Solecki's research interests lie in applications of mathematical logic (model theory, set theory and, primarily, descriptive set theory) to analysis and topology. The project consists of several parts. For example, Solecki proposes to investigate a problem of automatic continuity of universally measurable homomorphisms on Polish groups. A possible solution to this problem will certainly involve descriptive set theory of Polish groups. However, it is also likely to rely heavily, on the one hand, on set theoretic assumptions (like the Continuum Hypothesis) and, on the other hand, on algebraic properties of the underlying group (amenability, freeness). Indications of the connections between this problem and set theory and algebra can be found in earlier work of Christensen, Mokobodzki and Solecki. In another part of the project, Solecki will apply methods developed first in set theory (in the study of Borel equivalence relations) to problems in topology and model theory. First, he will use descriptive set theoretic techniques augmented by tools coming from model theory to investigate the structure of an important topological space---the pseudo-arc. Second, he will apply methods from descriptive set theory to study groups associated canonically with countable complete theories---Lascar's Galois groups.
In the project, Solecki will apply techniques and notions developed in mathematical logic to problems in other areas of mathematics. For example, he will investigate a certain topological space---a hereditarily indecomposable continuum called the pseudo-arc. These types of topological spaces were first introduced in mathematics in the first quarter of the twentieth century as curious examples of extremely complicated curves. Today, we know that they appear naturally in many mathematical contexts, e.g., in fluid dynamics, in smooth dynamical systems living in Euclidean spaces, among topological groups, in the study of continuous functions, etc. Solecki intends to gain a better understanding of the pseudo-arc by using methods from descriptive set theory, a branch of mathematical logic, which were originally developed to study in an effective way the size of abstract sets. It is hoped that crossing boundaries of subfields of mathematics by applying set theoretic methods in analysis and topology should lead to new and substantial insights.
