## Slawomir Solecki

### METRIC SPACES AND COMBINATORICS

During his Center appointment, Professor Solecki will investigate two mathematical problems related to metric structures.

The first problem, proposed in the 1930s by mathematicians Banach and Ulam, concerns the existence of a notion of size of subsets of a compact metric space with the property that any two definable subsets A and B have the same size provided that points of A can be matched with points of B by a distance preserving matching. The second problem concerns the algebraic structure of a universal Polish group--the group of isometries of the Urysohn metric space. (Polish groups form a subclass of topological groups studied in set theory, model theory, and topology.)

Recently, Professor Solecki realized that results from the seemingly unrelated fields of combinatorics and finite model theory might help to reduce the first problem to a special case, which can then possibly be addressed through methods of harmonic analysis. The reduction will likely call on recent methods in continuous logic and the theory of the Urysohn metric space to prove a far-reaching generalization of a model theoretic result known as the Herwig-Lascar theorem. This proof should also yield a solution to the second problem.

The project combines diverse mathematical approaches in unusual ways. Professor Solecki has run a learning seminar on material related to these topics, preparing his graduate students to participate in the research.