Our program will focus on the deformation theory of geometric structures on manifolds, and the resulting geometry and dynamics. Formally a subfield of differential geometry and topology, with a heavy infusion of Lie theory, its richness stems from close relations to dynamical systems, algebraic geometry, representation theory, Lie theory, partial differential equations, number theory, and complex analysis.
Hyperbolic structures on surfaces provide the first nontrivial examples, and the classical Teichmuller space is the prototype of a deformation space of locally homogeneous structures. More general deformation spaces arise from the space of representations of the fundamental group of a manifold in a Lie group, which appears also as the moduli space of flat connections on the manifold. These "character varieties'' have played an important role in developing topological invariants of manifolds, particularly in dimensions 3 and 4.
Teichmuller space can be realized as subset of the space of representations of a surface group into PSL(2,R). What has recently been called "higher Teichmuller theory" by Fock and Goncharov concerns certain deformation spaces arising from subsets of the space of representations of a surface groups into Lie groups of higher rank, e.g.PSL(n,R), which share some of the properties of classical Teichmuller space.
Recent interest in this subject has also come from mathematical physics, through Witten's suggestion relating representations in the Hitchin components, which furnish examples of higher Teichmuller spaces, to W_n-algebras, and applications of Hitchin representations to the geometric Langlands program. These unexpected inter-relationships underscore this subject's richness, timeliness and diversity. A central goal of the program will be to bring together researchers who work in the more fully developed areas of Teichmuller geometry and deformation spaces of hyperbolic structures in low dimensions with researchers studying more general deformation spaces in order to explore these new connections.