Applications of Descriptive Set Theory to Ideals of Closed Sets and Indecomposable Continua

Solecki研究了描述性集合论在不可分解连续体和闭集理想中的可能应用。该项目的第一部分是利用Borel等价关系研究中发展起来的技巧和概念来研究不可分解连续体上的合成等价关系。Solecki建立在他之前关于不可分解连续体的工作基础上。他主要研究了不可分解连续体的可分解子集上的合成等价关系是否通过保持同构的稀疏集同构于两个特殊的Borel等价关系之一的问题。这个问题的肯定答案将解决Kuratowski的一个老问题，即使是特殊不可分解连续体的部分结果也将使文献中的几个定理变得尖锐。在项目的第二部分，Solecki研究了波兰空间的闭子集的理想。他研究了紧集的简单可定义理想的某种非常具体的表示。这与这一数学领域中的几个公开问题有关。此外，他继续研究波兰群的Haar零子集的理想。这里的特别目的是发展所有非交换波兰群的理论(该理论适用于具有不变度量的波兰群)，并充分理解局部紧群的无限乘积中的Haar零集与因子群的顺从性之间的联系。Solecki项目的主题之一是研究不可分解连续体。这些都是迷人的几何物体，其复杂的拓扑性质自上个世纪初以来就吸引了数学家的兴趣。然而，直到最近，人们才意识到这种连续体是多么普遍，以及它们在动力系统和拓扑学的各种背景下扮演着多么重要的角色。由于库拉托夫斯基提出了一个古老的猜想，这个猜想至今仍未解决，它的证实将完全揭示不可分解连续体的更精细结构。索莱茨基研究了这一假说的特别重要的例子以及与之相关的其他问题。Solecki项目的另一个主题是研究某些微小的概念。这些在数学的各个分支中都很重要，用来衡量所考虑的集合的大小。这里的出发点是他观察到，有一大类这样的家庭，由小集合组成，承认令人惊讶的和非常具体的类型的表现。以这种方式表示一个小集合族的可能性对这种族的结构有着深刻的影响，如果实现的话，还可以回答关于这种结构的一些古老的问题。Solecki研究了这种表示可以建立的程度，这些类型的表示之间的相互联系和小概念的性质，以及其他与小概念有关的问题。

Solecki studies possible applications of descriptive set theory to indecomposable continua and to ideals of closed sets. The first part of the project is concerned with studying the composant equivalence relation on indecomposable continua using techniques and notions developed in the study of Borel equivalence relations. Solecki builds on his prior work on indecomposable continua. He primarily investigates the question whether on a comeager subset of an indecomposable continuum the composant equivalence relation is Borel isomorphic to one of two special Borel equivalence relations via an isomorphism preserving meager sets. The affirmative answer to this question would solve an old problem of Kuratowski and even partial results for special indecomposable continua would sharpen several theorems from the literature. In the second part of the project, Solecki studies ideals of closed subsets of a Polish space. He investigates a certain very concrete representation of simply definable ideals of compact sets. This is connected with several open problems in this area of mathematics. Additionally, he continues his study of the ideal of Haar null subsets of a Polish group. Particular aims here are to develop the theory for all non-abelian Polish groups (the theory works fine for the class of Polish groups with invariant metrics) and to fully understand the connection between Haar null sets in infinite products of locally compact groups and amenability of the factor groups.One of the themes of Solecki's project is the investigation of indecomposable continua. These are fascinating geometrical objects whose intricate topological properties attracted interest of mathematicians since the beginning of the (last) century. However, only quite recently it was realized how ubiquitous such continua are and how important a role they play in various contexts in dynamical systems and topology. There is an old conjecture, due to Kuratowski, which is still unresolved and whose confirmation would completely reveal the finer structure of indecomposable continua. Solecki works on particularly important instances of this hypothesis and other problems related to it. Another theme of Solecki's project is the study of certain notions of smallness. These are important in various branches of mathematics to measure the size of sets under consideration. The starting point here is his observation that a vast class of such families of small sets admit surprising and very concrete type of representations. The possibility of representing a family of small sets in this fashion has deep implications for the structure of such families and, if realized, answers some old questions regarding this structure. Solecki studies the extent to which such representations can be established, interconnections between these type of representations and properties of notions of smallness, and other problems related to notions of smallness.