Multiple Recurrence and Convergence Along Polynomials in Ergodic Ramsey Theory

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Abstract 9706057 Bergelson/Leibman This project will investigate problems of multiple recurrence and convergence with the emphasis on the behavior of dynamical systems along polynomials. The recurrence problems considered are tied to problems in combinatorial number theory and set theory and the results can be used to prove the existence of patterns with the conventional methods of classical analysis, combinatorics and number theory fail to discern. The PIs recently proved that the ergodic polynomial Szemeredi theorem extends to operators generating a nilpotent group. This naturally leads to more general questions and conjectures about multiple recurrence for nilpotent group actions. This grant is to fund the study of the long term behavior of dynamical systems and stochastic processes. Many important physical phenomena, as varied as the behavior of the stock market, lines in the supermarket, and data being received from a distant transmission source, exhibit locally erratic behavior which can be understood better by looking at the long term average behavior instead. The study of such stochastic processes and their connections with one another, as well as their connections in terms of their fine structure with other less random processes, is a central one in modern mathematics. It is only through such studies that we will be able to understand better and thus be able to control or predict the evolution of the physical phenomena which this mathematics can model.