Next quarter, starting on Wednesday January 9, I will be teaching a graduate course entitled “Topics in Ergodic Theory“. As an experiment, I have decided to post my lecture notes on this blog as the course progresses, as it seems to be a good medium to encourage feedback and corrections. (On the other hand, I expect that my frequency of posting on non-ergodic theory topics is going to go down substantially during this quarter.) All of my class posts will be prefaced with the course number, 254A, and will be placed in their own special category.
The topics I plan to cover include
- Topological dynamics;
- Classical ergodic theorems;
- The Furstenberg-Zimmer structure theory of measure preserving systems;
- Multiple recurrence theorems, and the connections with Szemerédi-type theorems;
- Orbits in homogeneous spaces (and in particular, in nilmanifolds);
- (Special cases of) Ratner’s theorem, and applications to number theory (e.g. the Oppenheim conjecture).
If time allows I will cover some other topics in ergodic theory as well (I haven’t decided yet exactly which ones to discuss yet, and might be willing to entertain some suggestions in this regard.)
If this works out well then I plan to also do the same for my spring class, in which I will cover as much of Perelman’s proof of the Poincaré conjecture as I can manage. (Note though that this latter class will build upon a class on Ricci flow given by my colleague William Wylie in the winter quarter, which will thus be a de facto prerequisite for my spring course.)