We present an extension of the Wasserstein L2 distance to the space of positive Radon measures as an infimal convolution between the Wasserstein L2 metric and the Fisher-Rao metric. In the work of Brenier, optimal transport has been developed in its study of the incompressible Euler equation. For the Wasserstein-Fisher-Rao metric, the corresponding fluid dynamic equation is known as the Camassa-Holm equation (at least in dimension 1), originally introduced as a geodesic flow on the group of diffeomorphisms. This point of view provides an isometric embedding of the group of diffeomorphisms endowed with this right-invariant metric in the automorphisms group of the fiber bundle of half densities endowed with an L 2 type of cone metric. As a direct consequence, we describe a new polar factorization on the automorphism group of half-densities which can be seen as a constrained version of Brenier’s theorem. The main application consists in writing the Camassa-Holm equation on S^1 as a particular case of the incompressible Euler equation on a group of homeomorphisms of R^2 that preserve a radial density which has a singularity at 0, the cone point.

L’exposé portera sur la limite quasineutre pour les équations de Vlasov. Il s’agit d’une limite singulière qui permet de dériver, au moins formellement, les équations d’Euler généralisées à la Brenier. On expliquera les phénomènes d’instabilité qui permettent de comprendre quand la limite formelle est valable ou ne l’est pas.

To every convex function $psi$ tending to infinity at infinity we can associate its moment measure, which is the image by the gradient of $psi$ of the measure with density $e^{-psi}$. We aim at characterizing all the measures that can be obtained as moment measure of some convex function. This will be done by studying a variational problem that is closely related to the one of optimal transportation theory. This variational problem can be studied using tools from the geometry of log-concave measures.

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.