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Traces and categorification

Bastiaan Cnossen (Bonn)

Abstract: The trace of a linear operator is simple to define, yet appears all over mathematics in many disguises: from characters of representations, through fixed-point formulas, to various geometric transfer maps. The theory of oo-categories and higher algebra allows one to organize many of these occurrences of the trace within a formal unified calculus. This calculus is more intricate and elaborate than one might expect, because some of its fundamental features are revealed only by categorification, leading to investigations of traces in (oo,n)-categories.

In this talk, I will describe a joint work with Shachar Carmeli, Maxime Ramzi and Lior Yanovski 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). The interaction between traces and categorification plays a key role in our approach. I will also explain how this theory can be applied to the study of the Becker-Gottlieb transfer and of topological Hochschild homology of Thom spectra.