Dynamical Aspects of Ramsey Theory

动力系统的数学分析--随时间演变的系统--在从经济学和天气预报到工程和天文学的各个领域都有不同的应用。在这类应用中产生的许多系统往往表现出混沌行为，并且可能很难直接分析。然而，通过研究它们的长期行为，总是有可能获得关于这类系统的有用信息。这个项目的总体目标是开发新的方法来研究动力系统在理论环境下的长期行为，并探索其在组合数学问题中的应用。1977年，Furstenberg用遍历理论证明了Szmeredi定理，此后利用动力系统的思想和方法得到了Ramsey理论和组合数论中的几个结果。与组合学的相互作用同样丰富了遍历理论和拓扑动力学的领域，带来了大量有趣的新问题。本项目的两个主要研究方向是：1.在动力系统的框架内发展新的技术来解决Ramsey理论问题，重点是关于非线性方程组的划分正则性的问题。通过探索多重遍历平均沿多项式和其他自然序列类的多次递归和收敛现象，提高对保持测量系统的长期行为的理解。

The mathematical analysis of dynamical systems -- systems which evolve through time -- has found diverse applications in fields from economics and weather forecast to engineering and astronomy. Many systems arising from such applications tend to exhibit chaotic behavior and can be very difficult to analyze directly. Nevertheless, it is always possible to obtain useful information about such systems by studying their long term behavior. The overarching goal of this project is to develop new methods to study the long term behavior of dynamical systems in a theoretical setting and to explore applications to the mathematical subject of combinatorics.Inspired by Furstenberg's proof of Szemeredi's theorem using ergodic theory in 1977, several results in Ramsey theory and combinatorial number theory were since obtained using ideas and methods from dynamical systems. The interplay with combinatorics has likewise enriched the fields of ergodic theory and topological dynamics with a wealth of interesting new problems. The two main research directions of this project are:1. To develop new techniques within the framework of dynamical systems to solve Ramsey theoretic problems, with emphasis on those concerning partition regularity of (systems of) non-linear equations.2. To improve the understanding of the long term behaviour of measure preserving systems by exploring the phenomena of multiple recurrence and convergence of multiple ergodic averages along polynomials and other natural classes of sequences.