Ergodic Ramsey Theory and Dynamical Systems on Nilmanifolds

摘要本课题主要研究遍历理论中的多重递归性和收敛性问题，重点研究了与零流形上的动力系统的联系。所考虑的问题可以看作是经典结果的深远扩展。同时，这些问题也使得遍历理论在组合学、数论和代数中得到了广泛的应用，这是迄今为止传统方法所无法达到的。近年来pi们得到的多项式Szemeredi定理、多项式Hales-Jewett定理及其推广，推动了多次递归理论的进一步发展。这些发展提供了对多项式多次递归现象的更好理解，并带来了新的研究前景。幂零群的引入与pi的多项式结果开辟了一个有趣而重要的研究方向。我们所熟悉的处理交换群的结果不仅可以自然地推广到幂零设置，而且幂零动力学允许我们得到关于保测度变换的单参数群的收敛/递归性质的新信息。文中提出的相关猜想为幂零动力学与遍历理论、组合学和均匀分布等重要问题的联系提供了新的线索。提案中提出的问题和猜想连接了数学的不同领域（遍历论、组合学、数论），并对每个领域都有所贡献。提出的研究旨在更好地理解在多项式（和更一般）函数值对应的时刻采样的动力系统行为的规律性。虽然该提案侧重于这种现象在组合学和数论中的强大应用，但物理学家也可能对此感兴趣。例如，多重递归理论的一个推论是，沿着多项式（而不是线性）的时间实例测量物理系统的状态可以揭示系统的很多信息。

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.