Program

Much recent work in ergodic theory has been motivated by interactions with combinatorics and with number theory. A particular is example is Szemerédi's Theorem, which states that a set of integers with positive upper density contains arbitrarily long arithmetic progressions. The original argument was an intricate use of combinatorics; a second proof was given by Furstenberg using ergodic theory and more recently, Gowers gave a third proof based on Fourier analysis. In the last few years, methods of combinatorics, number theory, harmonic analysis, and ergodic theory have been combined to attack old problems on patterns, such as arithmetic progressions, in the prime numbers. Furstenberg's proof uncovered the connection between combinatorial results and ergodic theory, and his ergodic theoretic proofs of combinatorial statements had unforeseen consequences within ergodic theory itself. Furstenberg and others introduced certain classes of dynamical systems and ergodic theoretic structures, and their study has become of independent interest. Moreover, the better understanding of the underlying ergodic theory has provided new combinatorial results, some of which have yet to be proven by any other method. The recent result of Green and Tao on arbitrarily long arithmetic progressions in the set of primes immediately attracted the attention of ergodic theorists. The Green-Tao proof, similar to Furstenberg's proofs, is based on a philosophy used in ergodic theory since Riesz's proof of the Mean Ergodic Theorem: prove a structure theorem, showing that a given object can be decomposed into "structured" and "negligible" parts. However, the interest runs deeper. Finer analysis of structures in the primes has relied on "non-commutative" methods; objects similar to those in the structure theorem for multiple ergodic averages have now arisen in a combinatorial setting. These non-commutative objects have also arisen in harmonic analysis, in the context of Bourgain's result on subsequence ergodic theorems. One aim of the proposed program is to understand the meaning of these recent results for ergodic theory. As the history of Szemerédi's Theorem shows, such an understanding benefits both ergodic theory and other fields, such as probability, combinatorics, number theory and harmonic analysis. A difficulty facing researchers in this area is the need to be fluent in several felds of mathematics: number theory, ergodic theory, combinatorics, and harmonic analysis. A few of the leading researchers are already able to use tools from the various fields and pass from one field to another with ease. A systematic survey of the fields benefits a wider audience. This program brings together researchers in ergodic theory, harmonic analysis, number theory, and combinatorics who are interested in similar problems from very diffrent perspectives. While the main theme of the program is ergodic theory (and in this it differs from other similar proposed programs), we plan to highlight the unity between ergodic theory and these related areas; we welcome researchers in any of these areas. For information how to apply please go to: Member Application