Abstracts for Conference on "Algebraic Topology, in memory of Hans-Joachim Baues", October 17 - 21, 2022

Seiberg-Witten theory has played a central role in the study of smooth low-dimensional manifolds since their introduction in the 90s. Parallel to this, Cohen, Jones, and Segal asked the question of whether various types of Floer homology could be upgraded to the homotopy level by constructing (stable) homotopy types encoding Floer data. In 2003, Manolescu constructed Seiberg-Witten Floer spectra for rational homology 3-spheres, and in particular used these to settle the triangulation conjecture once and for all.

The trace of a linear operator is simple to define, yet it is a surprisingly interesting and mysterious operation. From characters of representations, through fixed-point formulas, to various geometric transfer maps it appears all over mathematics. The theory of oo-categories and higher algebra allows one to organize many of these occurrences of the trace within a formal unified calculus. However, this calculus itself is more intricate and elaborate than one might expect.

(joint with Nima Rasekh)

While weak versions of n-categories are much more widespread in nature, strict version allows for modelling specific shapes of higher categories. For n=2, we have a complete understanding of their relationship. After introducing the terms and setting up the stage, I will report about this and - time permitting - about work in progress towards analogous results for n>2. This is joint work with Martina Rovelli.