Recurrence for Measure Preserving Systems and Infinitary Ramsey Theory

提案编号：DMS-0200700 PI：兰德尔McCutcheon摘要这个项目的重点是遍历理论中的多重递归，与组合数论的应用，特别是密度拉姆齐理论。在沿着组合良性轨迹（例如多项式或IP系统的范围）的测度保持作用下，对正样本集的回报进行分析，得出了许多其他方法无法获得的强定理。沿着组合良性轨迹（例如多项式或IP系统的范围）的测度保持作用下的正样本集的回报分析，得出了许多其他方法无法获得的强定理。理论动力学起源于对物理系统随时间演化的长期平均行为的研究，抽象遍历理论就是从这样的发展中产生的。研究这一理论对看似不相关的非物理实体的适用性，如组合数学中出现的实体，加强了我们对整个数学中的联系和统一性的认识。

Proposal Number: DMS- 0200700PI: Randall McCutcheon ABSTRACTThis project focuses on multiple recurrence in ergodic theory, with applications to combinatorial number theory, specificallydensity Ramsey theory. The analysis of the returns of positivemeasure sets under measure preserving actions along combinatoriallybenign trajectories, such as the ranges of polynomials or IP-systems,has led to a variety of strong theorems not accessible by othermethods. Theoretical dynamics originated with the investigation of the long term average behavior of physical systems evolving over time, andabstract ergodic theory grew out of such developments. Study of theapplicability of this theory to seemingly unrelated, non-physical entities, such as those arising in combinatorics, strengthens oursense of the connectedness of and the unification of the landscapein the whole of mathematics.