MAT 656 Topics in Dynamical Systems

An introduction to homogeneous dynamics, focusing on a collection of theorems named after M. Ratner on unipotent flows. We will also discuss many applications of Ratner's theorems in dynamics, geometry and number theory.

Lectures: Mondays and Wednesdays 9:30 -- 10:50am in Physics P112

Office hours: Mondays and Wednesdays 11:00am -- 12:00pm

Office: Math Tower 5D148C

Main reference: Ratner's theorems on unipotent flows, by Dave Witte Morris. You can find publisher's version and arXiv version.

Here is a rough plan for the semester:

1. Introduction: preliminary examples; general statements of Ratner's Orbit Closure Theorem, Equidistribution Theorem, and Measure Classification Theorem

2. Applications of Ratner's theorems

   • Geometry: unipotent flows on hyperbolic manifolds, geodesic submanifolds in hyperbolic manifolds, etc.

   • Number theory: Oppenheim conjecture, lattices, quadratic forms, etc.

   • ......

3. A proof for the simplest nontrivial case SL(2,R)

4. Background needed for the general case

   • Lie theory

   • Ergodic theory and entropy

   • Algebraic groups

5. A sketch of the proof of the general case

6. Related results, generalizations, and current directions

[WM]=Ratner's theorems on unipotent flows, by Dave Witte Morris

Mon Jan 27: Preliminary examples on tori (Chapter 1 of [WM])
Wed Jan 29: Preliminary examples on tori (Chapter 1 of [WM])
Mon Feb 3: Some Lie theory; general statements of Ratner's theorems (Chapter 1 of [WM])
Wed Feb 5: More on the general statements; some basic hyperbolic geometry (Chapter 1 of [WM])
Mon Feb 10: Geodesic and horocycle flows on hyerbolic surfaces
Wed Feb 12: Cancelled
Mon Feb 17: Cancelled
Wed Feb 19: Geodesic and horocycle flows, higher-dimensional examples; Oppenheim conjecture
Mon Feb 24: Double cosets; polynomial divergence and shearing (Sec 1.5 of [WM])
Wed Feb 26: Polynomial divergence and shearing; minimal sets (Sec 1.5 of [WM])|
Mon Mar 3: Proof of Hedlund theorem
Wed Mar 5: Sketch of measure-theoretic version for PSL(2,R)
Mon Mar 10: Entropy -- examples of two dynamical systems (irrational rotation and symbolic shift) (Sec 2.1 of [WM])
Wed Mar 12: Entropy -- definitions and some properties (Sec 2.2 of [WM])