Quantum Teichmuller spaces of bordered Riemann surfaces and generalized Frobenius Manifolds

This is a project in Integrable Systems, to attract an outstanding scientist, Prof. L. Chekhov, for one year in Manchester.Physical phenomena are generally described by differential equations. These are usually very difficult or impossible to solve. Nevertheless there is a special class of differential equations which are solvable in some sense. They are called integrable systems. When we manage to describe a physical phenomenon by an integrable system, we can understand and often predict its behavior. Recently the theory of integrable systems has been reformulated in the language of Frobenius manifolds. The theory of Frobenius manifolds lies at the crossroad of many disciplines in Pure, Applied Mathematics and Theoretical Physics. One of the beauties of this theory consists in its universality: results proved for a special class of Frobenius manifolds turn out to be true also for other classes of Frobenius manifolds. For example the isomorphy of certain Frobenius manifolds in quantum cohomology and in singularity theory is one version of mirror symmetry.In this project we plan to explore anew link between the theory of Frobenius manifolds and another two fascinating branch of mathematics: the problem of quantization of Teichmuller space known in quantum gravity and the theory of Cluster Algebras recently intruduced by Fomin and Zelevinsky.This research will open up new lines of ground breaking research. In fact, it is always the case that when two rich branches of mathematics are unified, many interesting new question will arise and many unexpected result will be proved.