In the past 15 years, set theory has been used in a broad array of applications to other branches of mathematics including algebra, topology, dynamics, and geometry. These applications are varied both in method and in consequence. Rosendal has developed a concept of large scale geometry applicable to transformation groups such Homeo$_o$$(M)$ for a compact manifold and the mapping class group of certain surfaces of infinite type. Bergfalk and Lambie-Hanson established that certain derived limits arising in algebraic topology vanish in carefully chosen models of set theory. Marks and Unger demonstrated that it is possible to decompose a square into finitely many Borel pieces and reassemble them into a circle using isometries.

 

What these results share in common is a realization that fundamental techniques (e.g. Birkhoff-Kakutani metrization), concepts (e.g. gaps and coherent families), and heuristics (e.g. finite combinatorics informs Borel combinatorics) within general topology and set theory capture essential difficulties in problems arising outside the subject. Furthermore, these applications expose new and challenging problems within pure set theory which help facilitate its own development.

 

This Thematic Program will bring fields such as algebra, topology, dynamics, and geometry together with those working in set theory in order to further develop current applications of set-theoretic methods and to find and explore new applications. All of these interactions are ongoing and some are still in an early state of development. As such, a single program bringing together those with complementary perspectives and talents will be timely and greatly amplify the continued success of these interactions.