Ergodic Ramsey Theory, Polynomials, and Actions of Nilpotent Groups

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ABSTRACTV. Bergelson and A. Leibman will study multiple recurrence andconvergence in ergodic theory with emphasis on the polynomial aspects ofbehavior. The proposers earlier showed that sampling along polynomial times was suffient to reveal deep ergodic phenomina. The problems considered may be viewedas far reaching extensions of classical results. In addition to demonstrating new properties of dynamical systems, theseproblems lead naturally to strong applications of ergodic theory to combinatorics,number theory and algebra which until now have been inaccessible conventionalmethods. The polynomial Szemeredi theorem and the polynomialHales-Jewett theorem, obtained by the proposers in recent years, already serve asan impetus for further developments in the theory of multiple recurrence.This provides better understanding of multiple recurrence along polynomials and bring to light new vistas of research. An important feature of the research opened by theproposers's earlier work is the appearance of nilpotent nilpotent groups into the picture.Not only does this lead to strong applications in combinatorics and numbertheory, but the computational flexibility stemming from this theory may be of significance in applications.