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.

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.

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!

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.

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.

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.

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

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.

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.

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.