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统和闭集理想中的可能应用。该项目的第一部分是有关研究的composant等价关系的不可分解连续使用的技术和概念，在博雷尔等价关系的研究。索莱茨基建立在他之前关于不可分解连续统的工作之上。他主要研究的问题是否在comeager子集的不可分解连续的composant等价关系是博雷尔同构的两个特殊的博雷尔等价关系之一，通过同构保持贫乏集。肯定的回答这个问题将解决一个老问题的Kuratowski和甚至部分结果的特殊不可分解连续将锐化几个定理从文献。在该项目的第二部分，Solecki研究理想的封闭子集的波兰空间。 他调查了一定的非常具体的代表性简单定义理想的紧凑集。这与数学这一领域的几个开放问题有关。此外，他继续他的研究理想的哈尔空子集的波兰集团。特别是在这里的目的是发展理论的所有非阿贝尔波兰团体（理论工程罚款类波兰团体不变度量），并充分了解之间的联系哈尔空集在无限产品的地方紧群体和顺从的因素groups.One的主题Solecki的项目是调查不可分解的连续。这些都是迷人的几何对象，其复杂的拓扑性质吸引了数学家的兴趣，因为（上）世纪开始。然而，直到最近才意识到这种连续统是多么的普遍存在，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。有一个古老的猜想，由于库拉托夫斯基，这仍然是未解决的，其确认将完全揭示了不可分解连续统的精细结构。索莱茨基的工作是研究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研究某些关于小的概念。这些在数学的各个分支中都很重要，可以用来测量所考虑的集合的大小。这里的出发点是他的观察，一个巨大的类，这样的家庭的小集承认令人惊讶的和非常具体的类型的代表性。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，如果实现了，就回答了关于这种结构的一些老问题。索莱茨基研究在何种程度上可以建立这样的表示，这些类型的表示和属性的概念小，以及其他问题有关的概念小之间的相互联系。

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.