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

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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.