
20072008 Milliman Lectures TERENCE TAO
December 4th, 5th, and 6th Recent Developments in Arithmetic Combinatorics Lecture I: Additive Combinatorics and the Primes 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. Lecture II: Additive Combinatorics and Random Matrices 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 wellunderstood 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 LittlewoodOfford 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. Lecture III: Sumproduct Estimates, Expanders, and Exponential Sums 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 sumproduct 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.
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). Click here for the Milliman Lecture homepage. 



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