2005-2006 Milliman Lectures


		2005-2006 Milliman Lectures Department of Mathematics University of Washington TIMOTHY GOWERS Rouse Ball Professor of Mathematics Cambridge University April 4th, 5th, and 6th 4:00 - 5:00pm Smith Hall, Room 120 The lectures will be aimed at a wide, non-specialist audience, but some proofs will appear, especially in the later lectures. The third lecture will depend to some extent on familiarity with ideas discussed in the second. The first lecture is not a logical prerequisite for the later ones, but it will provide considerable motivation for them. Lecture I: Some Theorems and Open Problems in Arithmetic Combinatorics (Tuesday, April 4th) Arithmetic combinatorics is the name that has been given to a thriving new area of mathematics, or, to be more accurate, a new and very interesting fusion of old areas. I shall give an overview of the area and try to explain what combinatorics, harmonic analysis, and number theory have to offer each other. Lecture II: Discrete Fourier Analysis: Its Uses and Limitations (Wednesday, April 5th) This lecture will be about one of the main techniques in arithmetic combinatorics: the use of the discrete Fourier transform. I shall outline how it can be used to prove two central theorems in the subject, namely Roth's theorem on arithmetic progressions, and Freiman's theorem on the structure of sets with small sumsets. Both these theorems result in major open problems, and I shall try to explain their difficulty by highlighting what it is that we do not yet understand about Fourier analysis. Lecture III: The Potential of "Polynomial" Fourier Analysis (Thursday, April 6th) In this lecture I shall try to give some idea about the proofs of two celebrated results about arithmetic progressions: Szemerédi's theorem, which states that every dense set of integers contains arbitrarily long arithmetic progressions, and the Green-Tao theorem, which tells us that the same is true of the set of primes. Both these results involve situations where conventional Fourier analysis is inadequate, but recently discovered "polynomial" generalizations can be used instead. Much remains to be understood: if time permits, I shall speculate about what arithmetic combinatorics may look like in twenty years' time. Timothy Gowers is the Rouse Ball Professor of Mathematics at Cambridge University. He works in combinatorics, combinatorial number theory, and the theory of Banach spaces, and has made fundamental contributions to each of these fields. Before Gowers' work, most mathematicians would have viewed these as being unrelated, but Gowers has shown otherwise, to great success: in 1998 he was awarded a Fields Medal. In 1996 he received the Prize of the European Mathematical Society, and in 1999 he was elected Fellow of the Royal Society. Banach spaces are important in quantum physics, as well as in mathematics, and mathematicians and physicists study their inner structure and their symmetries. When Gowers began working on Banach spaces, many of the most important problems were rather old, dating from the work of the eponymous Polish mathematician Stefan Banach (1892-1945). Solving one fifty-year old problem is significant, but Gowers has in fact settled a number of these. In combinatorics Gowers has worked on problems involving arithmetic progressions and randomness in graph theory. One notable result was a beautiful new proof of a famous theorem of Endre Szemerédi about arithmetic progressions. He has also studied extremal graph theory, improving on another of Szemerédi's results, his regularity lemma: Gowers showed that what appeared in the lemma to be a very weak lower bound was in fact tight. Gowers also wrote the wonderful book, Mathematics: A Very Short Introduction. Click here for the Milliman Lecture homepage.

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