Treeings, quasi-invariance, and ergodic combinatorics

该建议属于波兰空间上可定义等价关系的广泛范围，这是描述集合论的现代焦点。这个理论为理解数学对象分类的本质提供了一个总体框架，直到一些等价的概念，而且，由于它的广泛范围，它与数学的许多领域有自然的相互作用。该理论的中心位置是可数Borel等价关系（cber），它通过可数群和局部可数图的作用产生。等价关系、群作用和图之间的这些联系在描述集合论、遍历论、测量群论和描述图组合学之间创造了极其富有成效的相互作用。该提案的总体目标是加深对这些联系的理解，并进一步发展以重大开放问题为指导目标的cber理论。提案的项目可以分为三个主题。树与子树在cber的研究中，可树的树起着主要作用，即存在一个无环Borel图（树），其连接的组件正是等价类。它们不仅形成了cber的一个特别有趣的子类，而且是研究所有cber的关键工具。一个项目涉及可树cber类的闭包属性，而其他项目则需要开发子树、子图和子林的新结构。后者关注的是在结构上见证一个群体在其在概率空间上的自由行动中的不可顺从性的努力。为了解决这里的主要问题，我提出了在测量图上发展渗透理论。这本身就很有趣，并开辟了新的前景和问题。当CBER $E$是概率测度保持（pmp）时，可以更好地理解，$E$的Borel自同构保持了概率测度。这是因为来自不同领域的许多技术可用于pmp cber，包括成本理论、$\ well ^2$-(co)同调、算子代数和渗透理论。然而，当CBER仅仅是拟pmp时，即$E$的每一个Borel自同构只保留集合的非零性时，这些技术都是不可用的。在我最近的工作中，我开发了处理准不变性的新工具，我建议使用这些工具将pmp设置中已知的几个结果推广到准pmp设置中。此外，我对这种情况下成本的定义有一个想法，应该研究一下。如果有希望，我建议发展一种准pmp成本理论。最后一个主题致力于逐点遍历定理，即使用我最近证明Birkhoff遍历定理的风格的逐点组合平铺论证来证明新实例并找到已知实例的新证明。

GENERAL SCOPE The proposal lies within the broad scope 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 proposal is to deepen the understanding of these connections and further the theory of CBERs having major open questions as guiding targets. The projects of the proposal can be grouped into three topics. TREEINGS AND SUBTREEINGS In the study of CBERs, a principal role is played by those that are treeable, i.e. there is an acyclic Borel graph (a treeing) whose connected components are exactly the equivalence classes. These not only form a particularly interesting subclass of CBERs, but also serve as a critical tool for studying all CBERs in general. One project concerns a closure property of the class of treeable CBERs, while others require the development of new constructions of subtreeings, subgraphings, and subforests. The latter concerns the endeavor of structurally witnessing the nonamenability of a group within its free action on a probability space. To attack the main problems here, I propose the development of percolation theory on measured graphs. This is interesting on its own and opens new prospects and questions. QUASI-INVARIANCE A CBER $E$ is better understood when it is probability measure preserving (pmp), Borel automorphism of $E$ preserve a probability measure. This is because a number of techniques from various areas is available for pmp CBERs, including the theory of cost, $\ell^2$-(co)homology, operator algebras, and percolation theory. However, when the CBER is merely quasi-pmp, that is, every Borel automorphism of $E$ only preserves the non-nullness of sets, none of these techniques are available. In my recent work, I develop new tools for dealing with quasi-invariance, using which, I propose to generalize several results known in the pmp setting to the quasi-pmp setting. Furthermore, I have an idea for a definition of cost in this setting, which should be investigated. If promising, I propose developing a theory of quasi-pmp cost. ERGODIC THEOREMS The last topic 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 my recent proof the Birkhoff ergodic theorem.