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

索莱茨基研究了描述集合论在不可分解连续体和闭集理想中的可能应用。项目的第一部分是利用Borel等价关系研究中发展起来的技术和概念研究不可分解连续体上的组成等价关系。索莱茨基建立在他之前关于不可分解连续体的研究之上。他主要研究了在不可分解连续体的可分解子集上，组成等价关系是否通过同构保持微集与两个特殊的Borel等价关系中的一个Borel同构。这个问题的肯定答案将解决库拉托夫斯基的一个老问题，甚至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更加尖锐。在项目的第二部分，Solecki研究了波兰空间的封闭子集的理想。他研究紧集的单可定义理想的一个非常具体的表示。这与这个数学领域的几个悬而未决的问题有关。此外，他还继续研究了波兰群的Haar零子集的理想。这里的具体目标是发展所有非阿贝尔波兰群的理论（该理论适用于具有不变度量的波兰群类），并充分理解局部紧群无穷积中的Haar零集与因子群的可调性之间的联系。索莱茨基项目的主题之一是对不可分解连续体的研究。自上世纪初以来，这些令人着迷的几何物体以其复杂的拓扑特性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，它们在动力系统和拓扑的各种情况下发挥着多么重要的作用。有一个古老的猜想，由于库拉托夫斯基，它仍然没有得到解决，它的证实将完全揭示不可分解连续体的更精细的结构。索莱茨基研究了这一假设的特别重要的实例以及与之相关的其他问题。索莱茨基项目的另一个主题是对小的某些概念的研究。在数学的各个分支中，这些对于测量所考虑的集合的大小是很重要的。这里的出发点是他的观察，即大量这样的小集合的家族承认令人惊讶和非常具体的表示类型。以这种方式表示小集合族的可能性对这种族的结构具有深刻的含义，如果实现，将回答有关这种结构的一些老问题。Solecki研究了这种表征可以建立的程度，这些表征类型与小概念属性之间的相互联系，以及与小概念相关的其他问题。

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.