An Anosov representation of a word hyperbolic group Γ into a semisimple Lie group G is a dynamically defined strengthening of a quasi-isometric embedding of Γ into G, which serves as a flexible higher rank analogue of the notion of convex-cocompactness. In particular, Anosov representations yield interesting discrete subgroups of G. Guichard-Wienhard and Kapovich-Leeb-Porti constructed co-compact domains of proper discontinuity for these discrete subgroups lying in generalized flag manifolds G/P where P<G is a parabolic subgroup. Distinct domains of discontinuity are indexed by certain special subsets (ideals) in the Weyl group of G with respect to the Bruhat order. In this talk, we discuss the calculation of the homology groups of the quotient manifolds in the case when Γ is a closed surface group, and G is a complex simple Lie group. The formulas express the Betti numbers explicitly in terms of the combinatorial properties of the corresponding subset of the Weyl group of G. This yields a sufficient condition to distinguish the homotopy type of two quotient manifolds obtained from different ideals in the Weyl group. Time permitting, we will present some interesting special cases where the Poincaré polynomial can be expressed as a particularly simple rational function with the degree of the numerator and denominator depending on the genus of the surface.

Hadamard Lectures 2018

 

Retrouvez toutes les informations sur le site de la Fondation Mathématique Jacques Hadamard :

 

https://www.fondation-hadamard.fr/fr/financements-accueil-206-cours-avances/accueil-lecons-hadamard

Hadamard Lectures 2018

 

Retrouvez toutes les informations sur le site de la Fondation Mathématique Jacques Hadamard :

 

https://www.fondation-hadamard.fr/fr/financements-accueil-206-cours-avances/accueil-lecons-hadamard

The central problem of Noncommutative Geometry is constructing differential calculus on a given noncommutative algebra. Some known approaches to this problem will be mentioned in my talk.

Also, I shall exhibit a new approach to constructing such a calculus on the enveloping algebras of Lie algebras gl(n) and their super-analogs. This approach is based on a new form of the Leibniz rule. As a result, the corresponding differential algebra can be treated as a quantization (deformation) of its commutative counterpart, namely, the differential algebra on the symmetric algebra of a given Lie algebra gl(n).

The role of braided algebras (i.e., those related to the corresponding quantum groups) in constructing this calculus will be exhibited. Applications to quantization of some dynamical models by means of so-called « quantum spherical coordinates » will be also exhibited.

The Chow group of zero-cycles of a smooth and projective variety defined over a field k is an invariant of an arithmetic and geometric nature which is well understood only when k is a finite field (by higher-dimensional class field theory). In this talk, we will discuss the case of local and strictly local fields. We prove in particular the injectivity of the cycle class map to integral l-adic cohomology for a large class of surfaces with positive geometric genus over p-adic fields. The same statement holds for semistable K3 surfaces over C((t)), but does not hold in general for surfaces over C((t)) or over the maximal unramified extension of a p-adic field. This is a joint work with Hélène Esnault.

Page web du séminaire

Thurston conjectured that quasi-Fuchsian manifolds are uniquely determined by the induced hyperbolic metrics on the boundary of their convex core and Mess extended this conjecture to the context of globally hyperbolic anti de-Sitter spacetimes. In this talk I will discuss a universal version of Thurston and Mess' conjectures: any quasisymmetric homeomorphism from the circle to itself is obtained on the convex hull of a quasicircle in the boundary at infinity of the 3-dimensional hyperbolic (resp. anti-de Sitter) space. We will also discuss a similar result for convex domains bounded by surfaces of constant curvature K in (-1,0) in the hyperbolic setting and of curvature K in (-∞,-1) in the anti de-Sitter setting with a quasicircle as their asymptotic boundary.

(This is joint work in progress with F. Bonsante, J. Danciger and J.-M. Schlenker.)

During pre-implantation development, the mammalian embryo forms the blastocyst, which will implant into the uterus. The architecture of the blastocyst is essential to the specification of the first mammalian lineages and to the implantation of the embryo. Consisting of an epithelium enveloping a fluid-filled cavity and the inner cell mass, the blastocyst is sculpted by a succession of morphogenetic events. These deformations result from the changes in the forces and mechanical properties of the tissue composing the embryo. Using microaspiration, live-imaging, genetics and theoretical modelling, we study the biophysical and cellular changes leading to the formation of the blastocyst. In particular, we uncover the crucial role of acto-myosin contractility, which generates periodic waves of contractions, compacts the embryo, controls the position of cells within the embryo and influences fate specification.

I will talk about the "cohesive module" introduced by Jonathan Block. Concepturally it is a complex of vector bundles with a superconnection. I will describe the geometric nature of this object. In particular, on complex manifolds, the cohesive module gives a DG enhancement of the bounded derived category of coherent sheaves. Nevertheless, it applies to many other places.

I will discuss the physical background of compressible Euler equations, its mathematical difficulties, and some important research problems.

I will discuss a method, based on p-adic height pairings, for determining the integral solutions of certain hyperelliptic equations.

The method produces, for a hyperelliptic curve over the rational numbers whose rank equals its genus, an explicit function on the p-adic points on the curve that takes finitely many explicitly determined values on the integral points. The proof goes via p-adic Arakelov theory.

Time permitting I will explain how one can use the generated function to effectively find the integral points and suggest some potential applications.