Applications of Ergodic Theory to Combinatorics and Number Theory

该项目的重点是遍历理论中的多重递归问题，重点是与组合学和数论的相互丰富的联系。所考虑的问题可被视为深远的扩展经典递归结果的动态。与此同时，这些问题导致强大的应用遍历理论的组合学，数论和代数是无法访问的，到目前为止，通过传统的方法。近年来，一些多项式结果的提议者作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。其中一些远景导致有趣的新的连接和应用理论的素数和有限组合。本提案中考虑的新问题反映了新方法和新技术的出现。这些包括几何方法利用动力系统的nilmanifolds和方法涉及拓扑代数的斯通-切赫紧化。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。该提案中提出的一些理论为幂零动力学与遍历理论和组合学的重要问题之间的联系提供了新的思路。另一组图解涉及借助于拓扑代数技巧改进递推结果。然而另一组aceturtures是基于新的成果和想法，其中链接遍历理论，数论在功能领域的有限特征，和combinatorics.The领域的遍历拉姆齐理论与其多样性的问题，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。本提案中提出的问题和猜想将数学的各个领域（遍历理论、组合学、代数、数论）联系起来，并对每个领域做出贡献。例如，近年来的方法，结果和思想起源于多重递归理论（其中一些是由于提议者）已经带来了壮观的进步，在理论的素数。另一个有趣的研究方向，已出现在最近几年链接遍历理论的多重递归与组合在有限领域。这些发展与理论计算机科学有关。拟议的研究旨在更好地了解在对应于多项式（和更一般的）函数值的时刻采样的动力系统的行为的规律性。虽然该提案侧重于这种现象在组合学和数论中的应用，但它可能也会引起物理学家的兴趣。

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.