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.
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