Ergodic Ramsey Theory, Polynomials, and Actions of Nilpotent Groups

ABSTRACTV。Bergelson和A. Leibman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。先前的提出者表明，沿多项式次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深远扩展。除了证明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，这些应用到目前为止一直是传统方法无法实现的。近年来提出的多项式Szemeredi定理和多项式hales - jewett定理，已经为多次递归理论的进一步发展提供了动力。这提供了对多项式多次递归的更好理解，并带来了新的研究前景。该研究的一个重要特征是，在早期的工作中出现了幂零群。这不仅导致了在组合学和数论中的强大应用，而且源于该理论的计算灵活性可能在应用中具有重要意义。

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.