The project is focused on the problems of multiple recurrence in ergodic theory with emphasis on the mutually enriching connections with combinatorics and number theory. The problems considered may be viewed as far reaching extensions of classical recurrence results in dynamics. At the same time, these problems lead to strong applications of ergodic theory to combinatorics, number theory and algebra which are inaccessible, so far, by conventional methods. Some of the polynomial results obtained by the proposers 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 along polynomials and bring new vistas of research to light. Some of these vistas lead to interesting new connections with and applications to the theory of prime numbers and finitary combinatorics. The new problems considered in this proposal reflect entrance of new methods and techniques into the picture. These include the geometric method utilizing dynamical systems on nilmanifolds and methods involving the topological algebra in Stone-Cech compactifications. Not only are most of the familiar results dealing with commutative groups naturally extendible to the nilpotent setup, but also it turns out that nilpotent dynamics allows one to get new information about convergence and recurrence properties of abelian groups of measure preserving transformations. Some of the conjectures formulated in the proposal shed new light on the connections of nilpotent dynamics with important problems of ergodic theory and combinatorics. Another group of conjectures deals with refining of the recurrence results with the help of the topological algebra techniques. Yet another group of conjectures is based on new results and ideas which link together ergodic theory, number theory in function fields of finite characteristics, and combinatorics.
The area of Ergodic Ramsey Theory with its diversity of problems, techniques and applications is an excellent medium for attracting undergraduates to mathematics and graduate students to an area of active research. The problems and conjectures that are posed in this proposal connect diverse areas of mathematics (ergodic theory, combinatorics, algebra, number theory) and contribute to each. For example, in recent years the methods, results and ideas originating in the theory of multiple recurrence (some of which are due to proposers) have brought spectacular advancements in the theory of prime numbers. Another interesting direction of research that have emerged in recent years links together ergodic theory of multiple recurrence with combinatorics in finite fields. These developments have connections with the theoretical computer science. The proposed study aims at better understanding of the regularity of the behavior of dynamical systems sampled at moments of time corresponding to values of polynomial (and more general) functions. While the proposal focuses on applications of this phenomenon in combinatorics and number theory, it may be of interest to a physicist as well.
