Abstracts for Workshop on Homotopy theory, manifolds, and field theories

The abstract for the entire Felix Klein lecture series can be found here. The talks during the workshop will start over with a new topic.

The passage from the category of spaces to the category of spectra can be desribed by universally `inverting' spheres or equivalently imposing stability. This turns out to be technically very usefuly and philosopically explains why spectra play such an important role. A similar description can be given in equivariant homotopy theory for a fixed Lie group G. We will explain in this talk how such a description can be given for the category of global spectra (which are in some sense equivariant for all Lie groups at the same time) recently studied by Schwede.

A relative (also called twisted) quantum field theory should be some transformation between quantum field theories, which themselves are symmetric monoidal functors out of a space-time category. In examples, the notion of natural transformation turns out to be too strong, making it necessary to relax it. In joint work with Theo Johson-Freyd we provide a framework for both lax and oplax transformations and their higher analogs called transfors between strong $(\infty, n)$-functors.

I will survey recent advances in our understanding of extended 3-dimensional topological field theories. I will describe recent work (joint with B. Bartlett, C. Douglas, and J. Vicary) which gives an explicit ”generators and relations” classification of partially extended 3D TFTS (assigning values only to 3-manifolds, surfaces, and 1-manifolds). This will be compared to the fully-local case (which has been considered in joint work with C. Douglas and N. Snyder).