Almost 40 years ago, Stanley noticed that some of the deep theorems of algebraic geometry have powerful combinatorial applications. Among other things, he used the hard Lefschetz theorem to rederive Dynkin’s theorem, and to characterize face numbers of simplicial polytopes.
Since then, several more deep combinatorial and geometric problems were discovered to be related to theorems surrounding the Lefschetz theorem. One first constructs a ring metaphorically modelling the combinatorial problem at hand, often modelled on constructions for toric varieties, and then tries to derive the combinatorial result using deep results in algebraic geometry. For instance:
– a Lefschetz property for implies that a simplicial complex PL-
embedded in R4 cannot have more triangles than four times the number of it’s edges. (Kalai/A.)
– a Hodge-Riemann type property implies the log-concavity of the coefficients of the chromatic polynomial. (Huh)
– a decomposition type property implies the positivity of the
Kazhdan-Lusztig polynomial. (Elias-Williamson)
At this point one can then hope that indeed, algebraic geometry provides the answer, which is often only the case in very special cases, when there is a sufficiently nice variety behind the metaphor.
It is at this point that purely combinatorial techniques can be attempted to prove the desired. This is the modern approach to the problem, and I will discuss the two main approaches used in this area: Firstly, an idea of Peter McMullen, based on local modifications of the ring and control of the signature of the intersection form. Second, an approach based on a theorem of Hall, using the observation that spaces of low-rank linear maps are of special form.

Séminaire « Equations différentielles »

To a large extent the weight 3 motives attached to one-parameter families of Calabi-Yau threefolds can be studied from the associated Picard-Fuchs operator alone. I review how this can be used to find Calabi-Yau threefolds associated with modular forms and give examples with elliptic modular forms, Bianchi modular forms and Hilbert modular forms.

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IHES Covid-19 regulations:

– all the participants who will attend the event in person will have to keep their mask on in indoor spaces
and where the social distancing is not possible;
– speakers will be free to wear their mask or not at the moment of their talk if they feel more comfortable
without it;
– Up to 70 persons in the conference room, every participant will be asked to be able to provide a health pass
– Over 70 persons in the conference room, every participant will be asked to provide a health pass which will
be checked at the entrance of the conference room.

==================================================================

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.

Conformal field theory is a vast subject intensively studied in theoretical physics since the 80s. In this talk I will explain how one can use probabilistic methods, analytic methods and tools from Teichmüller spaces and the geometry of Riemann surfaces to construct rigorously (in the mathematical sense) an important conformal field theory in dimension 2, called the Liouville conformal field theory. This theory is a theory of random Riemannian metrics on surfaces and its correlation functions can be computed explicitly and decomposed into two quantities: the so-called structure constant (the 3 point function on the sphere) and the Virasoro conformal blocks. The conformal blocks are holomorphic functions of the moduli of surfaces linked to the representation theory of the Virasoro algebra.

This is based on joint works with Kupiainen, Rhodes and Vargas, and an ongoing work with the same authors together with Baverez.

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IHES Covid-19 regulations:

– all the participants who will attend the event in person will have to keep their mask on in indoor spaces
and where the social distancing is not possible;
– speakers will be free to wear their mask or not at the moment of their talk if they feel more comfortable
without it;
– Up to 25 persons in the conference room, every participant will be asked to be able to provide a health pass
– Over 25 persons in the conference room, every participant will be asked to provide a health pass which will
be checked at the entrance of the conference room.

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A metric space is called injective if any family of pairwise intersecting balls has a non-empty intersection. Injective metric spaces enjoy many properties typical of nonpositive curvature. In particular, when a group acts by isometries on such a space, we will review the many consequences this has. We will also present numerous groups admitting an interesting action on an injective metric space, such as hyperbolic groups, cubulable groups, lattices in Lie groups, mapping class groups, some Artin groups…

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IHES Covid-19 regulations:

– all the participants who will attend the event in person will have to keep their mask on in indoor spaces
and where the social distancing is not possible;
– speakers will be free to wear their mask or not at the moment of their talk if they feel more comfortable
without it;
– Up to 25 persons in the conference room, every participant will be asked to be able to provide a health pass
– Over 25 persons in the conference room, every participant will be asked to provide a health pass which will
be checked at the entrance of the conference room.

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Workshop on Quantum Geometry

The main topic of this conference will be some recent progress in enumerative geometry, emphasizing its connection with mathematical physics and quantization. There will be a mini-series of lectures by Pierrick Bousseau and Hülya Argüz on their recent work (also joint with Pandharipande-Zvonkine), together with research seminars by other early-career speakers working on similar topics.

