Archives du séminaire de Géométrie et Théorie Ergodique

Automne 2009

This seminar is a joint seminar for the Chair of Geometry (IGAT), the Chair of Ergodic and Geometric Group Theory (IMB) and the Geometric Analysis Unit (IGAT). The seminar is intended to cover a wide range of topics including (but not exclusively) geometric analysis, geometric group theory, low-dimensional geometry and topology, and geometry in a broad sense. People who wish to receive emails concerning the seminars are invited to contact one of the organizers.

The seminars are generally in English or French.

Le séminaire se déroule  les jeudis à 11h30 en salle  MAA110

Organisation : Marc Troyanov 

Programme :

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Séminaire GTE - Jeudi 24 septembre 2009, 11h15

Judit Abardia :  Integral geometry in complex space forms

Abstract: Classicaly, integral geometry in Euclidean space $\R^n$ deals with two problems: the study of the measure of planes meeting a convex domain in terms of the geometry of the convex domain, and the study of the kinematic formula of Blashcke-Chern-Santaló.

In order to study the first problem, it is used an invariant (with respect to the isometry group) density on the space of planes. 

If we interpret the measure of planes meeting a convex domain as a functional of the space of convex domains in $\R^n$ to $\R$, then this functional satisfies an additive property (the same as the volume), and it is called valuation. Moreover, it is invariant under the isometry group of $\R^n$. The space of functionals satisfying these properties has structure of finite dimensional vector space.

If now we change $\R^n$ by $\C^n$, we can study the same questions (we can also consider other manifolds such as $\CP^n$ or $\CH^n$). In this talk, using a basis of the space of invariant valuations under the isometry group of $\C^n$, we will give analogous results in $\C^n$, and also an expression for the Gauss-Bonnet formula in $\CP^n$ and $\CH^n$.

 Séminaire GTE - Jeudi 15 octobre 2009, 11h30

 Marc Troyanov : Surfaces with bounded integral curvature in the sense of Alexandrov

Abstract: In the years 1940-1970, Alexandrov and the "Leningrad School" has developed a rich theory of singular surfaces. These are topological surfaces, with an intrinsic metric for which we can define a notion of curvature which is a Radon measure. This class of surfaces has nice convergence properties and it is remarkably stable with respect to various geometric constructions (such as glueing etc.). It includes polyhedral surfaces and C^2 Riemannian surfaces, both classes being dense in the space of Alexandrov surfaces. Any singular surface one can reasonably think of is an Alexandrov surface, yet many geometric properties of smooth surfaces do extend to Alexandrov surfaces.
The aim of this lecture is to give a non-technical introduction to Alexandrov's theory, to give examples and some of the fundamental facts from the theory. We will also discuss a classification theorem of (compact) Alexandrov surfaces.

(The paper is downloadable at http://sma.epfl.ch/~troyanov/publications.html)

Séminaire GTE - Jeudi 22 octobre 2009, 11h30

Stephen Ducret : L_{qp}-cohomology of Riemannian manifolds and simplicial complexes and bounded geometry

Abstract:  We investigate conditions on Riemannian manifolds under which the existence of Sobolev inequalities is a quasi-isometry invariant. By some mean, this leads us to defining an integrable cohomology for manifolds (where the measure comes from the Riemannian structure), and a similar cohomology for simplicial complexes. We prove a de Rham-type theorem, and show how this gives the quasi-isometry invariance for integrable cohomology of manifolds, under suitable assumptions.

Séminaire GTE - Jeudi 29 octobre 2009, 11h30

Alberto Setti : Stochastic methods and Gradient Ricci Solitons

The aim of this talk is to describe some results in stochastic analysis on manifolds, such as stochastic completeness (in the form of the weak maximum principle at infinity), parabolicity  and $L^p$-Liouville type results for weighted Laplacians and show how they may be used to obtain triviality, rigidity results, and scalar curvature estimates for  gradient Ricci solitons under $L^p$ conditions on the relevant quantities.

Séminaire GTE - Jeudi 12 novembre 2009, 11h30

Frédéric Mangolte : Cremona transformations and diffeomorphisms of topological surfaces

Résumé : We show that the action of Cremona transformations on the real points of quadrics exhibits the full complexity of the diffeomorphisms of the sphere, the torus, and of all non-orientable surfaces. The main result  says that if X is a surface birational to the real projective plane, then Aut(X), the group of algebraic automorphisms, is dense in Diff(X), the group of self-diffeomorphisms of X. (Joint work with János Kollár.)

Séminaire GTE - Jeudi 26 novembre 2009, 11h30

M. Fujii : Growth functions of Artin groups

Abstract: In this talk, I will talk on the relationship between the growth functions of Artin groups and those of Artin monoids. Also I will talk about some word accepter for Artin monoids.

Séminaire GTE - Jeudi 3 décembre 2009, 11h30

Athanase Papadopoulos : Actions du groupe modulaire d'une surface.

Abstract : Je vais passer en revue plusieurs propriétés de rigidité d'actions du groupe modulaire (ou groupe de Teichmüller) d'une surface de type fini. Je vais ensuite présenter un nouveau résultat (obtenu en commun avec Mustafa Korkmaz) sur l'acion de ce groupe sur le complexe des triangulations idéales de la surface.

Séminaire GTE - Jeudi 10 décembre 2009, 10h15

Alexandre Mednykh :  Holomorphic maps between Riemann surfaces of small genera.

Abstract:  We will talk about upper bound for the number of surgective holomorphic maps from the genus three Riemann surfece onto genus two Riemann surface. By classical de Franchis theorem (1913) this number is finite. We show that the sharp bound is exactly 48.

Séminaire GTE - Jeudi 10 décembre 2009, 11h30

Christopher M. Judge : Subconics in translation surfaces.

Abstract:  We consider the space of conic sections---or more precisely the complements of conic sections---immersed in Euclidean surfaces with conical singularities.  In the special case where the holonomy is trivial---translation surfaces---we show that this space of `subconics' can be used to encode the Teichmueller geodesic flow on moduli space. In particular, closed orbits of this flow can be characterized.

Séminaire GTE - Jeudi 17 décembre 2009, 11h30

A. Arnold: Polygones Hyperboliques Larges

Abstract: Dans un papier intitulé « Hyperbolic Geometry » par A. Aigon-Dupuis, P. Buser et K.-D. Semmler, on trouve une description intéressante des surfaces hyperboliques de genre 2. Celle-ci a pour domaine fondamental un octogone, lui-même construit à partir de 6 points p_1, ., p_6 vérifiant la relation p_1.p_6=+id (écrit comme produit de matrices). Ces points correspondent aux points de Weierstrass de la surface et cette construction se généralise pour les surfaces hyperelliptiques compactes de genre g>2. Le but de cet exposé est de rappeler le formalisme utilisé (notation matricielle) dans ce papier, et de donner l'idée générale de cette construction. En partculier, une définition d'un "polygone large et orienté" sera donnée. Cette notion est intéressante, car de tels polygones donnent une alternative aux paramètres de Fenchel-Nielsen dans la description de l'espace de Teichmüller des surfaces hyperelliptiques compactes de genre g>1.