Title and Abstract

1. Speaker : Francesco Bonsante (Pisa, Italy)

Anti de Sitter geometry is Lorentzian geometry of constant curvature -1. It can be thought as the analogue of the hyperbolic geometry in Lorentzian setting. After Mess seminal work, it turned out there are many deep connections between Anti de Sitter geometry and hyperbolic geometry in dimension 3. In these lectures I will introduce the basic notions of Anti de Sitter geometry, and explain the main results of Mess work. In particular I will describe the connection between Thurston hyperbolic earthquakes and bent surfaces in Anti de Sitter spaces. Roughly speaking this connection can be regarded as the analogue in the Lorentzian setting of the classical connection between grafting and bent surfaces in hyperbolic spaces. Finally I will show more recent developments of Mess machinery, in particular explaining the relation between minimal surfaces in Anti de Sitter spaces and minimal Lagrangian maps between hyperbolic surfaces.

2. Speaker : Patrick Foulon (Strasbourg, France)

The aim of the talks will be to present some recent results concerning some typical examples of Anosov flows. On the one hand,  we will focus on Hilbert geometry of strictly convex divisible sets and discuss some volume and entropy rigidity results. This will provide the opportunity to present concrete Anosov flows which are also very interesting in the non-smooth case. We will then investigate the smooth contact Anosov flows family and the longitudinal cocycle.

On the other hand, we will observe some nice flexibility properties by presenting recent works carried out with Hasselblatt, where we produce many examples of new contact Anosov Flows, not topologically equivalent to algebraic flows. For those examples constructed on hyperbolic 3–manifolds, we will investigate the surprising behaviour of periodic orbits based on the works of Barthelmé and Fenley.

1- Hyperbolic geometry

2-Hilbert geometry and geodesic flow

3- Divisible convex sets - volume and entropy rigidity

4- KAM cocycle longitudinal rigidity

5- Contact Anosov flows

6- Anosov flows on Seifert bundles - topological rigidity

7- New contact Anosov flows in dimension 3

8- Some open questions

Tight maps were first introduced by Clerc and Orsted. They were extensively studied by Burger, Iozzi and Wienhard and applied in their work on surface group representations with maximal Toledo invariant. In this talk I will discuss some properties of tight maps and some of the techniques used in my work classifying tight maps.

TBA

5. Speaker : Genkai Zhang (Gothenbourg, Sweden)

Title : Eichler-Shimura isomorphism and applications

Abstract :

I shall present some joint work with Inkang Kim in progress. We give a short elementary proof of the Eichler-Shimura isomorphism for hyperbolic lattices. We shall apply the isomorphism to study the metric structures on Hitchin components.