Videos

Introduction to Ratner's Theorems on Unipotent Flows

Presenter
February 2, 2015
Keywords:
  • quasi-immersed manifolds
  • dense embedding
  • Ratner's theorem
  • discrete group actions
  • Riemannian geometry
  • actions on Lie groups
MSC:
  • 53C42
  • 53Cxx
  • 53C35
  • 53C30
  • 22D40
  • 37-XX
Abstract
Let f be the obvious covering map from Euclidean n-space to the n-torus. It is well known that if L is any straight line in n-space, then the closure of f(L) is a very nice submanifold of the n-torus. In 1990, Marina Ratner proved a beautiful generalization of this observation that replaces Euclidean space with any Lie group G, and allows L to be any subgroup of G that is ``unipotent.'' We will discuss the statement of this theorem and related results, some of the ideas in the proofs, and a few of the important consequences.
Supplementary Materials