This project focuses on the problems of multiple recurrence and convergence in ergodic theory, with emphasis on their mutually enriching connections with dynamical systems on nilmanifolds. The problems considered may be viewed as far-reaching extensions of classical results. At the same time, these problems lead to strong applications of ergodic theory to combinatorics, number theory, and algebra that are inaccessible, so far, by conventional methods. Some of the polynomial results obtained by the principal investigators in recent years have 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 avenues of research lead to interesting new connections with and applications to the theory of prime numbers and finitary combinatorics. An interesting and important direction of research opened up by the results of the principal investigators is connected to the entrance of nilpotent groups into the picture. Not only are most of the familiar results dealing with commutative groups naturally extendible to the nilpotent setting, but it also turns out that nilpotent dynamics allow one to get new information about convergence and recurrence properties of abelian groups of measure-preserving transformations. The related conjectures that underlie the project shed new light on the linkages between nilpotent dynamics and important problems in ergodic theory, combinatorics, and uniform distribution.
The problems and conjectures that are under investigation in this project 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 principal investigators) have engendered spectacular advances in the study of prime numbers. Another interesting direction of research that has emerged recently links ergodic theory of multiple recurrence with combinatorics in the algebraic area of "finite fields." These developments have connections with theoretical computer science. The goal of this project is to obtain a 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 project focuses on applications of this phenomenon in the mathematical areas of combinatorics and number theory, it may be of interest to physicists as well. Finally, it is worth mentioning that the area of so-called 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 research performed by the principal investigators lies in the mathematical realm of dynamical systems. A dynamical system, as a mathematical object, is comprised of a state space and a rule of evolution for the elements of this space. The branch of dynamical systems known as ergodic theory is mainly concerned with the study and applications of dynamical systems where the volume (calculated using a so-called measure on the state space) occupied by sub-collections of objects is unmodified by the transformations of the space which implement the evolutionary rule. While the roots of ergodic theory are in the kinetic theory of gases and celestial mechanics, modern ergodic theory stands at the junction of many areas. Problems that were considered and advanced by the principal researchers included: * The derivation of various results on the long-term behavior and recurrence properties of systems with continuous time from those pertaining to systems with discrete time. * Applying ergodic-theoretical techniques to the theory of prime numbers. * Studying dynamical systems with infinitely generated groups of measure preserving transformations and deriving from them new results in combinatorics and the theory of uniform distribution. * Obtaining new results on generalized polynomials (important algebraic objects obtained from conventional polynomials by using, in addition to the regular algebraic operations, the operation of taking the integer part) from dynamical results about systems on special kinds of topological surface, the so-called nil-manifolds. * Utilizing the topological algebra of the Stone-Cech compactification to obtain new results in combinatorics. Results obtained by the principal investigators were widely disseminated via publications, web postings and presentations at various forums.
