I shall begin with an introduction to the deep and fascinating analogy between lattices such as SL(n,Z), mapping class groups, and automorphism groups of free groups. This analogy has served remarkably well as a guiding principle in the study of these groups in recent years (despite being dented occasionally by some uncooperative facts). In this talk, I shall present results that develop aspects of the analogy concerning curvature, rigidity and superrigidity. In this scheme, Teichmüller space emerges as an analogue of the symmetric space for SL(n,Z), and the corresponding object for the outer automorphism group of a free group is Outer Space. I shall end with a list of open problems.
Certain geometric structures on surfaces arise naturally in the Higgs bundle treatment of the Teichmüller component of representations of a surface group in a split real form of a Lie group. The talk will describe some of these and their links with integrable systems.
Let G be a finite group and V a G-variety, i.e. an irreducible algebraic variety with a regular action of G. A compression of V is a G-equivariant dominant morphism f : V --> X such that G acts faithfully on X. The basic questions are: (a) How much can one compress a given action? (b) What are the incompressible G-varieties? We first discuss this concept from two rather different point of view: (i) Generic structure of Galois-coverings and (ii) Equations for field extension. We then define the covariant dimension of G which measures how much a representation of G can be compressed. This has to be compared with the essential dimension of G which was introduced by Buehler and Reichstein in order to study the number of parameters of equations. Finally, we will give a short overview on known results, work out a few interesting examples and discuss some open questions. (This is mostly joint work with G.W. Schwarz.)
Cluster algebras (introduced jointly with S.Fomin) have found applications in a diverse variety of settings: total positivity, representation theory, quiver representations, Teichmüller theory, Poisson geometry, discrete dynamical systems, tropical geometry, and algebraic combinatorics. An algebraic structure of a cluster algebra is encoded by a family of Laurent polynomials expressing distinguished generators (cluster variables) in terms of an "initial cluster" consisting of finitely many algebraically independent cluster variables. We will discuss an interpretation of these Laurent polynomials (due to F.Chapoton and P.Caldero) in terms of the geometry of Grassmannians of quiver representations.
[LMS Regional Meeting]
[Teichmüller Theory and Cluster Algebras]
[London Mathematical Society]
The University of Leicester Mathematics