# Abstracts

## Dr Anna Felikson, Durham University

### Cluster algebras, mutations and triangulated surfaces

Cluster algebras were introduced by Fomin and Zelevinsky in 2002. Since then it turned out that this notion is connected to numerous different fields in mathematics (such as Poisson geometry, representation theory, integrable systems, combinatorics of polytopes, Teichmuller theory, dilogarithm identities, probability theory and many others). In this talk we will introduce cluster algebras and show some connections to triangulated surfaces. More precisely, we will see how an experience in the classification of hyperbolic Coxeter polytopes helped to solve some classification problem in cluster algebras. This work is joint with Pavel Tumarkin and Michael Shapiro.

## Professor Jing Ping Wang, University of Kent

### Classification of Integrable equations and Number Theory

Among nonlinear partial differential equations(PDE) there is an exceptional class of integrable equations, which can be studied with the same completeness as linear PDEs, at least in principle. They possess a rich set of exact solutions and many hidden properties. Classification of integrable equations of a given family of nonlinear PDEs is an important topic in the field of soliton theory. There are many approaches to this problem, among which the symmetry approach has proved to be very efficient and powerful. In this talk, I’ll give a brief account of recent development of the symmetry approach. The progress has been achieved mainly due to a symbolic representation of the ring of differential polynomials, which enables us to use results from algebraic geometry and number theory.