Polynomial ergodic theorems and Ramsey theory

摘要：本项目主要研究动力系统的多重递归和收敛性，重点是阿贝尔群和幂零群的多项式作用。考虑的问题导致不同的应用ofergodic理论的组合，数论和代数arenaccessible迄今为止的传统方法。多项式Szemeredi定理和Hales-Jewett定理近年来被提出者证明，此后被提出者及其同事从不同的方向推广，每一个都提供了对沿沿着多项式多次递归现象的更好理解，并提供了新的研究前景。本建议涉及的研究方向包括（但不限于）幂零群作用的多重常返性的深入研究以及在IP-系统框架下对多项式多重常返性的研究。另一个有趣的领域，在这个建议中所考虑的是与遍历平均的收敛性，自然出现在多重递归理论及其应用。最近得到的结果的提议者表明adichotomy行为的遍历平均依赖于whitheracting集团有多项式或指数增长。该提案最后一部分提出的相关猜想为这些问题提供了新的线索。 我们在这个项目中关注的多重递归理论使用了来自数学的几个不同领域的思想，旨在提高我们对动力系统的内在性质的认识，这些性质与它们的长程行为有关。一个例子，一个高度非平凡的事实源于我们的调查是规则性的行为的动力系统沿着多项式时间的测量。反过来，这个事实在数论和组合学看似遥远的领域也有很强的应用。

ABSTRACT:The project concentrates on multiple recurrence and convergence indynamical systems with a focus on polynomial actions of abelian andnilpotent groups. The problems considered lead to diverse applications ofergodic theory to combinatorics, number theory and algebra which areinaccessible so far by conventional methods. The polynomial Szemeredi andHales-Jewett theorems proved by the proposers in the recent years havesince been extended by the proposers and their colleagues in differentdirections, each providing a better understanding of the phenomenon ofmultiple recurrence along polynomials and offering new vistas of research.The directions of study touched upon in this proposal include(but are not limited to) the deeper study of multiple recurrence fornilpotent group actions and the investigation of polynomial multiplerecurrence in the framework of IP-systems. Another interesting areaconsidered in this proposal has to do with convergence of ergodic averagesnaturally appearing in the theory of multiple recurrence and itsapplications. The results recently obtained by the proposers indicate adichotomy between the behavior of ergodic averages depending on whetherthe acting groups have polynomial or exponential growth. Therelated conjectures formulated in the last section of the proposal areshedding new light on these questions. The theory of multiple recurrence that we focus on in this project usesthe ideas from several diverse areas of mathematics and aims to advanceour knowledge about the intrinsic properties of dynamical systems whichare related to their long range behavior. An example of a highlynontrivial fact stemming from our investigations is the regularity of thebehavior of dynamical systems along polynomial time measurements. Thisfact, in its turn, has strong applications to seemingly distant areas ofnumber theory and combinatorics.