I will describe the proof of the crossing property for the 4-point function.

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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

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Séminaire « Equations différentielles »

The Mahler measure of multivariate polynomials appears in several fields of mathematics including number theory. In particular there are deep conjectural links between Mahler measures of integer polynomials and special values of L-functions. The first theorem in this direction is due to Smyth (1981), who computed the Mahler measure of the polynomial 1+x+y in terms of the L-function of the Dirichlet character of conductor 3. This can be generalised to a class of 2-variable polynomials called exact polynomials. I will explain work in progress with Riccardo Pengo, where we introduce the degree of exactness of a polynomial P(x_1,…,x_n). Under certain conditions, the Mahler measure of P can then be interpreted as a Deligne period on a subvariety of the hypersurface P=0, whose codimension is equal to the exactness degree of P.

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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.

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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 »
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In 2006, Chapoton defined a class of Tamari intervals called « new intervals » in his enumeration of Tamari intervals, and he found that these new intervals are equi-enumerated with bipartite planar maps. We present here a direct bijection between these two classes of objects using a new object called « degree tree ». Our bijection also gives an intuitive proof of an unpublished equi-distribution result of some statistics on new intervals given by Chapoton and Fusy.

A long-standing problem proposed by David Aldous consists in giving a sharp upper bound for the mixing time of the so-called “triangulation walk”, a Markov chain defined on the set of all possible triangulations of the regular n-gon. A single step of the chain consists in performing a random edge flip, i.e. in choosing an (internal) edge of the triangulation uniformly at random and, with probability 1/2, replacing it with the other diagonal of the quadrilateral formed by the two triangles adjacent to the edge in question (with probability 1/2, the triangulation is left unchanged).
While it has been shown that the relaxation time for the triangulation walk is polynomial in n and bounded below by a multiple of n^{3/2}, the conjectured sharpness of the lower bound remains firmly out of reach in spite of the apparent simplicity of the chain. For edge flip chains on different models — such as planar maps, quadrangulations of the sphere, lattice triangulations and other geometric graphs — even less is known.
We shall discuss results concerning the mixing time of random edge flips on rooted quadrangulations of the sphere, partly obtained in joint work with Alexandre Stauffer. A “growth scheme” for quadrangulations which generates a uniform quadrangulation of the sphere by adding faces one at a time at appropriate random locations can be combined with careful combinatorial constructions to build probabilistic canonical paths in a relatively novel way. This method has immediate implications for a range of interesting edge-manipulating Markov chains on so-called Catalan structures, from “leaf translations” on plane trees to “edge rotations” on general planar maps. Moreover, we are able to apply it to flips on 2p-angulations and simple triangulation of the sphere, via newly developed “growth schemes”.

Une carte serrée est une carte avec des sommets marqués, telle que tous ses sommets de degré 1 sont marqués. Pour un jeu donné de faces étiquetées de 1 à n de degrés prescrits (et en interprétant les sommets marqués comme des faces de degré 0), le nombre de cartes serrées distinctes que l’on peut construire sur la sphère est, comme l’a montré Norbury, un quasi-polynôme de degré n-3 dans les carrés des degrés des faces. Je montrerai comment obtenir la formule explicite de ce quasi-polynôme de manière purement bijective par une décomposition des cartes serrées planaires en tranches (« slices »). Je montrerai enfin comment en déduire une extension de la « formule des slicings » de Tutte (1962) au cas de cartes planaires ayant un nombre arbitraire de faces de degrés impairs.

(avec V. Delecroix, E. Goujard et P. Zograf)

Nous commencerons par le comptage de graphes à rubans, fait par M. Kontsevich et P. Norbury, et par le comptage de multi-géodésiques fermées simples sur les surfaces hyperboliques, effectué par M. Mirzakhani. En utilisant ces résultats, nous allons décrire la structure d’une multi-géodésique typique sur une surface de grand genre. Cette description est basée en partie sur les formules asymptotiques pour les corrélateurs de Witten-Kontsevich et pour les volumes de Masur-Veech de l’espace de modules de différentielles quadratiques en grand genre (les deux notions seront expliquées). Ces dernières formules, conjecturées par Delecroix-Goujard-Zograf-Zorich, ont été récemment démontrées par Amol Aggarwal.

We will discuss topological and ergodic aspects of the horocyclic flow on families of examples. This is joint work with Fernando Alcalde Cuesta.

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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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The Selberg trace formula provides a link between the length spectrum and the Laplacian spectrum of a hyperbolic surface. I will speak about a joint project with Maxime Fortier Bourque in which we are using this formula to probe extremal problems in hyperbolic geometry. These are questions of the form: what is the hyperbolic surface of a given genus with the largest kissing number or the largest spectral gap? Concretely, I will explain the general principle of our method, which is inspired by ideas from the world of Euclidean sphere packings. Moreover, I will explain why the Klein quartic, the most symmetric Riemann surface in genus 3, solves one of our extremal problems.

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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.

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

Séminaire « Equations différentielles »

Gaudin model is the simplest case of the quantum Hitchin system corresponding to the rational curve with marked points. Algebraic Bethe ansatz provides a general way to solve the eigenproblem of this quantum integrable system, it expresses the eigenvectors and the eigenvalues as some explicit functions on the auxiliary parameters which satisfy  Bethe ansatz equations. The solutions of Bethe ansatz equations depend on the parameters of the integrable system, and this dependence is highly multivalued. We describe the monodromy of solutions of the Bethe ansatz equations and relate it to the RSK combinatorics of Young tableaux.

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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.

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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 »
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Séminaire « Equations différentielles »

The aim of this talk is to give an overview on Frobenius Manifolds and its different facets. Frobenius manifolds arose in the process of axiomatisation of Topological Field Theory (TFT) and play a central role in the mathematical vision of mirror symmetry. However, recent results allow to go beyond this panorama.  In particular, we will discuss the existence of a new source of Frobenius Manifolds, which shows an unexpected bridge between algebraic geometry—involving moduli spaces of curves, Gromov—Witten like invariants, Saito’s space—and statistical manifolds, coming from information geometry.

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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.

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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 Laurent Schwartz — EDP et applications

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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.

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

Séminaire Laurent Schwartz — EDP et applications

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

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.

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