This course is an introduction to
Ergodic Theory, leading to
ergodic theoretical proofs of
Szemeredi's theorem (the existence of
arbitrarily long arithmetic progressions in
sequences of integers of
positive upper density)
and of its multidimensional generalizations.
Prerequisites:
1. Basic measure theorywhat is covered in course 205a or equivalent.
Additional topics in measure theory will be part of the course.
2. Some very basic facts from functional analysis, some
covered concurrently in 205b.
Here again, the needed prerequisite material is minimal.
Additional needed material will be done in class; in particular
the spectral theorem for unitary
operators and, if time permits, of unitary actions of locally compact abelian groups (with ergodic theoretical applications to Z^{d} and R^{d}).
The course aims to show how elementary methods from measure theory and functional analysis, from the material essentially covered by Math 205a and 205b, combine in ``real life'', and yield some highly nontrivial mathematics.
Lecture notes for the entire program will be posted here. The first installment
can be found here:
Notes part 1.
Spectral theorem et al.
Additional material:
Bulletin article:
1978 article:
``Ergodic Szemer\'edi theorem for commuting transformations''
Erdos Volume article:
``Chromatic numbers of Cayley graphs on $\bbz$ and recurrence''
