Dans ce cours nous étudierons plusieurs modèles de graphes aléatoires allant du plus classique (le modèle d'Erdös-Renyi introduit en 1960) aux plus récents (les cartes planaires aléatoires étudiées depuis le début des années 2000). Le fil conducteur du cours sera la notion de convergence locale et les propriétés des graphes limites dits dilués.

Contenu du cours :

– Modèle d'Erdös-Renyi, transition de phase et propriétés de base
– Convergence locale et "méthode objective" d'Aldous et Steele
– Arbre couvrant minimum et théorème de Frieze 
– Graphes aléatoires unimodulaires
– Limites locales d'arbres aléatoires
– Limites locales de cartes aléatoires (construction, épluchage, théorème de Benjamini-Schramm)

By quantum matrix algebras I mean these related to braidings (solutions to Quantum Yang-Baxter Equation) and in a sense similar to the classical matrix algebras. In first turn, I am interested in the so-called Reflection Equation algebra. By using it, me (in collaboration with P.Saponov) have introduced the notion of partial derivatives on the enveloping algebra U(gl(m)). This leads to a new type of Noncommutative Geometry (we call it Quantum Geometry), which is deformation of the classical one. In my talk I plan to consider a way of defining some dynamical models on U(u(2)) background.

Consider a reductive group G over a p-adic field F. The local Langlands conjecture relates the irreducible smooth representations of G(F) with the set of (local) L-parameters, which are maps from the Weil group of F to the L-group of G; refinements of the conjecture relate the fibres of this map with the automorphism group of the L-parameter. Based on ideas from V. Lafforgue's work in the global function field case, I outlined a strategy for attaching (semisimple) L-parameters to irreducible smooth representations of G(F) in my 2014 Berkeley course. At the same time and place, L. Fargues formulated a conjecture relating the local Langlands conjecture with a geometric Langlands conjecture on the Fargues-Fontaine curve. The goal of this course will be to discuss some of the developments since then. On the foundational side, this concerns basics on the etale cohomology of diamonds including smooth and proper base change and Poincare duality, leading up to a good notion of "constructible" sheaves on the stack of G-bundles on the Fargues-Fontaine curve. On the applied side, this concerns the construction of (semisimple) L-parameters, the conjecture of Harris (as modified by Viehmann) on the cohomology of non-basic Rapoport-Zink spaces, and the conjecture of Kottwitz on the cohomology of basic Rapoport-Zink spaces.

 

Retrouvez toutes ces informations sur le site de la Fondation Mathématique Jacques Hadamard :

https://www.fondation-hadamard.fr/fr/evenements/lecons-hadamard

 

ERC Advanced Grant : AAMOT (Arithmetic of Automorphic Motives)

PI : Michael HARRIS

 

Soit G un groupe réductif sur un corps global de caractéristique positive.  Vincent Lafforgue a montré comment associer un paramètre de Langlands à une représentation automorphe cuspidale de G.  Le paramètre est un homomorphisme du groupe de Galois global à valeurs dans le  L-groupe  LG of G.   Mon exposé porte sur mes travaux en cours avec Böckle, Khare, et Thorne sur la méthode de Taylor-Wiles-Kisin dans le cadre de la correspondance de Lafforgue.  De nouvelles questions se manifestent des deux côtés de la correspondance lorsque nous essayons de généraliser les travaux antérieurs de Böckle and Khare sur le cas de GL(n).  Sous certaines hypothèses sur le paramètre de Langlands, nous pouvons appliquer les arguments de modularité de manière inconditionnelle ;  alors nous obtenons un théorème de modularité potentiel pour un groupe déployé adjoint quelconque.

Consider a reductive group G over a p-adic field F. The local Langlands conjecture relates the irreducible smooth representations of G(F) with the set of (local) L-parameters, which are maps from the Weil group of F to the L-group of G; refinements of the conjecture relate the fibres of this map with the automorphism group of the L-parameter. Based on ideas from V. Lafforgue's work in the global function field case, I outlined a strategy for attaching (semisimple) L-parameters to irreducible smooth representations of G(F) in my 2014 Berkeley course. At the same time and place, L. Fargues formulated a conjecture relating the local Langlands conjecture with a geometric Langlands conjecture on the Fargues-Fontaine curve. The goal of this course will be to discuss some of the developments since then. On the foundational side, this concerns basics on the etale cohomology of diamonds including smooth and proper base change and Poincare duality, leading up to a good notion of "constructible" sheaves on the stack of G-bundles on the Fargues-Fontaine curve. On the applied side, this concerns the construction of (semisimple) L-parameters, the conjecture of Harris (as modified by Viehmann) on the cohomology of non-basic Rapoport-Zink spaces, and the conjecture of Kottwitz on the cohomology of basic Rapoport-Zink spaces.

