## Math 7853 - Topics in Topology

Class meeting time: MW 12:30 - 1:50 in JWB 333 (for now)

We will occasionally have class on Friday to make up some lectures that
I will miss because I'll be out of town.

This course will focus on various topics on ergodic theory and dynamics
on surfaces. Here are some of the (possible) topics we will cover:

- definitions of ergodicity, mixing,
entropy
- equivalent formulations of ergodicity
- the von Neumann and Birkhoff ergodic
theorems
- 2-dimensional hyperbolic geometry
- the geodesic and horocyle flows
- classification of Bernoulli shifts
- unique ergodicity of interval exchange
maps
- translations surfaces
- Ratner's theorem and its application
to the Oppenheim conjecture

We will definitely cover the first 5 topics. The remaining topics are
more advanced and we certainly will not be able to get to all of them.
Much of what we do will be very concrete and won't require much
background except for some measure theory. Math 6210 will be more than
sufficient and even here I will give a brief review of measure theory
as we use it.

References:

Ergodic Theory and Topological Dynamics of Group Actions on
Homogeneous Spaces, M. Bachir Bekka & Matthias Mayer

Ratner's Theorems on Unipotent Flows, David Witte Morris

Notes
on Ergodic Theory, Curt McMullen