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.)
4 comments
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14 December, 2007 at 4:22 pm
Anonymous
Hello,
Ratner’s theorem is quite different from the other theme of your course. The only connection I see is that both apply to NT. That’s the reason for the choice ?
A
14 December, 2007 at 4:42 pm
Terence Tao
Dear Anonymous,
I am planning to focus mainly on Ratner’s theorem for nilmanifolds, which does have some connections to Furstenberg’s multiple recurrence theorem due to the emergence of nilmanifolds as characteristic factors for this type of multiple ergodic average. Of course, the real applications of Ratner’s theorem are for more sophisticated homogeneous spaces than nilmanifolds; I haven’t yet decided how much of those I will cover.
14 December, 2007 at 6:37 pm
Doug
Hi Terence,
Two questions about ergodic theory possibly relating to other mathematical concepts:
1 – Is there a link to the Richard Bellman concept of dynamic programming?
2 – Can the 2006 Gauss prize concepts of Kiyosi Itô, stochastic analysis, be utilized as they were for explaining Brownian motion in terms of probability as opposed to randomness?
15 December, 2007 at 2:53 am
Adam
A more pertinent question is how the ergodic hypothesis is
related to the dynamics of finite systems of spherical particles,
where a fraction of the kinetic energy may be lost during collisions.
After a sufficient time such a system decouples into maximal subsystems
(clusters). This was officially conjectured by van der Waals in his Nobel
lecture (pseudoassociations) and shown by L. N. Vaserstein in Commun. Math. Phys. 69, 31-56 (1979).