuracy templates for the gravitational wave signals emitted by compact binaries moving in inspiralling eccentric orbits. By contrast to the simpler problem of modeling the gravitational wave signals emitted by inspiralling {\it circular} orbits, which contain only two different time scales, namely those associated with the orbital motion and the radiation reaction, the case of {\it inspiralling eccentric} orbits involves {\it three different time scales}: orbital period, periastron precession and radiation-reaction time scales. By using an improved method of variation of constants', we show how to combine these three time scales, without making the usual approximation of treating the radiativetime scale as an adiabatic process. We explicitly implement our method at the 2.5PN post-Newtonian accuracy. Our final results can be viewed as computing new post-adiabatic' short period contributions to the orbital phasing, or equivalently, new short-period contributions to the gravitational wave polarizations,$h_{+,\times}$, that should be explicitly added to the post-Newtonian' expansion for$h_{+,\times}$, if one treats radiative effects on the orbital phasing of the latter in the usual adiabatic approximation. Our results should be of importance both for the LIGO/VIRGO/GEO network of ground based interferometric gravitational wave detectors (especially if Kozai oscillations turn out to be significant in globular cluster triplets), and for thefuture space-based interferometer LISA. 27/05/2004 PDF Conformal invariance and rationality in an even dimensional quantum field theory P/04/21 21/05/2004 PDF Vertex Operators for Closed Superstrings P/04/20 Voir plus > Voir moins > We construct an iterative procedure to compute the vertex operators of the closed superstring in the covariant formalism given a solution of IIA/IIB supergravity. The manifest supersymmetry allows us to construct vertex operators for any generic background in presence of Ramond-Ramond (RR) fields. We extend the procedure to all massive states of open and closed superstrings and we identify two new nilpotent charges which are used to impose the gauge fixing on the physical states. We solve iteratively theequations of the vertex for linear x-dependent RR field strengths. This vertex plays a role in studying non-constant C-deformations of superspace. Finally, we construct an action for the free massless sector of closed strings, and we propose a form for the kinetic term for closed string field theory in the pure spinor formalism. 13/05/2004 PDF Self-similar Fractals in Arithmetic M/04/18 Voir plus > Voir moins > We define a notion of self-similarity on algebraic varieties by considering algebraic endomorphisms as "similarity" maps. Self-similar objects are called fractals, for which we present several examples and define a notion of dimension in many differentcontexts. We also present a strong version of Roth's theorem for algebraic points on a variety approximated by fractal elements. Fractals provide a framework in which one can unite several important conjectures in Diophantine geometry. 30/04/2004 PDF The residues of quantum field theory - numbers we should know M/04/17 Voir plus > Voir moins > We discuss in an introductory manner structural similarities between the polylogarithm and Green functions in quantum field theory. 20/04/2004 PDF Noncentral extension of the$AdS_{5} \times S^{5}$superalgebra : supermultiplet of brane charges P/04/15 Voir plus > Voir moins > We propose an extension of the su(2, 2|4) superalgebra to incorporate the F1/D1 string charges in type IIB string theory on the$AdS_{5} \times S^{5}$background, or the electro-magnetic charges in the dual super Yang-Mills theory. With the charges introduced, the superalgebra inevitably undergoes a noncentral extension, as noted recently in [1]. After developing a group theoretical method of obtaining the noncentral extension, we show that the charges form a certain nonunitary representation of the original unextended superalgebra, subject to some constraints. We solve the constraints completely and show that, apart from the su(2,2|4) generators, there exist 899 complex brane charges in the extended algebra. Explicitly we present all the super-commutation relations among them. 18/04/2004 PDF D-brane charges on SO(3) P/04/14 Voir plus > Voir moins > In this letter we discuss charges of D-branes on the group manifold SO(3). Our discussion will be based on a conformal field theory analysis of boundary states in a Z_2-orbifold of SU(2). This orbifold differs from the one recently discussed by Gaberdieland Gannon in its action on the fermions and leads to a drastically different charge group. We shall consider maximally symmetric branes as well as branes with less symmetry, and find perfect agreement with a recent computation of the corresponding K-theory groups. 