For a given measure mu on the Euclidean space, we can look for a convex function u such that the image of the density f(u) through Du is mu (here f is a given function from R to R_+, typically decreasing enough). In the case where f(t)=e^{-t} we have the moment measures problem, but for negative powers we have interesting problems, linked to other questions in convex and affine geometry. In the case where mu is uniform on a convex set K in the plane and f(t)=t^{-4}, the convex function u can be used to construct an affine hemi-sphere based on the polar convex set K*, for instance. Also, in the case where mu is uniform on a set, the problem is equivalent to finding a suitable solution of Det(D^2u)=f(u). In the talk, I will briefly explain how to cast these problems as JKO-like optimization problems involving optimal transport, and explain a first idea of how to use the semidiscrete numerical methods which have been used for steps of the JKO scheme to get an approximation of the solutions of these problems. Then, I will explain how to improve this approach in a way which better fits the problem, thus obtaining a true discretization of this moment measure problem, where the measure mu has simply been replaced by a finitely supported approximation of it, which is no more specifically linked to optimal transport. For this discretization, I will present numerical results and proofs of convergence, coming from an ongoing work in collaboration with B. Klartag and Q. Mérigot

MINI-COURS

 

Conformal bootstrap is a mathematically well-defined framework for performing non-perturbative computations in strongly coupled conformal field theories, including theories of real physical interest like the critical point of the 3d Ising model.  In these lectures I will describe the recent advances in this field and the challenges it faces.

We consider a matching problem between a population of consumers and a population of producers, we look for equilibrium prices that is prices for which the distribution of demand and supply coincide. Producers minimize production cost minus price which can be described by means of optimal transport. But on the consumers’ side, the picture is slightly different, indeed a realistic assumption is that consumer maximize their utility under a price constraint. I will prove existence of an equilibrium and, formally, discuss connections with some (nonconvex) optimal transport problems which somehow mix L^1 and L^infty criteria. This is a joint work in progress with Alfred Galichon and Ivar Ekeland.

I will briefly describe, by means of a few examples, some of the ways in which proof-theoretic methods are being applied in mathematics today. I focus on a particular tool – the proof interpretation, although more generally my aim is to illustrate how ideas and techniques from proof theory have an impact outside of the foundations of mathematics.

From a graph (e.g., cities and flights between them) one can generate an algebra which captures the movements along the graph. This talk is about one type of such correspondences, i.e., Leavitt path algebras. Despite being introduced only 8 years ago, Leavitt path algebras have arisen in a variety of different contexts as diverse as analysis, symbolic dynamics, noncommutative geometry and representation theory. In fact, Leavitt path algebras are algebraic counterpart to graph C*-algebras, which has become an area of intensive research. There are strikingly parallel similarities between these two theories. Even more surprisingly, one cannot (yet) obtain the results in one theory as a consequence of the other; the statements look the same, however the techniques to prove them are quite different (as the names suggest, one uses Algebra and other Analysis). These all suggest that there might be a bridge between Algebra and Analysis yet to be uncovered. In this talk, we introduce Leavitt path algebras and then try to understand the behaviour and to classify them by means of (graded) K-theory. We will ask nice questions!

ERC Advanced Grant : AAMOT (Arithmetic of Automorphic Motives)

PI : Michael HARRIS

 

I will outline the construction of two new Euler systems, living in the motivic cohomology of the Shimura varieties attached to GSp4 and GU(2,1). This is joint work with David Loeffler and Chris Skinner.

Plus d’informations sur : http://www.proba.jussieu.fr/pageperso/anr-graal/

Let X be a smooth connected algebraic curve over an algebraically closed field, let S be a finite closed subset in X, and let F_0 be a lisse l-torsion sheaf on X-S. We study the deformation of F_0. The universal deformation space is a formal scheme. Its generic fiber has a rigid analytic space structure. By studying this rigid analytic space, we prove a conjecture of Katz which says that if a lisse $overline{Q}_ell$-sheaf F is irreducible and physically rigid, then it is cohomologically rigid in the sense that chi(X,j_*End(F))=2, where j:X-S–> X is the open immersion.

ERC Advanced Grant : AAMOT (Arithmetic of Automorphic Motives)

PI : Michael HARRIS

 

Inspiré par la conjecture de Deligne, on peut conjecturer une relation entre les valeurs spéciales de fonctions L de Rankin-Selberg et les périodes automorphes pour GLn * GLm sur un corps CM. Dans cet exposé, je vais introduire une nouvelle approche proposé par Michael HARRIS pour étudier cette relation. Je vais d’abord présenter les résultats connus. Puis, je vais expliquer comment la conjecture de Ichino-Ikeda implique les cas manquant. Il s’agit d'un travail en commun avec H. Grobner et M. Harris.

Soit X une courbe projective lisse et G un groupe réductif au dessus d'un corps de base k.

 

La correspondance de Langlands géométrique vise à comparer la catégorie des D-modules sur le champ Bun_G des G-fibrés sur X avec la catégorie des faisceaux quasi-cohérents sur le champ LocSys_{G^L}, où G^L est le groupe dual de G au sens de Langlands. On peut imaginer une telle équivalence comme une transformation de Fourier non-abélienne. Pour cette raison, elle est extrêmement difficile à démontrer. 

 

Cependant, si k est de caractéristique positive, les deux côtés deviennent assez proches de leurs limites quasi-classiques (à savoir, les espaces de Hitchin correspondants) que l'on peut comparer grace à la transformation de Fourier-Mukaï habituelle (c'est-à-dire, abélienne). Cette idée a été suggérée dans l'article de Bezrukavnikov-Braverman et puis développée par Bezrukavnikov, Chen, Travkin et Zhu.

 

Dans cet exposé j'expliquerai les idées principales de cette théorie, ainsi qu'une application à la théorie en caractéristique 0, à savoir la construction de faisceaux automorphes à partir des *opers*, selon un article récent de Bezrukavnikov-Travkin.

I will start by an introduction to Donaldson Thomas theory and some of the statements about its modularity properties, as well as its connection to S-duality conjecture in superstring theory, made formerly by physicists Gaiotto, Strominger, Yin. I will then provide an algebraic geometric approach to prove this conjecture for DT invariants of sheaves supported on hyperplane sections of the quintic Calabi-Yau threefold.