Vladimir Fock will speak in the geometry seminar on **Wednesday** 16 November at 13h30, in the Salle de Profs (9th floor, building NO). Please note the unusual day! His title is *“Teichmuller space as the main example of a cluster variety”* and his abstract is below.

We will review a combinatorial approach to the Teichmüller space of complex structures on Riemann surfaces and show that such approach can be generalized to a large class of simple Lie groups related manifolds. The main advantage of this approach is to be able to use constructions natural in the Teichmüller spaces theory to study Lie groups or integrable systems and visa versa. In the second part of the talk we will discuss two main conjectures concerning cluster varieties in general. The duality conjecture (recently proved by Gross, Hacken, Keel and Kontsevich) claiming the existence of a kind of Fourier transform on a cluster variety. The second conjecture suggests a canonical hyperkähler structure on a double of a cluster variety.