### GEORGE MASON
UNIVERSITY

DEPARTMENT OF MATHEMATICAL
SCIENCES

COLLOQUIUM

JANUARY 31, 2020

**Speaker: **
Sara Maloni, University of Virginia

**Title: ***Dynamics on character varieties
*

**Abstract: **
Given a surface S, the mapping class group Mod(S) is the group of isotopy
classes of homeomorphisms of S. There is a natural action of this group on
the set of conjugacy classes of representations from the fundamental group of S
into the group PSL(2,R) of orientation preserving isometries of hyperbolic
space. A famous theorem of Fricke proves that this action is properly
discontinuously ("discrete/nice") on the representations which are discrete
and faithful. Note that such representations correspond to marked isotopy
classes of hyperbolic structures on S and constitute the famous Teichmueller
space T(S) of S. Goldman conjectured that the action is ergodic ("chaotic") on
the complement of T(S). One can generalize this discussion by considering the
action of the outer automorphism group Out(H) on the set X(H, G) of conjugacy
classes of representations from a group H into a Lie group G. In this talk,
I will survey some of the results known, some of the many open conjectures in
the field and some recent results in this direction. I will define the objects
I will introduce and provide examples, so no advanced previous knowledge is
required for the talk.

**Time:** Friday, January 31, 2020, 11:30-12:20 p.m.
Note the unusual time!!

**Place:** Exploratory Hall, room 4106

**Refreshments** will be served at 11:00 a.m.

Department of Mathematical Sciences

George Mason University

4400 University Drive, MS 3F2

Fairfax, VA 22030-4444

http://math.gmu.edu/

Tel. 703-993-1460, Fax. 703-993-1491