This event will be held at IHES, Bures-sur-Yvette, and is open to all interested researchers. Ph.D. students and postdoctoral fellows are especially welcome to participate.

 

Registration is free and open until April 21, 2022.

Note that the conferences will take place from 2:00 pm to 6:00 pm every day.

Organised by : Veronica Fantini (IHES) & Alex Takeda (IHES)

Invited speakers:

Nezhla Aghaei (SDU/QM Center)
Hülya Argüz (IST Austria)
Anna Barbieri (University of Milano Statale)
Pierrick Bousseau (ETH Zurich)
Pierre Descombes (Sorbonne Université UPMC)
Maxime Fairon (University of Glasgow)
Elba Garcia-Failde (IMJ-PRG)
Alessandro Giacchetto (IPhT)
Oscar Kivinen (EPFL)
Joshua Lam (IHES)
Mingkun Liu (IMJ-PRG)
Riccardo Ontani (SISSA)
Gabriele Rembado (Hausdorff Centre for Mathematics, Bonn)
Alexander Soibelman (IHES)

The Coon amplitude is a deformation of the Veneziano amplitude with logarithmic Regge trajectories and an accumulation point in the spectrum, which interpolates between string theory and field theory. With string theory, it is the only other solution to duality constraints explicitly known and it constitutes an important data point in the modern S-matrix bootstrap. Yet, its basics properties are essentially unknown. In this paper we fill this gap and derive the conditions of positivity and the low energy expansion of the amplitude. On the positivity side, we discover that the amplitude switches from a regime where it is positive in all dimensions to a regime with critical dimensions, that connects to the known d = 26, 10 when the deformation is removed. En passant, we find that the Veneziano amplitudes can be extended to massive scalars of masses up to m^2 = 1/3, where it has critical dimension 6.3. On the low-energy side, we compute the first few couplings of the theory in terms of q-deformed analogues of the standard Riemann zeta values of the string expansion. We locate their location in the EFT-hedron, and find agreement with a recent conjecture that theories with accumulation points populate this space. We also discuss their relation to low spin dominance.

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IHES Covid-19 regulations:

– all the participants who will attend the event in person will have to keep their mask on in indoor spaces
and where the social distancing is not possible;
– speakers will be free to wear their mask or not at the moment of their talk if they feel more comfortable
without it;
– Up to 25 persons in the conference room, every participant will be asked to be able to provide a health pass
– Over 25 persons in the conference room, every participant will be asked to provide a health pass which will
be checked at the entrance of the conference room.

==================================================================

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 quantum_encounters_seminar PRENOM NOM »
(indiquez vos propres prénom et nom) et laissez le corps du message vide.

I will talk about a new type of critical quantum liquids, dubbed Stiefel liquids, that can emerge in quantum magnets. Our theory is based on 2+1 dimensional nonlinear sigma models on target space SO(N)/SO(4), supplemented with Wess-Zumino-Witten terms. We argue that the Stiefel liquids form a class of 3d CFTs with extraordinary properties, such as large emergent symmetries, a cascade structure, and nontrivial quantum anomalies. We show that the well known deconfined quantum critical point and U(1) Dirac spin liquid (i.e. Nf=4 QED3) are unified as two special examples of Stiefel liquids, with N=5 and N=6, respectively. Furthermore, we conjecture that Stiefel liquids with N>6 are non-Lagrangian, in the sense that under renormalization group they flow to infrared (conformally invariant) fixed points that cannot be described by any renormalizable continuum Lagrangian. I will also discuss a physical way to realize Stiefel liquids (both the Dirac spin liquid and N=7 non-Lagrangian Stiefel liquid) in quantum spin systems, for example, on triangular or kagome lattice, through the intertwinement of symmetry breaking orders.

Participer à la réunion Zoom
https://us02web.zoom.us/j/82757702598?pwd=RnBJZWRZWFIvd3lGU0ZRMG9OUEVhQT09

ID de réunion : 827 5770 2598
Code secret : 498440

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IHES Covid-19 regulations:

– all the participants who will attend the event in person will have to keep their mask on in indoor spaces
and where the social distancing is not possible;
– speakers will be free to wear their mask or not at the moment of their talk if they feel more comfortable
without it;
– Up to 70 persons in the conference room, every participant will be asked to be able to provide a health pass
– Over 70 persons in the conference room, every participant will be asked to provide a health pass which will
be checked at the entrance of the conference room.

==================================================================

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 quantum_encounters_seminar PRENOM NOM »
(indiquez vos propres prénom et nom) et laissez le corps du message vide.

