2007-2008 Milliman Lectures
Department of Mathematics
University of Washington

TERENCE TAO
Professor of Mathematics
University of California, Los Angeles

December 4th, 5th, and 6th
4:00 - 5:00pm in Physics/Astronomy Auditorium, Room A118

Recent Developments in Arithmetic Combinatorics

Lecture I: Additive Combinatorics and the Primes
(Tuesday, December 4th)

Abstract: Given a set A of N integers, how many additive patterns (e.g. arithmetic progressions of length three) does A have? The answer depends of course on the nature of A. At one extreme are sets with very strong additive structure, such as the arithmetic progression {1,..., N}, which tend to have many additive patterns. At the other extreme are "random" sets A, which tend to have relatively few additive patterns. Understanding and classifying these two extremes, and the extent to which an arbitrary set lies between these extremes, is a major topic of additive combinatorics today. Recent progress on these questions has led to applications in number theory, particularly regarding the question of detecting additive patterns in the primes; in particular, I will discuss my result with Ben Green that the primes contain arbitrarily long arithmetic progressions.
View a streaming video of Lecture I using the free RealPlayer.

Lecture II: Additive Combinatorics and Random Matrices
(Wednesday, December 5th)

Abstract: The theory of random continuous matrices (such as the gaussian unitary ensemble or GUE), and in particular the study of their eigenvalues, is an intensively studied and well-understood subject. In contrast, the theory of random discrete matrices, such as the Bernoulli ensemble in which each entry of the matrix equals +1 and -1 with equal probability, is only just now being developed. Much of the recent progress relies on an understanding of the distribution of discrete random walks, and in particular on solving the inverse Littlewood-Offord problem. This in turn requires the use of tools from additive combinatorics, such as the geometry of multidimensional arithmetic progressions, and Freiman's inverse theorem. I will survey these developments, which include joint work of myself with Van Vu.
View a streaming video of Lecture II using the free RealPlayer.

Lecture III: Sum-product Estimates, Expanders, and Exponential Sums
(Thursday, December 6th)

Abstract: Consider a finite set A of elements in a ring or field. This set might be "almost closed under addition", which for instance occurs when A is an arithmetic progression such as {1,..., N}. Or it may be "almost closed under multiplication", which for instance occurs when A is a geometric progression such as {1, a,..., a^N}. But it is difficult for A to be almost closed under addition and multiplication simultaneously, unless it is very close to a subring or subfield. The emerging field of sum-product estimates in arithmetic combinatorics seeks to establish strong inequalities to quantify this phenomenon. Recent breakthroughs in this area by Bourgain and coauthors, based primarily on Fourier analysis and additive combinatorial tools, have yielded new exponential sum and sieve estimates in number theory, new randomness extractors in computer science, and new constructions of expander graphs in combinatorics. This lecture will be a survey of several of these developments.
View a streaming video of Lecture III using the free RealPlayer.

Terence Tao was born in Adelaide, Australia in 1975. He has been a professor of mathematics at UCLA since 1999, having completed his PhD under Elias Stein at Princeton in 1996. Tao's areas of research include harmonic analysis, PDE, combinatorics, and number theory. He has received a number of awards, including the Salem Prize in 2000, the Bochner Prize in 2002, the Fields Medal and SASTRA Ramanujan Prize in 2006, and the MacArthur Fellowship and Ostrowski Prize in 2007. Terence Tao also currently holds the James and Carol Collins chair in mathematics at UCLA, and is a Fellow of the Royal Society and the Australian Academy of Sciences (Corresponding Member).