Treeability, Quasi-Invariance, and Ergodic Combinatorics

该奖项支持首席研究员的研究项目在波兰空间，这是一个现代的重点描述集理论的可定义的等价关系的一般框架。这一理论提供了一个一般框架，用于理解数学对象的分类性质，直到某种等价概念，并且由于其广泛的范围，它与许多数学领域有着自然的相互作用。在这个理论中的一个中心位置被可数博雷尔等价关系（CBER）占据，它通过可数群的作用以及通过局部可数图产生。等价关系、群作用和图之间的这些联系在描述性集合论、遍历理论、度量群理论和描述性图组合学之间创造了极其富有成效的相互作用。本研究的总体目标是加深对这些联系的理解，并进一步发展以主要开放问题为指导目标的CBER理论。第一个问题是封闭属性的类树CBER，即，是否抽象的准等距和有限的指数扩展保持treatment在博雷尔或可测量的上下文中。其次，研究了准不变设置，旨在发展准不变成本理论。第三个主题是CBER图的理想子图的存在性，例如，代价逼近子图或子树以及极大超有限自由因子的存在性。最后一部分是致力于逐点遍历定理，即证明新的实例，并找到新的证明已知的使用逐点组合平铺参数的风格，PI的最近证明伯克霍夫遍历theorety.This奖反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

This award supports the principal investigator's research projects within the general framework of definable equivalence relations on Polish spaces, which is a modern focus of descriptive set theory. This theory provides a general framework for understanding the nature of classification of mathematical objects up to some notion of equivalence, and, due to its broad scope, it has natural interactions with many areas of mathematics. A central place in this theory is occupied by countable Borel equivalence relations (CBERs), which arise via actions of countable groups as well as via locally countable graphs. These connections between equivalence relations, group actions, and graphs create an extremely fruitful interplay between descriptive set theory, ergodic theory, measured group theory, and descriptive graph combinatorics. The overarching goal of the reseach is to deepen the understanding of these connections and further the theory of CBERs having major open questions as guiding targets.The projects can be grouped into four topics. The first concerns closure properties of the class of treeable CBERs, namely, whether abstract quasi-isometries and finite index extensions preserve treeability in the Borel or measurable contexts. Next is the investigation of the quasi-invariant setting aiming to develop a theory of quasi-invariant cost. The third topic is the existence of desirable subgraphs of graphings of a CBER, e.g., the existence of cost-approximating subgraphings or subtreeings, as well as that of maximal hyperfinite free factors. The last part is devoted to the pointwise ergodic theorems, namely, proving new instances and finding new proofs of known ones using a pointwise-combinatorial tiling argument in the style of the PI's recent proof the Birkhoff ergodic theorem.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.