In order to apply V. Lafforgue's ideas to the study of representations of p-adic groups, one needs a version of the geometric Satake equivalence in that setting. For the affine Grassmannian defined using the Witt vectors, this has been proven by Zhu. However, one actually needs a version for the affine Grassmannian defined using Fontaine's ring B_dR, and related results on the Beilinson-Drinfeld Grassmannian over a self-product of Spa(ℚ_p). These objects exist as diamonds, and in particular one can make sense of the fusion product in this situation; this is a priori surprising, as it entails colliding two distinct points of Spec(ℤ). The focus of the talk will be on the geometry of the fusion product, and an analogue of the technically crucial ULA (Universally Locally Acyclic) condition that works in this non-algebraic setting.

Séminaire Laurent Schwartz — EDP et applications

The sheaf of chiral differential operators is a sheaf of vertex algebras defined by Gorbounov, Malikov, and Schechtman in the early nineties that exists on any manifold with vanishing second component of its Chern character. Later on it was proposed by Witten to be related to the chiral operators of the (0,2)-supersymmetric sigma-model. Recently, we have proved this using an approach to QFT developed by Costello: the BV-quantization of the holomorphic twist of the (0,2) theory is isomorphic to the sheaf of chiral differential operators. Along with Gorbounov and Gwilliam, we prove this using the language of holomorphic factorization algebras in one complex dimension. In this talk I will sketch the proof of this result while also motivating a family of BV theories that produce sheaves of higher dimensional holomorphic factorization algebras that deserve to be called “higher” CDOs. We discuss the meaning of the OPE for these theories as encoded by the higher dimensional factorization structure.

Le but de l'exposé est de présenter un résultat d'existence et d'unicité globales pour le système magnétohydrodynamique incompressible en 3D pour des données initiales assez proches de l'état d'équilibre e3=(0,0,1).

Les problématiques de formation de singularité en équations aux dérivées partielles non-linéaires sont au coeur d'une dynamique de recherche intense ces vingt dernières années. Je présenterai quelques résultats récents concernant la construction de solutions explosives dans le cadre dit énergie sur critique. Dans le cas parabolique, je présenterai notamment la construction de toutes nouvelles bulles a géométrie fortement anisotrope.

In their paper, "On the motivic class of the stack of bundles",  Behrend and Dhillon were able to derive a formula for the class of a stack of vector bundles on a curve in a completion of the K-ring of varieties.  Later, Mozgovoy and Schiffmann performed a similar computation in order to obtain the number of points over a finite field in the moduli space of twisted Higgs bundles.  We will briefly introduce motivic classes.  Then, following Mozgovoy and Schiffmann's argument, we will outline an approach for computing motivic classes for the moduli stack of vector bundles with connections on a curve. This is joint work with Roman Fedorov and Yan Soibelman.

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/