Topics in Applications of Set Theory

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Solecki's research interests lie in applications ofmathematical logic (model theory, set theory and, primarily,descriptive set theory) to analysis and topology. The projectconsists of several parts. For example, Solecki proposes toinvestigate a problem of automatic continuity of universallymeasurable homomorphisms on Polish groups. A possible solution tothis problem will certainly involve descriptive set theory ofPolish groups. However, it is also likely to rely heavily, on theone hand, on set theoretic assumptions (like the ContinuumHypothesis) and, on the other hand, on algebraic properties of theunderlying group (amenability, freeness). Indications of theconnections between this problem and set theory and algebra can befound in earlier work of Christensen, Mokobodzki and Solecki. Inanother part of the project, Solecki will apply methods developedfirst in set theory (in the study of Borel equivalence relations)to problems in topology and model theory. First, he will usedescriptive set theoretic techniques augmented by tools comingfrom model theory to investigate the structure of an importanttopological space---the pseudo-arc. Second, he will apply methodsfrom descriptive set theory to study groups associated canonicallywith countable complete theories---Lascar's Galois groups.In the project, Solecki will apply techniques and notionsdeveloped in mathematical logic to problems in other areas ofmathematics. For example, he will investigate a certaintopological space---a hereditarily indecomposable continuum calledthe pseudo-arc. These types of topological spaces were firstintroduced in mathematics in the first quarter of the twentiethcentury as curious examples of extremely complicated curves.Today, we know that they appear naturally in many mathematicalcontexts, e.g., in fluid dynamics, in smooth dynamical systemsliving in Euclidean spaces, among topological groups, in the studyof continuous functions, etc. Solecki intends to gain a betterunderstanding of the pseudo-arc by using methods from descriptiveset theory, a branch of mathematical logic, which were originallydeveloped to study in an effective way the size of abstract sets.It is hoped that crossing boundaries of subfields of mathematicsby applying set theoretic methods in analysis and topology shouldlead to new and substantial insights.