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6th Annual Thomas Wolff Memorial Lecture
Topics in Additive Combinatorics.
The lectures will be accessible to a general mathematical audience.
Lecture 1 (May 9): Freiman's Theorem.
Abstract: Suppose that I have a set A in some abelian group. We write A + A for the set of elements of the form x + y, where x and y both lie in A. If the size of A + A is not much larger than A then we say that A has "small doubling". What can we say about the structure of sets with small doubling? Freiman's theorem and related theorems by other people provide a pretty decent answer to this question. We will discuss these results together with some interesting open problems.
Lecture 2 (May 11): Gowers Norms and Nilsequences.
Abstract: The Gowers U^k-norms, k = 2,3,4... are a family of norms which may be defined for functions on any finite abelian group. The Gowers U^k-norm of a function f : G -> C measures, in a certain sense, how close f is to being a "random" function. We will discuss these norms and their connection with so-called nilsequences, which are objects arising from nilpotent Lie groups. We will explain how these objects might play a role in a kind of "non-abelian harmonic analysis" which has yet to be fully understood.
Lecture 3 (May 15): Arithmetic Progressions.
Abstract: We will discuss some special cases of the famous theorem of Szemeredi, which states that any set of integers with "positive density" contains arbitrarily long arithmetic progressions.
Lecture 4 (May 16): Patterns of Primes.
Abstract: We will describe how ideas in the first three lectures may be used to prove the existence of certain patterns of prime numbers. In particular we will indicate how it is possible to prove that there are arbitrarily long arithmetic progressions of primes, and how one may give an asymptotic count of the number of 4-tuples p_1 < p_2 < p_3 < p_4 of arithmetic progressions of primes.
Ben Green was recognized for his joint work with Terry Tao on arithmetic progressions of prime numbers. These are equally spaced sequences of primes such as 31, 37, 43 or 13, 43, 73, 103. Results in the area go back to the work of Lagrange and Waring in the 1770's. A major breakthrough came in 1939 when the Dutch mathematician Johannes van der Corput showed that there are an infinite number of three-term arithmetic progressions of primes. Green and Tao showed that for any n, there are infinitely many n-term progressions of primes. Their proof, which relies on results of Szemerédi (1975) and Goldston and Yildirim (2003), uses ideas from combinatorics, ergodic theory, and the theory of pseudorandom numbers. The Green-Tao result is a major advance in our understanding of the primes.
Green was born in 1977 in Bristol, England. He was educated at Trinity College, Cambridge first as an undergraduate and later as a research student of Fields Medalist Tim Gowers. Since 2001 he has been a Fellow of Trinity College, and in that time he has visited Princeton, the Renyi Institute in Budapest and University of British Columbia, Vancouver, for extended periods. Green held a chair in Pure Mathematics at Bristol University from January 2005 until his return to Cambridge in September 2006.
Green will hold an appointment as a Clay Research Fellow from July 1, 2005 to June 30, 2007.
You are invited to attend a dinner following theThomas Wolff Memorial Lectures in Mathematics at The Athenaeum on Wednesday, May 9, 2007
Host bar 5:45 p.m.
MENUCitrus Avocado Salad
Grilled Rosemary Marinated Free Range Chicken
Cinnamon Apple Galette
Please indicate if you require a vegetarian or kosher meal
For reservations, please contact Stacey V. Croomes at 626-395-4336 or send payment by May 4, 2007 made out to Caltech for $35.00 per person to:
Stacey V. Croomes
The Thomas Wolff lectures, sponsored by donations from his widow and his parents, memorialize Caltechs great analyst who was tragically killed at age 46 in an automobile accident in July 2000. Wolff was a specialist in analysis, particularly harmonic analysis. Professor Wolff made numerous highly original contributions to the mathematical fields of Fourier analysis, partial differential equations, and complex analysis. A recurrent theme of his work was the application of finite combinatorial ideas to infinite, continuous problems.
His early work on the Corona theorem, done as a Berkeley graduate student, stunned the mathematical community with its simplicity. Tom never wrote it up himself since several book writers asked for permission to include the proof in their books where it appeared not long after he discovered it. After producing a number of very significant papers between 1980 and 1995, he turned to the Kakeya problem and its significance in harmonic analysis, works whose impact is still being explored.
Peter Jones, professor of mathematics at Yale, described Tom’s contributions as follows: “The hallmark of his approach to research was to select a problem where the present tools of harmonic analysis were wholly inadequate for the task. After a period of extreme concentration, he would come up with a new technique, usually of astonishing originality. With this new technique and his well-known ability to handle great technical complications, the problem would be solved. After a few more problems in the area were resolved, the field would be changed forever. Tom would move on to an entirely new domain of research, and the rest of the analysis community would spend years trying to catch up. In the mathematical community, the common and rapid response to these breakthroughs was that they were seen not just as watershed events, but as lightning strikes that permanently altered the landscape.”
Tom Wolff was noted for his analytic prowess, the depth of his insights, and the passion with which he nurtured the talents of young mathematicians. We miss him.