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 higher categories 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 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.

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