Time

The regular meeting time is on Wednesday at 1:25pm in Room 813.
There will be NO CLASS on Wednesday, January 24 (second week of term). A make-up class will be scheduled later in the semester.

Syllabus

The main goals to discuss Szemeredi's theorem (every subset of the integers with positive upper density contains arithmetic progressions of any finite length)as well as Ratner's theorem.
Here is an approximate syllabus for the course.

Lecture notes

I am hoping to post (sometimes abridged) notes for the lectures here. They are not always carefully edited and are very likely to contain many mistakes! Please email me with any corrections.
  1. Lecture 1: Fourier-analytic proof of Roth's theorem
  2. Lecture 2: Dynamical formulation of Szemeredi's theorem.
  3. Lecture 3: Furstenberg's proof of Roth's thereom
  4. Lecture 4: Weak-mixing. Factors.
  5. Sorry, no lecture notes for Lecture 5. We discussed factors in more detail.
  6. Lecture 6: Completion of proof.
  7. Lecture 7: The hyperbolic plane, geodesic flow, horocycle flow.
  8. Lecture 8: mixing of the SL(2,R)-action and consequences. (Revised; rather, a better version of the latter half of the notes will appear in the Lecture 9 notes instead)
  9. Lecture 9.
  10. Lecture 10: basic spectral theory of SL(2,R)-action(revised to reflect the order in class; covers roughly lecture 10,11)
  11. Lecture 12: towards quantitative mixing.

Online references

  1. A book by Manfred Einsiedler and Tom Ward about ergodic theory with a view towards number theory.
  2. A book by David Witte about Ratner's theorems.
  3. A survey article by Terence Tao discussing different approaches to Szemeredi's theorem.
  4. Papers by Ziegler, Host and Kra, Green and Tao, referred to in Lecture 6.