Additive Combinatorics, Spring 2011


This will be a graduate-level introduction to the fascinating subject of additive combinatorics.

A list of topics. The first half is essentially fixed and the second half is very open to suggestions. We will spend half to two-thirds of the class discussing some foundational topics (including proofs of Roth's and Freiman's theorems) and will spend the remainder discussing additional topics, including some of the very recent work in the subject. Feedback and suggestions highly encouraged!

Some sources:

Tao and Vu, Additive Combinatorics (book).

Sound's notes from 2006.

Gowers' notes (which Sound's notes were modelled on).

See also Terry Tao's blog and Gowers's blog.

Meeting times. PLEASE READ. IMPORTANT.

This class will meet on an unusual schedule, as I am serving on a jury for an extended trial. The class will meet on Fridays, 9:30-10:45, beginning April 1, until the trial is complete. Currently the court is predicting the trial will conclude in April. After the trial finishes the class will be MWF 9:30-10:45. Additional classes may be held Mondays or Wednesdays if the court is out of session for any reason.

We will be meeting Monday, April 4.

If needed (looks like it won't be, fortunately), makeup classes will be held after the end of the term (avoiding the Calgary, Edmonton, and AIM conferences in June) and/or on Tuesdays and Thursdays after the conclusion of the trial (or on days the court is off). Attendance at makeups will be optional.

Prerequisites

Graduate standing (in the math department) or permission of instructor.

Formal prerequisites for the subject are hard to quantify. I will aim my lectures at the math grad students, but others may be able to follow as well. The lectures might assume a knowledge of: elementary abstract algebra; enumerative combinatorics; complex and Fourier analysis; analytic number theory. Fluency in manipulating sums, identities, and inequalities will be assumed.

If you are trying to determine if your background is sufficient, please read the proof of Roth's theorem on pp. 10-12 of Sound's notes. If it mostly makes sense after a close reading (it's okay if you have a couple of questions) then you are probably reasonably prepared.

Homework

Students will be asked to write a ten-page paper and give an hourlong presentation on a topic of their choice (I am happy to offer suggestions; the book, notes, and blogs above abound with interesting mathematics). This requirement may be waived for math Ph.D. students at the discretion of the instructor.

See also the homework exercises on Sound's website.