Séminaire Laurent Schwartz — EDP et applications

Séminaire Laurent Schwartz — EDP et applications

ERC Advanced Grant : AAMOT (Arithmetic of Automorphic Motives)

PI : Michael HARRIS

 

In studying the arithmetic of automorphic Galois representations, an important role is played by global cohomology classes coming from algebraic cycles on Shimura varieties, or more generally from algebraic K-theory; these are the building blocks of Euler systems. Unfortunately, it is surprisingly difficult to prove that these cohomology classes are non-zero! One of the key inputs for the recent progress in the theory of Euler systems was a new approach to solving such problems, developed by Bertolini, Darmon and Prasanna, in which the non-vanishing of Galois cohomology classes can be obtained by relating them to p-adic period integrals via Besser's rigid syntomic cohomology. I will explain some examples of this strategy, for Galois representations arising from products of modular curves and Hilbert modular surfaces, and survey the problems that must be solved to extend this method to more general Shimura varieties.

A review of results concerning the classical XY model in various dimensions is presented.

 

I start by showing that the XY model does not exhibit any phase transitions in a non-vanishing external magnetic field, and that connected spin-correlations have exponential decay. These results can be derived from the Lee-Yang theorem.

 

Subsequently, I study the XY model in zero magnetic field: The McBryan-Spencer upper bound on spin-spin correlations in two dimensions is derived. The XY model is then reformulated as a gas of vortices of integer vorticity (Kramers-Wannier duality). This representation is used to explain some essential ideas underlying the proof of existence of the Kosterlitz-Thouless transition in the two-dimensional XY model. Remarks on the existence of phase transitions accompanied by continuous symmetry breaking and the appearance of Goldstone modes in dimension three or higher come next.

 

Finally, I sketch the random-walk representation of the XY model and explain some consequences thereof – such as convergence to a Gaussian fixed point in the scaling limit, provided the dimension is > 4; and the behaviour of the inverse correlation length as a function of the external magnetic field.

 

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.