Ergodic Ramsey Theory, Polynomials, and Actions of Nilpotent Groups

遍历拉姆齐理论、多项式和幂零群的作用

• 批准号: 0245350
• 负责人: Vitaly Bergelson
• 金额: $ 23.7万
• 依托单位: Ohio State University Research Foundation -DO NOT USE
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0245350&HistoricalAwards=false
• 关键词: Ergodic; Ramsey; Theory; Polynomials; Actions

ABSTRACTV. Bergelson and A. Leibman will study multiple recurrence andconvergence in ergodic theory with emphasis on the polynomial aspects ofbehavior. The proposers earlier showed that sampling along polynomial times was suffient to reveal deep ergodic phenomina. The problems considered may be viewedas far reaching extensions of classical results. In addition to demonstrating new properties of dynamical systems, theseproblems lead naturally to strong applications of ergodic theory to combinatorics,number theory and algebra which until now have been inaccessible conventionalmethods. The polynomial Szemeredi theorem and the polynomialHales-Jewett theorem, obtained by the proposers in recent years, already serve asan impetus for further developments in the theory of multiple recurrence.This provides better understanding of multiple recurrence along polynomials and bring to light new vistas of research. An important feature of the research opened by theproposers's earlier work is the appearance of nilpotent nilpotent groups into the picture.Not only does this lead to strong applications in combinatorics and numbertheory, but the computational flexibility stemming from this theory may be of significance in applications.

摘要电视。 Bergelson 和 A. Leibman 将研究遍历理论中的多重递推和收敛，重点是行为的多项式方面。 提议者早些时候表明，沿多项式时间采样足以揭示深层遍历现象。所考虑的问题可以被视为经典结果的深远延伸。除了展示动力系统的新性质之外，这些问题自然而然地导致了遍历理论在组合学、数论和代数中的强有力的应用，而迄今为止，这些问题是传统方法无法实现的。提出者近年来提出的多项式Szemeredi定理和多项式Hales-Jewett定理已经为多重递推理论的进一步发展提供了动力，为人们更好地理解多项式上的多重递推提供了新的研究前景。提议者早期工作开启的研究的一个重要特征是幂零幂群的出现。这不仅在组合数学和数论中产生了强大的应用，而且源于该理论的计算灵活性在应用中可能具有重要意义。