Combinatorial Ergodic Theory

该奖项是根据2009年《美国复苏和再投资法案》(公法111-5)提供资金的。这个项目涉及一个遍历理论和加性组合数学交叉的研究项目。大多数问题都是在遍历理论范围内的，与分析系统的长期行为有关，这些系统的动力学太复杂或太混乱，无法在当地理解。该项目的研究部分建立在首席调查员过去几年在多次重复和收敛方面的结果，特别是在对某些系统(零系统)在分析中的作用的新的理解的基础上。这一领域的遍历理论与加法组合学和调和分析中的问题有很强的联系。该项目将继续探索这些深入的联系，开发遍历理论在这些领域的应用。该项目的外展部分涉及会议组织、指导、咨询和开发新课程。首席调查员一直积极参与与遍历理论有联系的领域的研究人员的教育，并将继续这样做，主要是通过组织跨学科会议。遍历理论是概率系统和动力系统相遇的一个数学领域。因此，它与物理科学、工程和经济学中出现的许多现象的数学建模有关。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). This project involves a program of research in the intersections of ergodic theory and additive combinatorics. Most of the problems are within ergodic theory and are related to the analysis of the long-term behavior of systems whose dynamics are too complicated or too chaotic to be understood locally. The research component of the project builds on the principal investigator's results of the past few years in multiple recurrence and convergence and, in particular, on the new unerstanding these results have provided on the role of certain systems (nilsystems) in the analysis. This area of ergodic theory has strong relations to problems in additive combinatorics and harmonic analysis. The project will continue to explore these deep links, developing the applications of ergodic theory to such areas.The outreach portion of the project involves conference organization, mentoring, advising, and development of new courses. The principal investigator has been actively involved in education of researchers in fields with links to ergodic theory and will continue to do so, primarily by organizing interdisciplinary meetings. Ergodic theory is an area of mathematics in which probability and dynamical systems meet. As such, it is relevant to the mathematical modeling of many phenomena that arise in the physical sciences, engineering, and economics.