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