# Homotopy theory, manifolds, and field theories

Hausdorff Trimester Program

May 4 - August 21, 2015

Organizers: Søren Galatius, Haynes Miller, Stefan Schwede, Peter Teichner

In recent years, there had been tremendous progress in the interaction between the research areas that we brought together at HIM. The nascent theory of higher categories had turned out to be perfectly suited to the study of various problems, for example that of field theories, especially with regard to matters of locality. Generalizations of ordinary homotopy theory appeared in several different directions and the goals of the trimester program were to understand these new developments and their interactions, particularly in the following areas:

- The model category of (∞; n)-categories (reducing to ordinary simplicial sets for n = 0) is a cornerstone for the modern approach to topological field theory (TFT). It unifies categorical considerations with those of homotopy and manifold theory.
- Hochschild homology turned out to be the case of dimension n = 1 in a whole sequence of constructions, now called factorization homology or blob homology, which have arisen from particular examples of n-dimensional TFTs. The algebra of observables in non-topological (perturbative) field theories leads to a more general notion of factorization algebras that are at the center of current investigations.
- The model category of smooth simplicial sets, or smooth stacks, is essential in the study of geometric field theories; the relationship to global equivariant homotopy theory is not yet fully understood but should shed light on questions such as the universal properties of homotopical equivariant bordism.
- The moduli space of n-manifolds arises in many flavors and various contexts, some of which are now understood but others are still outstanding.

The program ran a weekly HIM-seminar where the participants explained their current research projects.

Special activities:

- May 4-8: Introductory School for graduate students and postdocs with lecture series by D. Ayala, O. Randal-Williams, S. Schwede, P. Teichner
- June 1-19: Felix Klein Lectures by Charles Rezk
- June 15-19: Workshop at the Max Planck Institute for Mathematics

- August 17-21: Conference on Topology and Geometry