2-D Gaussian free field (GFF) is an intriguing mathematical object emerging in a wide range of contexts in probability theory and statistical physics. Several important properties of GFF have been explored. Among them are its various metric properties which have attracted a substantial amount of research in recent years. In this talk, we will discuss three of them, namely the Liouville FPP, the Liouville graph distance and an effective resistance metric. We will discuss the contexts in which they arise, state the current results, try to give rough sketches of the proofs and mention some open problems for future research. The content of this talk is based on joint works with Jian Ding and Marek Biskup. 

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

On considère le noyau de la chaleur p(t,x,y) sur le revêtement universel d'une variété compacte de courbure négative et on en donne un équivalent quand t tend vers l'infini. La démonstration introduit une nouvelle famille équivariante naturelle de mesures à l'infini, liée au bas du spectre du Laplacien λ0 sur le revêtement universel. Il s'agit d'un travail en commun avec Seonhee Lim.

L'exposé commencera par une introduction au sujet, de type colloquium. 

Je parlerai du théorème suivant. On considère une variété compacte M de dimension supérieure ou égale à 3 et de courbure négative. Si une horosphère de M est plate, alors M est de courbure constante. Il s’agit d’un travail en commun avec Gérard Besson et Sa'ar Hersonsky.

It was proved by Gabber in the early 1980's that RPsi commutes with duality, and that RPhi preserves perversity up to shift. It had been in the folklore since then that this last result was in fact a consequence of a finer one, namely the compatibility of RPhi with duality. In this talk I'll give a proof of this, using a method explained to me by A. Beilinson.

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.

What is the probability that the number of triangles in an Erdős–Rényi random graph exceeds its mean by a constant factor?

 

The order of the log-probability was already a difficult problem until its resolution a few years ago by Chatterjee and DeMarco-Kahn. We now wish to determine the exponential rate of the tail probability. Thanks for the works of Chatterjee-Varadhan (dense setting) and Chatterjee-Dembo (sparse setting), this large deviations problem reduces to a natural variational problem. I will discuss techniques for analyzing this variational problem, with the following focuses in mind:

 

(a) Replica symmetry: conditioned on an Erdős–Rényi random graph having lots of triangles, does it look like another Erdős–Rényi random graph with higher edge density?

 

(b) Computing the large deviation rate for sparse random graphs G(n,p), with p → 0 as n increases.

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