Traces and categorification

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. This is because some of its fundamental features are revealed only by /categorification, /leading to investigations in (oo,n)-categories. In this talk, I will describe a joint work with Bastiaan Cnossen, Shachar Carmeli, and Maxime Ramzi that sets up a general "character theory" for studying, among other things, the interaction of traces with colimits by an "induced character formula" (generalizing and refining work of Ponto-Shulman). Our approach utilizes topological Hochschild homology (THH), which itself can be viewed as a categorified trace construction, via the "microcosm principle". I will also explain how this theory can be applied to the study of the Becker-Gottlieb transfer, the  THH of Thom spectra, and higher representations in chromatic homotopy theory.