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University of Copenhagen

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.