MAT6640: Additive Combinatorics, Winter Semester 2014

Prof: Andrew Granville
Bureau: 6153 André Aisenstadt, Tel: 343-6583; Courriel: andrew@dms.umontreal.ca

Course Book: >An Introduction to Additive Combinatorics, by Andrew Granville and Ben Green. Available for course participants, on request, by emailing me.
The only published course book is
Terence Tao and Van Vu's "Additive Combinatorics", Cambridge Studies in Advanced Mathematics 105, Cambridge University Press. i

Classtimes: Monday 10h30-12h00, Wednesday 14h30-16h00, in Pav Aisenstadt 5183
Class dates: From Jan 8 to Apr 21, 2014

Homework 2 (for 5/2/14): Questions: 2.2, 2.3, 2.4, 2.5, 2.6, 2.7 (for number theory students), 2.8, 2.9, 2.11, 2.12, 2.14, 2.22, 2.23, 2.25
Homework 1 (for 20/1/14): pg 11-12, section 1.4: Questions: 1.1, 1.2, 1.3, 1.4, 1.6, 1.7 (Compulsory). Also do 1.8-1.13 which are more fun. You should try 1.5, which is not easy, but worth thinking about.

En Francais, et page web complet

Soundararajan's 2007 course notes from Stanford

Papers to read for background

Terry Tao's blog contains an amazing amount of remarkable mathematics, usually well explained. He is one of the key people in the development of Additive Combinatorics and it is well worth finding his discussions of many of the key topics in this course.

WARNING! Wikipedia articles often contain misunderstandings and misleading statements. Do not use this as a primary source of information!

Course Contents/Syllabus

This will be a first course in additive combinatorics. This is a subject which incorporates ideas from an enormous number of areas: harmonic analysis, combinatorics, number theory, ergodic theory, discrete geometry, graph theory, theoretical computer science, even topology and algebra... The list is long, and no student can be expected to have all of the pre-requisites. We will attempt to explain the main ideas needed from the different subjects as they are needed. Our approach will be developed primarily from the point-of-view of harmonic analysis and analytic number theory.

There have been some spectacular results in this new area, including Green and Tao's proof that there are infinitely many k-term arithmetic progressions of primes, Bourgain's bounds on very short exponential sums, and the theory of prime values of the co-ordinates of orbits of matrix groups as developed by Bourgain-Gamburd-Sarnak and others. We will discuss these results, though fall far short of proving any of them! The main goal of our course will be to understand some of the key results and Gower's first proof of Szemeredi's Theorem for four term arithmetic progressions.

Topics may include
  • Adding sets and additive bases for the integers (including Dyson transformations).
  • Adding finite sets and covering lemmas.
  • Van der Waerden's theorem and the Hales-Jewett Theorem.
  • Discrete Fourier transformations and Weyl's equidistribution Theorem.
  • Vinogradov's three primes Theorem.
  • Roth's theorem.
  • The Freiman-Ruzsa Theorem.
  • Sum-Product Formulas. In the integers, in the reals, in finite fields, and in groups of matrices.
  • The Balog-Szemeredi-Gowers Theorem.
  • Gowers' norms and the inverse conjecture.
  • Szemeredi's regularity lemma.
  • Szemeredi's Theorem for four term arithmetic progressions.