Combinatorics and Arithmetic for Physics: special daysTenth Anniversary EditionThe meeting focuses on questions of discrete mathematics and number theory, emphasizing computability. Problems are drawn mainly from theoretical physics: renormalization, combinatorial physics, geometry, evolution equations (commutative and noncommutative), or related to its models, but not only. Computations, based on combinatorial structures (graphs, trees, words, automata, semirings, bases), or classical structures (operators, Hopf algebras, evolution equations, special functions, categories) are good candidates for computer-based implementation and experimentation.Organized by: Gérard H. E. DUCHAMP, Maxim KONTSEVICH, Gleb KOSHEVOY, Sergei NECHAEV, and Karol A. PENSON.Speakers:Marek Bozejko (Institute of Mathematics Wroclaw University)Ricardo Buring (INRIA)Philippe Di Francesco (UIUC)Gérard H. E. Duchamp (LIPN, Université Paris Nord) Harold Erbin (CTP, MIT, USA)Stéphane Gaubert (INRIA, CMAP, École polytechnique)Volker Genz (IBS CGP)Darij Grinberg (Drexel University) Dimitry Gurevich (IITP, Moscou) Yuki Kanabuko (MPIM, Bonn)Rinat Kedem (UIUC)Maxim Kontsevich (IHES)Gleb Koshevoy (IITP, Moscow & IHES)Thomas Krajewski (CPT, Marseille)Marek Kus (Center for Theoretical Physics)Hiroaki Nakamura (Osaka University) Toshiki Nakashima (Sophia University Tokyo) Hadrien Notarantonio (Inria Saclay)Karol A. Penson (LPTMC, Sorbonne Université) Eric Pichon-Pharabod (Université Paris-Saclay)Sanjaye Ramgoolam (Queen Mary University of London)Travis Scrimshaw (Hokkaido University) Andrea Sportiello (LIPN, Université Paris Nord)Adrian Tanasa (Université de Bordeaux)Vasily Sazonov (CEA)Jean-Bernard Zuber (LPTHE, Sorbonne Université)Karol Życzkowski (Jagiellonian University)Sponsors: IHES – Math-STIC – LIPN (UMR-7030) – LPTMC (Univ-Paris 6) – INRIA – GDR EFI – CEAScientific Committee:Joseph Ben Geloun (LIPN-Paris XIII), Alin Bostan (INRIA), Marek Bozejko (Wroclaw University), Vincent Rivasseau (Orsay-CEA)

Séminaire informel sur les intersections atypiquesI will recall how understanding the geometry of jumping loci for algebraic cycles in families of smooth projective complex varieties can be reinterpreted as an unlikely intersection problem. I will then present joint work with David Urbanik using this point of view to give sufficient conditions for the analytic density of these loci in the base of the family. ========Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.

Period integrals are a fundamental concept in algebraic geometry and number theory. In this talk, we will study the notion of non-archimedean periods as introduced by Kontsevich and Soibelman.  We will give an overview of the non-archimedean SYZ program, which is a close analogue of the classical SYZ conjecture in mirror symmetry. Using the non-archimedean SYZ fibration, we will prove that non-archimedean periods recover the analytic periods for log Calabi-Yau surfaces, verifying a conjecture of Kontsevich and Soibelman. This is joint work with Jonathan Lai.  Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide. 

Séminaire informel sur les intersections atypiquesThis is part one of a two part lecture, the second of which will be given by Greg Baldi. In the first part we introduce a unified framework for studying « geometric » unlikely intersection problems, which in particular includes all such problems arising from (mixed) Hodge theory, and prove a general geometric Zilber-Pink theorem in this context, subsuming previous results of this nature. The proofs are also effective, in the sense that they give explicit algorithms to compute the relevant atypical loci. In the second part we will explain how this common framework also applies to characterise geometric unlikely intersection phenomena beyond the Hodge-theoretic setting, and in particular to orbit closures in strata of abelian differentials. ========Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.

Séminaire informel sur les intersections atypiquesThis is part two of a two part lecture, the first one of which  was  given by David Urbanik. In the first part we introduce a unified  framework for studying « geometric » unlikely intersection problems,  which in particular includes all such problems arising from (mixed)  Hodge theory, and prove a general geometric Zilber-Pink theorem in  this context, subsuming previous results of this nature. The proofs are also effective, in the sense that they give explicit algorithms to compute the relevant atypical loci. In the second part we will explain how this common framework also applies to characterise geometric unlikely intersection phenomena beyond the Hodge-theoretic setting, and in particular to orbit closures in strata of abelian differentials. ========Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.

