Math 569: Representations of surface groups

David Dumas

University of Illinois at Chicago
Spring 2017

genus three

General information

Instructor David Dumas (david@dumas.io)
Lectures Tue & Thu, 3:30–4:45pm in Taft Hall 300
CRN 39674
Office hours Mon & Thu 2–3pm
Note: No office hours on Monday March 27
Texts F. Labourie, Lectures on Representations of Surface Groups.
European Mathematical Society, 2013.   ISBN 978-3-03719-127-9
PDF  /  Purchase from AMS  /  Not available on Amazon  /  Not in UIC library

Course description

We will study the space of representations of the fundamental group of a surface (i.e. a real 2-manifold) into a Lie group G. We will investigate the local and global geometry of this space, including questions about connectedness, smoothness, singularities, and complex and symplectic structures. Following the main thread of the textbook, we will discuss a remarkable formula that gives the symplectic volume of the space of representations when G is compact.

After that, we will branch out and consider some special classes of surface group representations with nice geometric properties, such as representations in SL(2,R) coming from hyperbolic structures, representations in SL(2,C) associated to complex projective structures and to bending deformations of surfaces in hyperbolic 3-space, and representations in SL(3,R) associated to convex real projective structures. The idea of studying these examples is to see how some general phenomena play out in these concrete situations, and also to discuss the unique features of each one.

Course documents

Student presentations

As described in the syllabus, each student taking the course for credit must prepare a 30-minute final presentation on a topic to be approved by the instructor. The topic must be selected and agreed upon by email on or before March 14.

The presentations will be held in three sessions:

Links and resources

Here we collect citations for further reading about course material and links to relevant online materials.
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