Au début des années 90, R.E. Schwartz a montré que le théorème de Pappus permet de définir une famille d'actions du groupe modulaire sur le plan projectif avec des propriétés géométriques et dynamiques remarquables. Ces propriétés sont semblables à celles satisfaites par les représentations d'Anosov. Dans sa thèse (sous ma direction), V. Pardini Valerio a élucidé ce fait en montrant que les représentations de Schwartz, restreintes à un sous-groupe d'indice deux, sont limites de représentations d'Anosov.

 

Je présenterai ce travail de thèse, et les progrès récents obtenus conjointement avec Gye-Seon Lee.

We prove that when 3g − 3 + p > 3, the Teichmüller space of the closed surface of genus g with p punctures contains balls which are not convex in the Teichmüller metric. We analyze the quadratic differential associated to a Teichmüller geodesic and, as a key step, show that the extremal length of a curve (as a function of time) can have a local maximum. This is a joint work with Maxime Fortier Bourque.

A convex projective manifold C = Ω/Γ is the quotient of convex subset of projective space, Ω, by a discrete group of projective transformations Γ ⊂ PGL(n+1,R). A generalized cusp in dimension 3 is a convex projective manifold that is the product of a ray and a torus. The holonomy centralizes a 1-parameter subgroup of PGL(n,R). I have shown: A generalized cusp on a properly convex projective 3-dimensional manifold is projectively equivalent to one of 4 possible cusps.

 

For a generalized cusp C = Ω/Γ in dimension n, we require that ∂C is compact and strictly convex (contains no line segment) and that there is a diffeomorphism h : [0,∞) × ∂C → C. Together with Sam Ballas and Daryl Cooper we have classified generalized cusps in dimension n, and explored new geometries arising from such cusps. We show the holonomy of a generalized cusp is a lattice in one of a family of Lie groups G(λ) parameterized by a point λ = (λ1, …, λn) ∈ Rn. More generally a maximal-rank cusp in a hyperbolic n-orbifold is determined by the similarity class of lattice in Isom(En-1).

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/

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.