12/04/2004 PDF Weak Bézout inequality for D-modules M/04/16 Voir plus > Voir moins > A bound is obtained on the leading coefficient of the Hilbert-Kolchin polynomial of a D-module in terms of the degrees of its generators. 10/04/2004 PDF The Hopf algebra of rooted trees in Epstein-Glaser Renormalization P/04/12 Voir plus > Voir moins > We show how the Hopf algebra of rooted trees, in a somewhat modified presentation, encodes the combinatorics of Epstein-Glaser renormalization and position space renormalization in general. In particular we prove that the Epstein-Glaser time-ordered products can be obtained from the Hopf algebra by suitable Feynman rules, mapping trees to operator-valued distributions, and by twisting the antipode with a renormalization character, which formally solves the Bogoliubov recursion and provides local counterterms due to the Hochschild 1-closedness of the grafting operator$B_+$. 01/04/2004 PDF S-duality and Topological Strings P/04/09 Voir plus > Voir moins > In this paper we show how S-duality of type IIB superstrings leads to an S-duality relating A and B model topological strings on the same Calabi-Yau as had been conjectured recently: D-instantons of the B-model correspond to A-model perturbative amplitudes and D-instantons of the A-model capture perturbative B-model amplitudes. Moreover this confirms the existence of new branes in the two models. As an application we explain the recent results concerning A-model topological strings on Calabi-Yau and its equivalence to the statistical mechanical model of melting crystal. 31/03/2004 PDF Lusternik-Schnirelman theory and dynamics, II M/04/13 30/03/2004 PDF An introduction to arithmetic groups M/04/11 Voir plus > Voir moins > Arithmetic groups are groups of matrices with integral entries. We shall first discuss their origin in number theory (Gauss, Minkowski) and their role in the "reduction theory of quadratic forms". Then we shall describe these groups by generators and relations. The next topic will be: are all subgroups of finite index given by congruence conditions? Finally, we shall discuss rigidity properties of arithmetic groups. 20/03/2004 PDF Integrable Renormalization II: the general case P/04/08 Voir plus > Voir moins > We extend the results we obtained in an earlier. The cocommutative case of rooted ladder graphs is generalized to a full Hopf algebra of (decorated) rooted trees. For Hopf algebra characters with target space of Rota-Baxter type, the Birkhoff decomposition of renormalization theory is derived by using the Rota-Baxter double construction, respectively Atkinson's theorem. We also outline the extension to the Hopf algebra of Feynman graphs via decorated rooted trees. 17/03/2004 PDF Torsion cohomology classes and algebraic cycles on complex projective manifolds M/04/10 Voir plus > Voir moins > Atiyah and Hirzebruch gave examples of even degree torsion classes in the singular cohomology of a smooth complex projective manifold, which are not Poincaré dual to an algebraic cycle. We notice that the order of these classes are small compared to the dimension of the manifold. However, building upon a construction of Kollár, one can provide such examples with arbitrary high prime order, the dimension being fixed. This method also provides examples of torsion algebraic cycles, which are non trivial in the Griffiths' groups, and lie in a arbitrary high level of the H.Saito filtration on Chow groups. 10/03/2004 PDF Systèmes de Taylor-Wiles pour${\rm GSp}_4$M/04/07 Voir plus > Voir moins > On montre qu'une représentation galoisienne symplectique de degré quatre congrue modulo p à une représentation galoisienne provenant d'une forme de Siegel de genre deux provient elle-même d'une telle forme de Siegel. On suppose pour cela que l'image de Galois est grosse modulo p et que des hypothèses locales sont satisfaites (minimalité en dehors de p et ordinarité cristalline en p, de poids de Hodge-Tate adéquats). On utilise pour cela la technique des systèmes de Taylor-Wiles. 