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.

There is a natural dichotomy between telescopic(T(n)-local) and chromatic (K(n)-local) homotopy theory. Telescopic homotopy theory is more closely tied to the stable homotopy groups of spheres and through them to geometric questions, but is generally computationally intractable. Chromatic homotopy theory is more closely tied to arithmetic geometry and powerful computational tools exist in this setting. Ravenel’s telescope conjecture asserted that these two sides coincide. I will present a family of counterexamples to this conjecture based on using trace methods to analyze the algebraic K-theory of a family of K(n)-local ring spectra beginning with the K(1)-local sphere. As a consequence of this we obtain a new lower bound on the average rank of the stable homotopy groups of spheres. I will then describe the étale fundamental group of the T(n)-local sphere and how this informs our understanding of telescopic homotopy theory. This talk is based on projects joint with Carmeli, Clausen, Hahn, Levy, Schlank and Yanovski. 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. 

11e séminaire ITZYKSON :Dénombrement de cartes : entre combinatoire, probabilités et physique théoriqueLe 11e séminaire Itzykson est organisé par Maxim Kontsevich (IHES), Sylvain Ribault (IPhT Saclay) et Pierre Vanhove (IPhT Saclay). Depuis une dizaine d’années l’axe math-physique de la Fondation Mathématique Jacques Hadamard (FMJH) organise un séminaire Itzykson tous les ans à l’IHES. Il s’agit d’une journée consacrée à un thème de physique mathématique, avec un cours en français et deux ou trois exposés spécialisés en français ou en anglais.Les cartes – surfaces obtenues par recollement de polygones le long de leurs arêtes – intéressent depuis des décennies différents domaines des mathématiques, de l’informatique et de la physique. En particulier, si le recollement est aléatoire, on obtient des cartes aléatoires, qui permettent de décrire des processus stochastiques. Si de plus on décore des cartes aléatoires, on peut décrire des modèles de physique statistique comme le modèle de boucles O(n).Durant cette journée seront présentés divers aspects des cartes, des problèmes de dénombrement aux applications physiques, des idées fondamentales aux développements récents.Un cours et deux exposés auront lieu dans la journée, présentés par : Mireille Bousquet-Mélou, CNRS, LaBRI, Université de BordeauxIgor Kortchemski, CNRS, École polytechnique & ETH ZurichJérémie Bouttier, IMJ-PRG, Sorbonne UniversitéL’inscription est gratuite mais nécessaire et sera possible jusqu’au 13 novembre 2023. Un buffet-déjeuner sera offert aux participants qui s’y seront inscrits. Le séminaire sera filmé et diffusé en différé sur la chaîne YouTube de l’IHES.

French Japanese Conference on Probability & Interactions6-8 March 2024at IHES – Marilyn and James Simons Conference CenterHow to get to IHESThe conference aims at gathering French and Japanese researchers sharing common interests in probability theory related to physical phenomena. Various themes will be considered such as random matrices, stochastic PDEs, particle systems, models in disordered media. Although these domains are represented by different communities, the conference will be the occasion to analyze the connections that occur between those different thematics and to strengthen the collaborations between researchers of both countries.Speakers:Ismael Bailleul, Université de Bretagne OccidentaleQuentin Berger, Sorbonne UniversitéMireille Capitaine, Institut de Mathématiques de ToulouseNicolas Curien, Université Paris-SaclayNizar Demni, Aix-Marseille UniversitéClément Erignoux, Inria Lille Nord-Europe et Université de Lyon 1Masato Hoshino, Osaka UniversityTakashi Imamura (Chiba University)Naotaka Kajino, Kyoto UniversityMylène Maïda, Université de LilleKirone Mallick, CEA, IPhTShuta Nakajima, Meiji UniversityIzumi Okada, Chiba UniversityAkira Sakai, Hokkaido UniversityTomoyuki Shirai, Kyushu UniversityRyokichi Tanaka, Kyoto UniversityMilica Tomasevic, CNRS, École polytechniqueCristina Toninelli, CNRS, Université Paris Dauphine – PSLKenkichi Tsunoda, Kyushu UniversityJulien Vovelle, CNRS, ENS LyonScientific committee:Charles Bordenave (CNRS, Aix-Marseille Université), Benoît Collins (Kyoto University), Arnaud Debussche (ENS Rennes), Takashi Kumagai (Waseda University), Grégory Miermont (ENS Lyon), Tomohiro Sasamoto (Tokyo Institute of Technology).Organizing committee: Anne de Bouard (CNRS, École polytechnique), Thierry Bodineau (CNRS, IHES), Reika Fukuizumi (Waseda University).

An extensive mathematical model for HIV-HCV co-infection is developed. The positivity and boundedness of the model under investigation is established using well-known theorems. The next generation matrix method is used to construct the basic reproduction number for the model. The local and global stabilities of the model are shown using the linearization and Lyapunov function approaches, respectively. Bifurcation analysis and sensitivity analysis of the model are also presented. The findings from the simulations will be presented accordingly.