Ergodic Ramsey Theory, Polynomials, and Actions of Nilpotent Groups

ABSTRACTV。伯格森和A·莱布曼将研究遍历理论中的多重递归和收敛，重点是行为的多项式方面。早些时候的提出者表明，沿着多项式时间进行采样足以揭示深度遍历现象。所考虑的问题可以看作是经典结果的更深远的推广。除了演示动力系统的新性质外，这些问题自然导致遍历理论在组合学、数论和代数中的强大应用，这些到目前为止一直是无法接近的常规方法。近年来提出的多项式Szmeredi定理和多项式Hales-Juett定理已经成为多重递推理论进一步发展的推动力，这有助于更好地理解沿多项式的多重递推，并为研究带来新的前景。这项研究的一个重要特点就是研究中出现了幂零群，这不仅在组合学和数论中得到了很好的应用，而且这一理论所产生的计算灵活性在应用中也具有重要意义。

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.