Ergodic Ramsey Theory and Dynamical Systems on Nilmanifolds

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AbstractThe project is focused on the problems of multiple recurrence and convergence in ergodic theory with emphasis on the 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 which are inaccessible, so far, by conventional methods. The polynomial Szemeredi theorem, the polynomial Hales-Jewett theorem and extensions thereof, obtained by the PIs 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. An interesting and important direction of research opened up by the polynomial results of the PIs 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 setup, but also it turns out that nilpotent dynamics allows one to get new information about convergence/recurrence properties of one parameter groups of measure preserving transformations. The related conjectures formulated in the proposal shed new light on the connections of nilpotent dynamics with important problems of ergodic theory, combinatorics and uniform distribution.The problems and conjectures that are posed in the proposal connect diverse areas of mathematics (ergodic theory, combinatorics, number theory) and contribute to each. 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 strong applications of this phenomenon in combinatorics and number theory, it may be of interest to a physicist as well. For example, one of the corollaries of the theory of multiple recurrence is that measuring the status of a physical system along polynomial (rather than linear) instances of time reveals quite a lot about the system.