Spring 2025
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:
Introduction: preliminary examples; general statements of Ratner's Orbit Closure Theorem, Equidistribution Theorem, and Measure Classification Theorem
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.
......
A proof for the simplest nontrivial case SL(2,R)
Background needed for the general case
Lie theory
Ergodic theory and entropy
Algebraic groups
A sketch of the proof of the general case
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 2.3 of [WM])
Mon Mar 24: Entropy -- how to calculate them (Sec 2.4 of [WM])
Wed Mar 26: Pesin theory -- entropy of volume-preserving diffeomorphisms (papers by D. Ruelle, R. Mañé) (Sec 2.5 of [WM])
Mon Mar 31: Entropy estimates (cf paper by Margulis-Tomanov) (Sec 2.6 of [WM])
Wed Apr 2: Entropy estimates (Sec 2.6 of [WM])
Mon Apr 7: Some ergodic theory; Birkhoff ergodicity theorem (Ch 3 of [WM], see also Ch 2 in this set of notes by C. McMullen)
Wed Apr 9: Ergodic decompositions
Mon Apr 14: Ergodic decompositions; Mautner phenomenon, Moore ergodicity theorem (Ch 3 of [WM])
Wed Apr 16: Mautner phenomenon, Moore ergodicity theorem
Mon Apr 21: Hyperbolic 3-manifolds and PSL(2,C)
Wed Apr 23: Proof of Ratner for PSL(2,R)-orbits in PSL(2,C)
Mon Apr 28: Proof of Ratner for PSL(2,R)-orbits in PSL(2,C)
Wed Apr 30: Related directions and open questions -- the infinite volume case
Mon May 5: Related directions and open questions -- effective equidistribution
Wed May 7: Related directions and open questions -- inhomogeneous PSL(2,R)-action on moduli spaces