Geometric structures on surfaces and the fundamental group
William Goldman, University of Maryland
A geometric structure on a space consists of systems of coordinates modelled on your favorite
geometry. The topology of a surface constrains the kinds of geometric structures a surface can
support. For example, no atlas of the whole world represents distances accurately on flat pieces
of paper without distortion -- the 2-sphere doesn't support a Euclidean geometric structure. In
general the geometric structures on a space closely relate to the fundamental group of that space.
The various ways a geometry can be put on a space relates to the space of representations of the
fundamental group. This space itself has a rich geometry and supports interesting dynamical
systems. This talk will survey basic examples and how geometric and analytic techniques
provide insight to the algebraic study of group representations.