Heidelberg Junior Geometry Seminar

Winter 2018/19

Next upcoming talk:

Tue Feb 12  ∙  Seminarraum 3  ∙  11:15

Ferdinand Vanmaele (Heidelberg)    Isolated flats in the Davis complex

Let (W,S) be a Coxeter system and \Sigma the Davis complex on which W acts cocompactly and properly discontinously. In the second talk we consider flats in \Sigma, or convex subsets isometric to R^n. In particular, we discuss when \Sigma has a collection of flats with isolated elements, based on work by P. Caprace. We use virtually abelian groups, affine and hyperbolic Coxeter groups, and ''orthogonal'' diagrams J^\perp to make this statement precise, and give several examples.

Past talks:

Wed Jan 30  ∙  Seminarraum 3  ∙  11:15

Oskar Riedler (Heidelberg)    Cartan geometry and a recurrence argument

It is impossible for a compact Riemannian manifold to have a non-compact isometry group. For other geometric structures, like semi-Riemannian metrics, this is no longer true. However there do not seem to be very many counter-examples. In the literature one finds reference to a vague conjecture: Geometrics structures on a compact space which are large (i.e. have non-compact automorphism group) are ''almost classifiable''. For the talk I would like to give a simple introduction to Cartan geometries, which encompass a large amount of geometric structures, as well as prove a step in the classification problem of compact 3 dimensional Lorentz-manifolds with non-compact isometry groups.

Wed Feb 6  ∙  Seminarraum 3  ∙  11:15

Nguyen-Thi Dang (IRMA Rennes)    Some problems of dynamics in higher rank

In this talk, I will introduce my research topic. Let $G$ be a semisimple, real linear, connected Lie group and $\Gamma$ a Zariski dense, discrete subgroup of $G$. What can we say about the dynamical properties of the right action by multiplication of a closed subgroup $H$ of $G$ on $\Gamma \backslash G$? I will introduce a limit cone $\mathcal{C}(\Gamma)$ that contains all the information about the spectrum of elements in $\Gamma$. Then I will state a result on random walks on $\Gamma$. First, I will talk about the Law of Large Number obtained by Furstenberg-Kesten. Then I will prove that the interior of the limit cone is filled with the mean values given by the LLN, Lyapunov vectors. Time permitting, I will explain how the topological behavior of the right action of a one parameter subgroup of the maximal torus on $\Gamma \backslash G$ is encoded by the limit cone.

Wed Dec 5  ∙  Seminarraum 3  ∙  11:15

Mireille Soergel (Heidelberg)    Simplicial Complexes associated to 2-generated Artin groups

We will start by presenting different simplicial complexes of 2-generated groups. The comple Y was constructed by Huang and Osajda to show that large-type Artin groups are systolic and allows us to obtain a complex Z. The other X(G) is from Bestvina and is used to study non-positively curved aspects of Artin groups of finite type. We will describe the construction of these complexes and show how they relate to each other. In the last talk we show that Z is isomorphic to X(G). This time we will prove that Z x R is homeomorphic to Y.

Wed Nov 28  ∙  Seminarraum 3  ∙  11:15

Tam Nguyen-Phan (MPI Bonn)    The Nash C^1 isometric embedding theorem

I will state and explain the proof of the Nash C^1 isometric embedding theorem. As surprising as the theorem may sound, the proof is completely elementary.

Wed Oct 31  ∙  Seminarraum 3  ∙  11:15

Ferdinand Vanmaele (Heidelberg)    Introduction to Coxeter groups and the Davis complex

Coxter groups arose as a natural generalization of reflection groups. J. Tits defined them in a simple way using generators and relators, that is, using a group presentation. Coxeter groups have a wide range of applications; for example, every Weyl group may be realized as a finite, irreducible Coxeter group. The Davis complex is a geometric realization of Coxeter groups, which is CAT(0) for every Coxeter group. It has therefore been one of the first classes of examples for CAT(0) spaces. In this first talk we provide a general introduction to Coxeter groups and the Davis complex.