07/03/2004 PDF Classical/quantum integrability in AdS/CFT P/04/05 07/03/2004 PDF An inverse theorem for the restricted set addition in Abelian groups M/04/06 Voir plus > Voir moins > Let$A$be a set of$k\ge 5$elements of an Abelian group$G$in which the order of the smallest nontrivial subgroup is larger than$2k-3$. Extending a result of Dias da Silva and Hamidoune (Bull. London Math. Soc. 26 (1994) 140-146), in a recent paperwe proved that the number of different elements of$G$that can be written in the form$a+a'$, where$a, a'\in A$,$a\ne a'$, is at least$2k-3$. Here we prove that the bound is attained if and only if the elements of$A$form an arithmetic progression in$G$, thus completing the solution of a problem of Erd\H os and Heilbronn. The proof is based on the so-called Combinatorial Nullstellensatz'. 25/02/2004 PDF Spectral Asymmetry, Zeta Functions and the Noncommutative Residue M/04/02 Voir plus > Voir moins > In this paper, motivated by an approach developped by Wodzicki, we look at the spectral asymmetry of elliptic PsiDO's in terms of theirs zeta functions. First, using asymmetry formulas of Wodzicki we study the spectral asymmetry of odd elliptic PsiDO'sand of geometric Dirac operators. In particular, we show that the eta function of a selfadjoint elliptic odd PsiDO is regular at every integer point when the dimension and the order have opposite parities (this generalizes a well known result of Branson-Gilkey for Dirac operators), and we relate the spectral asymmetry of a Dirac operator on a Clifford bundle to the Riemmanian geometric data, which yields a new spectral interpretation of the Einstein action from gravity. We also obtain a large class of examples of elliptic PsiDO's for which the regular values at the origin of the (local) zeta functions can easily be seen to be independent of the spectral cut. On the other hand, we simplify the proofs of two well-known and difficult results of Wodzicki: (i) The independence with respect to the spectral cut of the regular value at the origin of the zeta function of an elliptic PsiDO; (ii) The vanishing of the noncommutative residue of a zero'th order PsiDO projector. These results were proved by Wodzickiusing a quite difficult and involved characterization of local invariants of spectral asymmetry, which we can bypass here. Finally, in an appendix we give a new proof of the aforementioned asymmetry formulas of Wodzicki. 30/01/2004 PDF The entropy of black holes: a primer P/04/04 Voir plus > Voir moins > After recalling the definition of black holes, and reviewing their energetics and their classical thermodynamics, one expounds the conjecture of Bekenstein, attributing an entropy to black holes, and the calculation by Hawking of the semi-classical radiation spectrum of a black hole, involving a thermal (Planckian) factor. One then discusses the attempts to interpret the black-hole entropy as the logarithm of the number of quantum micro-states of a macroscopic black hole, with particular emphasis on results obtained within string theory. After mentioning the (technically cleaner, but conceptually more intricate) case of supersymmetric (BPS) black holes and the corresponding counting of the degeneracy of Dirichlet-brane systems, one discusses in some detail the "correspondence" between massive string states and non-supersymmetric Schwarzschild black holes. 21/01/2004 PDF Cayley-Hamilton Decomposition and Spectral Asymmetry M/04/01 Voir plus > Voir moins > In this paper we derive Cayley-Hamilton decompositions, along some of their consequences, for compact operators and closed operators with compact resolvent on a (separable) Hilbert space. In particular, we make use these decompositions to give a spectralinterpretation of a projector found by Wodzicki to encode the spectral asymmetry of elliptic PsiDO's on a compact manifold. As another application we get a convenient definition of the partial inverse of a closed operator with compact resolvent. Finally,we work out the results of this paper in the examples of an elliptic PsiDO on a compact manifold and of an elliptic PsiDO on a spectral triple (i.e. on a noncommutative manifold in the sense of Connes's noncommutative geometry). 20/01/2004 PDF Unraveling the Fourier Law for Hamiltonian Systems P/04/03 Voir plus > Voir moins > We exhibit simple Hamiltonian and stochastic models of heat transport in non-equilibrium problems. Theoretical arguments are given to show that, for a wide class of models, the temperature profile obeys a universal law depending on a parameter$\alpha $. When$\alpha=1$, the law is linear, but, depending on the nature of the energy exchange mechanism by tracer particles, we find that$\alpha $is, in many cases, different from$1$. When$\alpha \neq 1\$, the temperature profile is not linear, although translation invariance and, in some models, local thermal equilibrium, hold.

14/01/2004 PDF