Dynamical systems on nilmanifolds, ultrafilters, and polynomial multiple correlation sequences

本程序中提出的问题和算法可以看作是动力学中经典递推结果的广泛推广。同时，这些问题连接了数学的不同领域（如遍历理论，组合数学，数论，拓扑代数），并对每个领域都有贡献。本提案中考虑的新问题涉及有趣和有前途的新联系，并反映了新方法和新技术的进入。这些包括，特别是，涉及动力系统的方法对nilmanifolds和方法利用拓扑代数紧化。在整个建议中所述的acetrutures支持的结果所取得的调查人员和其他工作人员在这一充满活力的领域在最近几年。多项式Szemeredi定理，多项式Hales-Jewett定理，以及各种额外的结果所取得的调查人员在最近几年担任推动进一步发展的理论多次复发。这些发展提供了更好的理解的现象，多次复发，并带来新的前景的研究。 在这个建议中所涉及的研究方向揭示了组合学，数论和动力系统理论中的递归和收敛的各个方面之间的强大和相互持久的联系。该领域的遍历拉姆齐理论，其丰富的问题和连接和多样性的技术和方法，是一个交汇点的几个分支的现代数学，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。

The problems and conjectures presented in this program can be viewed as far reaching extensions of classical recurrence results in dynamics. At the same time, these problems connect diverse areas of mathematics (such as ergodic theory, combinatorics, number theory, topological algebra) and contribute to each. The new problems considered in this proposal deal with interesting and promising new connections and reflect the entrance of novel methods and techniques in the picture. These include, in particular, the methods involving dynamical systems on nilmanifolds and methods utilizing the topological algebra in compactifications. The conjectures stated throughout this proposal are supported by the results obtained by the investigators and other workers in this vibrant area in recent years.The polynomial Szemeredi theorem, the polynomial Hales-Jewett theorem, and various additional results obtained by the investigators 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 and bring new vistas of research to light. The directions of study touched upon in this proposal reveal strong and mutually perpetuating connections between combinatorics, number theory and various aspects of recurrence and convergence in the theory of dynamical systems. The field of Ergodic Ramsey Theory, with its richness of problems and connections and diversity of techniques and methods, is a meeting point of several branches of modern mathematics and is an excellent medium for attracting undergraduates to mathematics and graduate students to an area of active research.