 

 

Retrouvez toutes ces informations sur le site de la Fondation Mathématique Jacques Hadamard :

https://www.fondation-hadamard.fr/fr/evenements/lecons-hadamard

ERC Advanced Grant : AAMOT (Arithmetic of Automorphic Motives)

PI : Michael HARRIS

 

Dans une situation non ramifiée d'endoscopie tordue sur un corps p-adique, on peut énoncer un lemme fondamental tordu pour toutes les fonctions biinvariantes par un compact maximal hyperspécial. J. Arthur a démontré l'existence du transfert spectral dans le cas non tordu et cela a été généralisé au cas tordu par C. Moeglin. Le lemme fondamental en question équivaut plus ou moins à déterminer le transfert spectral des paquets stables contenant une représentation non ramifiée. Dans un premier article, B. Lemaire, C. Moeglin et moi avons prouvé ce lemme en utilisant cette interprétation spectrale. La preuve utilise le cas particulier du lemme appliqué à la fonction caractéristique d'un espace hyperspécial, ce qui est le théorème de Ngo Bao Chau. Mais celui-ci impose des restrictions sur la caractéristique résiduelle. Dans un second article avec B. Lemaire, nous supprimons ces restrictions. En fait, nous n'avons besoin du théorème de Ngo Bao Chau que dans une situation très particulière où le groupe endoscopique est un tore non ramifié. On montre que, dans ce cas, l'assertion résulte du lemme fondamental non tordu mais avec caractère pour les groupes GL(n). Celui-ci est connu grâce à T. Hales. Dans l'exposé, j'expliquerai les grandes lignes des deux articles.

It has long been realized by microlocal analysts that an alternative, ´Euclidean-signature´ semi-classical ansatz for solutions to Schrödinger´s equation is much more natural, from a mathematical point of view, than the classical, physics textbook, W.K.B. ansatz, even though the latter has the conceptual advantage of emphasizing the classical correspondence principle. For technical reasons though this microlocal approach has seemed to be limited to quantum mechanical problems and not to be applicable to quantum field theories. In this seminar I will review this method and discuss an expansive, ongoing project with A. Marini (Yeshiva, L´Aquila) and R. Maitra (Wentworth I.T.) to extend these influential ideas to bosonic quantum field theories including phip type scalar and Yang-Mills fields.

 

« Return of the IHÉS Postdoc Seminar »

 

Abstract: The mod-p Local Langlands Program aims to find a relationship between the mod-p representation theory of p-adic reductive groups and Galois groups, in the spirit of the classical Langlands correspondence on complex vector spaces.  I'll describe several aspects of this theory, and outline how techniques from the classical case fail in the mod-p setting.  Finally, I'll describe how modules over certain Hecke algebras can be used as an avatar for this correspondence in certain situations.

« Return of the IHÉS Postdoc Seminar »

 

Abstract: The Andre-Oort conjecture is an important problem in arithmetic geometry concerning subvarieties of Shimura varieties. It attempts to characterise those subvarieties V for which the special points lying on V constitute a Zariski dense subset. When the ambient Shimura variety is the moduli space of principally polarised abelian varieties of dimension g (in which case, a special point is a point corresponding to the isomorphism class of a CM abelian variety), the conjecture has been obtained by Pila and Tsimerman via the so-called Pila-Zannier strategy, reliant on the Pila-Wilkie counting theorem on o-minimal structures. In this talk, we will outline the Pila-Zannier strategy, providing some introduction to Shimura varieties and Andre-Oort, and explain the state of the art for the full conjecture. In particular, we will mention certain height bounds obtained jointly with Orr.

« Return of the IHÉS Postdoc Seminar »

 

Abstract: The theory of skeletons gives a way to write a non-Archimedean analytic space as fibred generically in tori over a polyhedral complex of (real) dimension equal to that of the analytic space. One does this procedure in the hope that the polyhedral complex can eventually tell us something about the original space. This is a precise version of a still mainly conjectural picture in complex geometry.

I'll give a panoramic overview of the various structures appearing in the subject, the techniques involved in constructing them, and what they can do for us.

 

 

Let G be a semi-simple and simply connected group and X an algebraic curve. We consider Bun_G(X), the moduli space of G-bundles on X. In their celebrated paper, Atiyah and Bott gave a formula for the cohomology of Bun_G, namely H^*(Bun_G)=Sym(H_*(X)otimes V), where V is the space of generators for H^*_G(pt). When we take our ground field to be a finite field, the Atiyah-Bott formula implies the Tamagawa number conjecture for the function field of X.

The caveat here is that the A-B proof uses the interpretation of Bun_G as the space of connection forms modulo gauge transformations, and thus only works over complex numbers (but can be extend to any field of characteristic zero). In the talk we will outline an algebro-geometric proof that works over any ground field. As its main geometric ingredient, it uses the fact that the space of rational maps from X to G is homologically contractible. Because of the nature of the latter statement, the proof necessarily uses tools from higher category theory. So, it can be regarded as an example how the latter can be used to prove something concrete: a construction at the level of 2-categories leads to an equality of numbers.