Emanuele Dotto: Hermitian K-theory and trace methods

ABSTRACT: The Hermitian K-theory of a ring with anti-involution is the group-completion of its space of Hermitian forms and isometries. In recent work Hesselholt and Madsen describe this space as the Z/2-fixed-points of an involution on the algebraic K-theory spectrum of the underlying ring. The geometric fixed-points of this Z/2-spectrum are equivalent, at least when 2 is invertible, to the symmetric L-theory spectrum of the ring. I will discuss ongoing work on Hermitian and L-theoretical versions of topological Hochschild homology and on the corresponding trace map from Hermitian K-theory.