# Trace methods for real algebraic K-theory

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J. D. Quigley, Cornell University

Trace methods have long been used to study algebraic K-theory, but have been less successful in studying related invariants like Hermitian K-theory and L-theory. In this talk, I will give a progress report on the development of trace methods for these other theories. The key to this approach is real algebraic K-theory, which combines algebraic K-theory, Hermitian K-theory, and L-theory into a single genuine $C_2$-spectrum. In previous work, we developed a theory of real cyclotomic spectra which provided new tools for computing real topological cyclic homology, an approximation to real algebraic K-theory. I will recall our previous work, discuss recent generalizations, and share some new computations.