Instructor: David Kerr
TR 12:45-2:00, Milner 313
The course will treat two main
topics of current research interest in
the theory of measure-preserving group actions on a probability space.
The first is Furstenberg's dynamical approach to Szemeredi's theorem
in additive combinatorics, which says that every set of natural numbers
with positive upper density contains
arbitrarily long arithmetic progressions. The second is Popa's recent
pair of cocycle superrigidity theorems with applications to orbit equivalence.
Common themes will be stressed, notably weak mixing and compactness and
dichotomies based around these two notions. A discussion
of certain analytic aspects of the
theory of discrete groups such as amenability and Kazhdan's property (T)
will also be included.
No background in ergodic
theory is required, but some functional analysis will be assumed
(e.g., operators on Hilbert space).
Prerequisite: MATH 608 or permission of the instructor.
The main references for the course are the following:
- H. Furstenberg. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton Univerisity Press, Princeton, NJ, 1981.
- H. Furstenberg, Y. Katznelson, and D. Ornstein. The ergodic theoretical proof of Szemeredi's theorem.
Bull. Amer. Math. Soc. (N.S.) 7
- Terence Tao's online lecture notes
- S. Popa. Cocycle and orbit equivalence superrigidity for malleable actions of w-rigid groups.
Invent. Math. 170 (2007), 243-295.
- S. Popa. On the superrigidity of malleable actions with spectral gap. J. Amer. Math. Soc. 21 (2008), 981-1000.
A. Furman. On Popa's cocycle superrigidity theorem.
Int. Math. Res. Not., vol. 2007, article ID rnm073.
For basic references in ergodic theory the following books are recommended:
- Eli Glasner. Ergodic Theory via Joinings. American Mathematical Society, Providence, RI, 2003.
- Karl Petersen. Ergodic Theory. Cambridge University Press, Cambridge, 1989.
- Peter Walters. An Introduction to Ergodic Theory. Springer-Verlag, New York, 2000.