Lattices in semi-simple Lie groups of rank at least 2 — e.g. SL(n,Z) for n>2 — form a class of discrete groups known for having remarkable linear rigidity properties. Notably, their finite dimensional representations are determined by those of the ambient Lie group they live in — e.g. SL(n,R) in the case of SL(n,Z). This is Margulis’ super-rigidity theorem (1974). Motivated by an ergodic version of this theorem, an ambitious program initiated by Gromov and Zimmer in the 1980s aims to understand « non-linear representations » of such lattices into diffeomorphism groups of closed manifolds, or in other words, the differentiable actions of such lattices on closed manifolds.I will first discuss the history and geometric origins of this program. I will then focus on rigidity results about actions of lattices which preserve non-unimodular geometric structures, such as conformal or projective structures, and will mention open directions. The proofs build on recent advances on Zimmer’s conjectures, especially an invariance principle which provides existence of finite invariant measures in various dynamical contexts.

Approximate lattices are discrete subsets of locally compact groups that are an aperiodic generalisation of lattices. They are defined as approximate subgroups (i.e. subsets that are closed under multiplication up to a finite multiplicative error) that are discrete and have finite co-volume. They were first studied by Yves Meyer who classified them in locally compact abelian groups by means of the so-called « cut-and-project schemes ». Approximate lattices were subsequently used to model a diversity of objects such as aperiodic tilings (Penrose and the « hat »), Pisot numbers, and quasi-crystals.In non-abelian groups, however, their structure remained mysterious. I will explain how the structure of approximate lattices in linear algebraic groups can be understood thanks to a notion of cohomology that sits halfway between bounded cohomology and the usual cohomology, thus generalising Meyer’s theorem. Along the way, we will talk about Pisot numbers, extending a theorem of Lubotzky, Mozes and Raghunathan, amenability and (some) model theory.

Séminaire informel sur les intersections atypiquesI will present some recent results obtained with  Andrei Yafaev on André-Pink-Zannier and generalised Hecke orbits in Shimura varieties, such as moduli spaces of abelian varieties. ========Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.

Probability and analysis informal seminarWe discuss Langevin dynamics of N particles on Rd interacting through a singular repulsive potential, such as the Lennard-Jones potential, and show that the system converges to the unique invariant Gibbs measure exponentially fast in a weighted total variation distance.  The proof relies on an explicit construction of a Lyapunov function using a modified Gamma calculus (Bakry-Emery).  In contrast to previous results for such systems, our results imply geometric convergence to equilibrium starting from an essentially optimal family of initial distributions. This is based on joint work with F.Baudoin and D.Herzog.========Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.

Probability and analysis informal seminarThere exists several ways to discretize holomorphic functions. One of them is based on Schramm’s orthogonal circle patterns, and their generalization to so-called « cross-ratio maps » and « P-nets ». These systems are naturally associated with a discrete time dynamics. I will mention results and open problems about this dynamics, in particular the « Devron » property, that states that singularities cannot be escaped by reversing time. I will show that these questions can be tackled by identifying those (birational) dynamics with the dSKP equation, which itself can be identified with partition functions of (oriented) dimers, a famously integrable model of statistical mechanics. Based on joint works with Niklas Affolter, Béatrice de Tilière, Jean-Baptiste Stiegler.========Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.

Séminaire informel sur les intersections atypiquesIn joint work with Jacob Tsimerman we study when the primitive of a given algebraic function can be constructed using primitives from some given finite set of algebraic functions, their inverses, algebraic functions, and composition. When the given finite set is just {1/x} this is the classical problem of « elementary integrability » (of algebraic functions). I will discuss some results, including a decision procedure for this question, and further problems and conjectures. ========Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.

Probability and analysis informal seminarThe Parisi formula is a fundamental result in spin glass theory. It gives a variational characterization of the asymptotic  limit of the expected free energy. The upper bound is a consequence of an interpolation identity due to F. Guerra and the lower bound is a celebrated result of M. Talagrand. In this talk I will present a new approach to (an enhanced version of) Guerra’s identity using stochastic analysis, more specifically Brownian motion and Ito’s calculus. This approach is suggested by the form of the Parisi formula in which the solution of a Hamilton-Jacobi equation is involved. It helps in many ways to illuminate the original method of Guerra and suggests some possible approaches to the significantly deeper lower bound, which has been intensively studied since Talagrand’s work. Among the techniques from stochastic analysis we will use include path space integration by parts for the Wiener measure, Girsanov’s transform (i.e., exponential martingales), and probabilistic representation of solutions to (linear) partial differential equations. The key observation is that the nonlinear Hamilton-Jacobi partial differentiation equation figuring in Parisi’s variation formula becomes linear after differentiating with respect to Guerra’s interpolation parameter, thus bringing the full strength of stochastic analysis based on Ito’s calculus into play. It is hoped that this approach will shed some lights on the much more difficult lower bound in the Parisi formula. ========Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide.