Dynamical systems on nilmanifolds, ultrafilters, and polynomial multiple correlation sequences

Ohio State University, The, Columbus OH

Investigators

Abstract

The problems and conjectures presented in this program can be viewed as far reaching extensions of classical recurrence results in dynamics. At the same time, these problems connect diverse areas of mathematics (such as ergodic theory, combinatorics, number theory, topological algebra) and contribute to each. The new problems considered in this proposal deal with interesting and promising new connections and reflect the entrance of novel methods and techniques in the picture. These include, in particular, the methods involving dynamical systems on nilmanifolds and methods utilizing the topological algebra in compactifications. The conjectures stated throughout this proposal are supported by the results obtained by the investigators and other workers in this vibrant area in recent years. The polynomial Szemeredi theorem, the polynomial Hales-Jewett theorem, and various additional results obtained by the investigators in recent years served as an impetus for further developments in the theory of multiple recurrence. These developments provide better understanding of the phenomenon of multiple recurrence and bring new vistas of research to light. The directions of study touched upon in this proposal reveal strong and mutually perpetuating connections between combinatorics, number theory and various aspects of recurrence and convergence in the theory of dynamical systems. The field of Ergodic Ramsey Theory, with its richness of problems and connections and diversity of techniques and methods, is a meeting point of several branches of modern mathematics and is an excellent medium for attracting undergraduates to mathematics and graduate students to an area of active research.