Parameterized, Algebraic, and Chromatic Traces

Cyclic invariance is prominent among desirable properties for invariants. It gives the trace in linear algebra a significant amount of its power, it allows for the construction for the Dennis trace from algebraic $K$-theory to (topological) Hochschild homology, and it is fundamental to the definition of the symmetric monoidal and bicategorical traces. The projects in this proposal are all united in the centrality of cyclicity. A primary source of inspiration for all of the projects in this proposal is the rich family of invariants that have flowed from the Euler characteristic in topological fixed point theory. The Euler characteristic starts as a very explicit invariant in terms of number of cells, but then generalizes to spectral refinements for endomorphisms that satisfy additivity and multiplicativity. The goals of this proposal are to develop similar rich generalizations for related invariants that all share the centrality of cyclicity.