Abstract: Teichmuller space is a moduli space of Riemann surfaces, where two Riemann surfaces are equivalent if they are biholomorphic and are homotopically related in a certain sense. It can be thought of as the space of local deformations of Riemann surfaces. Conformal field theories are quantum mechanical or statistical field theories which are invariant under infinitesimal rotations and rescalings, and thus in two dimensions they are closely tied to complex analysis. A mathematical model of conformal field theory was sketched by Segal and Kontsevich. Attempts to realize this model rigorously has spawned a great deal of deep mathematics.

A certain moduli space in conformal field theory turns out to be the quotient of Teichmuller space by a discrete group action. This relation between the two moduli spaces leads to the solution of analytic problems in the rigorous formulation of conformal field theory, and new results in Teichmuller theory. In this talk, I will give a non-technical introduction to the ideas of Teichmuller theory, and sketch the correspondence between the moduli spaces. Joint work with David Radnell.