Ergodic Ramsey Theory, Polynomials, and Actions of Nilpotent Groups

Ohio State University Research Foundation -Do Not Use, Columbus OH

Investigators

Abstract

ABSTRACT V. Bergelson and A. Leibman will study multiple recurrence and convergence in ergodic theory with emphasis on the polynomial aspects of behavior. The proposers earlier showed that sampling along polynomial times was suffient to reveal deep ergodic phenomina. The problems considered may be viewed as far reaching extensions of classical results. In addition to demonstrating new properties of dynamical systems, these problems lead naturally to strong applications of ergodic theory to combinatorics, number theory and algebra which until now have been inaccessible conventional methods. The polynomial Szemeredi theorem and the polynomial Hales-Jewett theorem, obtained by the proposers in recent years, already serve as an 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 the proposers's earlier work is the appearance of nilpotent nilpotent groups into the picture. Not only does this lead to strong applications in combinatorics and number theory, but the computational flexibility stemming from this theory may be of significance in applications.