Ergodic Ramsey Theory and Polynomial Dynamics on Nilmanifolds

这个项目的重点是遍历理论中的多重递归和收敛问题，重点是它们与动力系统的相互丰富的联系。所考虑的问题可以被看作是经典结果的深远的扩展。与此同时，这些问题导致遍历理论在组合学、数论和代数中的强有力的应用，而这些应用迄今为止都是用传统方法无法达到的。一些多项式的主要研究人员在最近几年获得的结果已成为进一步发展的动力，在理论的多重递归。这些发展提供了更好的理解沿沿着多项式的多重递归现象，并带来新的研究前景。其中一些研究途径导致有趣的新的联系和应用的理论素数和有限组合。由主要研究者的结果所开辟的一个有趣而重要的研究方向与幂零群的进入有关。不仅是大多数熟悉的结果处理交换群自然可扩展到幂零设置，但它也证明了幂零动力学允许一个得到新的信息的收敛性和递归性质的阿贝尔群的测度保持变换。作为项目基础的相关理论揭示了幂零动力学与遍历理论、组合学和均匀分布中的重要问题之间的联系。该项目正在调查的问题和理论将数学的不同领域（遍历理论、组合学、代数、数论）联系起来，并对每个领域都有贡献。例如，近年来的方法，结果，和思想起源于理论的“多重递归”（其中一些是由于主要研究人员）已经产生了惊人的进展，在研究素数。最近出现的另一个有趣的研究方向是把多重递归的遍历理论与“有限域”代数领域的组合学联系起来。“这些发展与理论计算机科学有关。该项目的目标是更好地了解在对应于多项式（和更一般）函数值的时刻采样的动力系统行为的规律性。虽然该项目的重点是这种现象在组合数学和数论数学领域的应用，但它可能也会引起物理学家的兴趣。最后，值得一提的是，所谓的遍历拉姆齐理论领域，以其多样性的问题，技术和应用，是一个很好的媒介吸引本科生数学和研究生的积极研究领域。

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.