Multiple Recurrence and Convergence Along Polynomials in Ergodic Ramsey Theory

[摘要]9706057 Bergelson/Leibman这个项目将研究多重递归性和收敛性问题，重点研究动力系统沿多项式的行为。所考虑的递归问题与组合数论和集合论中的问题联系在一起，其结果可以用来证明经典分析、组合学和数论的传统方法无法辨别的模式的存在性。pi最近证明了遍历多项式Szemeredi定理推广到生成幂零群的算子。这自然导致了关于幂零群作用的多重递归的更一般的问题和猜想。该基金用于研究动力系统和随机过程的长期行为。许多重要的物理现象，如股票市场的行为、超市里的排队、从远距离传输源接收的数据等，都表现出局部的不稳定行为，而通过观察长期平均行为可以更好地理解这种不稳定行为。研究这些随机过程及其相互之间的联系，以及它们的精细结构与其他不那么随机的过程之间的联系，是现代数学的一个核心问题。只有通过这样的研究，我们才能更好地理解，从而能够控制或预测数学可以模拟的物理现象的演变。

Abstract 9706057 Bergelson/Leibman This project will investigate problems of multiple recurrence and convergence with the emphasis on the behavior of dynamical systems along polynomials. The recurrence problems considered are tied to problems in combinatorial number theory and set theory and the results can be used to prove the existence of patterns with the conventional methods of classical analysis, combinatorics and number theory fail to discern. The PIs recently proved that the ergodic polynomial Szemeredi theorem extends to operators generating a nilpotent group. This naturally leads to more general questions and conjectures about multiple recurrence for nilpotent group actions. This grant is to fund the study of the long term behavior of dynamical systems and stochastic processes. Many important physical phenomena, as varied as the behavior of the stock market, lines in the supermarket, and data being received from a distant transmission source, exhibit locally erratic behavior which can be understood better by looking at the long term average behavior instead. The study of such stochastic processes and their connections with one another, as well as their connections in terms of their fine structure with other less random processes, is a central one in modern mathematics. It is only through such studies that we will be able to understand better and thus be able to control or predict the evolution of the physical phenomena which this mathematics can model.