ICTP Mathematics Colloquium by Nigel Hitchin on 'Generalizations of Teichmüller space'

Nigel Hitchin, FRS (University of Oxford)

Abstract: The famous theorem of Narasimhan and Seshadri showed that the moduli space of representations of the fundamental group of a surface into a unitary group could be described holomorphically in terms of stable vector bundles on an algebraic curve. The theory of Higgs bundles extended this idea to noncompact groups and in particular to real forms like SL(n,R) and in this case there is a component which is described as a space of holomorphic differentials on the curve. The special case n=2 gives classical Teichmüller space. The talk concerns approaches to the case of SL(n,R) where n=∞, interpreted as the group of Hamiltonian diffeomorphisms of the cotangent bundle of the circle and leads to differential geometric questions about geodesics on surfaces.