Lars Hesselholt: Around topological Hochschild homology (Lecture 3)

The lecture was held within the framework of the (Junior) Hausdorff Trimester Program Topology: "Workshop: Hermitian K-theory and trace methods"

Introduced by Bökstedt in the late eighties, topological Hochschild homology is a manifestation of the dual visions of Connes and Waldhausen to extend de Rham cohomology to the noncommutative setting and to replace algebra by higher algebra. In this expanded setting, topological Hochschild homology takes the place of differential forms; the de Rham differential is replaced by an action of the circle group; and de Rham cohomology is replaced by the Tate cohomology of said circle action.

The resulting cohomology theory has had numerous applications to algebraic K-theory and, more recently, to integral p-adic Hodge theory. The goal of these lectures is to give an introduction to this theory and its applications, and to explore the involution on algebraic K-theory and topological Hochschild homology that the presence of duality generates.