## Séminaire Orléans

**Towards a transfer operator method for locally symmetric spaces**

Joachim Hilgert

Thursday 13 December 2007 14:00 - Orléans - Salle de Séminaire

**Résumé :**

The geodesic flow on the modular surface SL(2,Z)\SL(2,R)/SO(2) and its quantization, the spectral theory of the Laplace-Beltrami operator L, is a key example in the mathematical analysis of quantum chaos. Apart from the Selberg trace formula, which in this context is well understood, one has a second tool to study the relation between the geodesic flow and its quantization: The Ruelle-Mayer transfer operator which is based on ideas of statistical mechanics. Whereas it has far reaching applications, it is not easily generalized to other examples. In this talk we will explain the transfer operator and its applications for the modular surface and some generalizations to branched coverings of the modular surface. Moreover we will point out a number of mathematical problems which have to be solved in order to turn these examples into a true theory.

**Liens :**