Topological Hochschild Homology and Zeta Values

Masterclass - University of Copenhagen
30 January - 3 February 2023

The masterclass will present spectacular recent advances in motivic filtrations and their applications. To briefly put this in context, every cohomology theory, in arithmetic geometry and elsewhere, should arise as the graded pieces of a motivic filtration of some localizing invariant, algebraic K-theory and topological cyclic homology being notable examples. The definition of the appropriate motivic filtrations, however, was long elusive. Voevodsky received the 2002 Fields medal, in part, for his definition of the motivic filtration of algebraic K-theory of schemes smooth over a field. This definition, however, was based on algebraic cycles, which are notoriously difficult to handle. In 2018, Bhatt, Morrow, and Scholze defined a motivic filtration of p-complete topological cyclic homology in an entirely different way, which is much simpler and easier to employ elsewhere, and this breakthrough has led to numerous advances.

Building on work of Antieau, Morin, and, independently, Bhatt-Lurie, have refined the Bhatt-Morrow-Scholze filtration to a filtration of non-completed topological cyclic homology, and Morin discovered that its graded pieces precisely account for an archimedean factor in a conjectural formula for the special values of the Hasse-Weil zeta function of a regular scheme, proper over the integers. It is very exciting to see that such quantative archimedean information can be extracted from topological cyclic homology! More recently, Hahn-Raksit-Wilson introduced the even filtration, and showed that it accounts for both the Bhatt-Morrow-Scholze filtration and the Morin and Bhatt-Lurie refinements thereof. It might also recover and extend Voevodsky's filtration?