Séminaire « Equations différentielles »

In 1896 Frobenius and Fricke published two seemingly unrelated papers: Frobenius started to develop his theory of k-characters for finite groups motivated by Dedekind’s question about factorisation of the group determinant, while Fricke followed Klein’s approach to the uniformization theorem. I will explain that in fact these two works can be naturally linked and both are related to remarkable Markov’s paper of 1880 on arithmetic of binary quadratic forms. The talk is based on a joint work with V.M. Buchstaber.

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IHES Covid-19 regulations:

– all the participants who will attend the event in person will have to keep their mask on in indoor spaces
and where the social distancing is not possible;
– speakers will be free to wear their mask or not at the moment of their talk if they feel more comfortable
without it;
– Up to 70 persons in the conference room, every participant will be asked to be able to provide a health pass
– Over 70 persons in the conference room, every participant will be asked to provide a health pass which will
be checked at the entrance of the conference room.

==================================================================

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 « Equations différentielles »

Some standard differential equations with irregular singularities, like Airy or Kloosterman and their symmetric products, behave in a way similar to Gauss-Manin differential equations , and their de Rham cohomology underlie a mixed Hodge structure, possibly with finite monodromy, enabling the use of tame arithmetic methods to handle the associated exponential sums. The talk will mainly focus on the Airy case, after a joint work with Jeng-Daw Yu.

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IHES Covid-19 regulations:

– all the participants who will attend the event in person will have to keep their mask on in indoor spaces
and where the social distancing is not possible;
– speakers will be free to wear their mask or not at the moment of their talk if they feel more comfortable
without it;
– Up to 70 persons in the conference room, every participant will be asked to be able to provide a health pass
– Over 70 persons in the conference room, every participant will be asked to provide a health pass which will
be checked at the entrance of the conference room.

==================================================================

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 « Equations différentielles »

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IHES Covid-19 regulations:

– all the participants who will attend the event in person will have to keep their mask on in indoor spaces
and where the social distancing is not possible;
– speakers will be free to wear their mask or not at the moment of their talk if they feel more comfortable
without it;
– Up to 25 persons in the conference room, every participant will be asked to be able to provide a health pass
– Over 25 persons in the conference room, every participant will be asked to provide a health pass which will
be checked at the entrance of the conference room.

==================================================================

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.

Quantum representations form a family of representations of modular groups of surfaces with values in projective pseudo-unitary groups PU(p,q), sending Dehn twists to finite-order elements. Toledo invariants of these representations, and more general characteristic classes, extend to cohomology classes defined on the Deligne-Mumford compactification of moduli spaces, defining cohomological field theories (CohFT). We will give explicit formulae for the Toledo part of the latter in some cases, including Fibonacci quantum representations. This allows to construct/recover complex hyperbolic structures on some moduli spaces. Work in progress with Julien Marché.

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IHES Covid-19 regulations:

– all the participants who will attend the event in person will have to keep their mask on in indoor spaces
and where the social distancing is not possible;
– speakers will be free to wear their mask or not at the moment of their talk if they feel more comfortable
without it;
– Up to 25 persons in the conference room, every participant will be asked to be able to provide a health pass
– Over 25 persons in the conference room, every participant will be asked to provide a health pass which will
be checked at the entrance of the conference room.

==================================================================

Positivity is meant as a generalisation of the cyclic order on the circle. Associated to that is the notion of monotone maps from a cyclically ordered set in the circle.

Generalisations of the idea of positivity appeared to be crucial in understanding some connected components of the space of representations of a surface group in a Lie group G, although the common phenomenon was not figured out until recently.

In this talk, based on a preprint with Olivier Guichard and Anna Wienhard, I will start by examples generalizing this notion of cyclic order on the circle: convex curves or configurations in the plane, time like curve in Minkowski space. Then I will move to the general geometry of parabolic spaces and explain why the notion of positivity relates to special configurations of pairwise transverse triples and quadruples of points. This notion of positivity, which abides simple combinatorial properties, allows to define positive — or monotone — curves, then positive representations of surface groups.

I will then sketch the proof of our main result: positive representations are Anosov and fill up connected components of the space of representations.

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IHES Covid-19 regulations:

– all the participants who will attend the event in person will have to keep their mask on in indoor spaces
and where the social distancing is not possible;
– speakers will be free to wear their mask or not at the moment of their talk if they feel more comfortable
without it;
– Up to 25 persons in the conference room, every participant will be asked to be able to provide a health pass
– Over 25 persons in the conference room, every participant will be asked to provide a health pass which will
be checked at the entrance of the conference room.

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