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

在这个程序中提出的问题和猜想可以看作是经典递归结果在动力学中的深远扩展。同时，这些问题连接了数学的不同领域（如遍历论、组合学、数论、拓扑代数），并对每个领域都有所贡献。本提案中考虑的新问题涉及有趣和有前途的新联系，并反映了新方法和技术的进入。其中特别包括涉及零流形上的动力系统的方法和紧化中利用拓扑代数的方法。在这个充满活力的领域，研究者和其他工作人员近年来所获得的结果支持了本建议中所述的猜想。多项式Szemeredi定理、多项式Hales-Jewett定理以及近年来研究者获得的各种附加结果，推动了多次递归理论的进一步发展。这些发展有助于更好地理解多重复发现象，并为研究带来新的前景。本提案所涉及的研究方向揭示了组合学、数论和动力系统理论中递归和收敛的各个方面之间强有力的、相互延续的联系。遍历拉姆齐理论领域以其丰富的问题和联系以及技术和方法的多样性，是现代数学几个分支的交汇点，是吸引本科生和研究生到一个活跃研究领域的极好媒介。

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.