Geometry and Topology Seminar
K-theory is a means of probing geometric objects by studying their vector bundles. Algebraic K-theory, the version applying to varieties and schemes, is a particularly deep and far-reaching invariant, but it is notoriously difficult to compute. The primary means of computing it is through its "cyclotomic trace" map K→TC to another theory called topological cyclic homology. However, despite the enormous computational success of these so-called "trace methods" in algebraic K-theory computations, the algebro-geometric nature of the cyclotomic trace has remained mysterious. In the first talk, I will describe a new construction of TC that affords a precise interpretation of the cyclotomic trace at the level of derived algebraic geometry, which is joint work with David Ayala and Nick Rozenblyum. By the end of the talk, you will be able to take home with you a very nice and down-to-earth fact about traces of matrices. No prior knowledge of algebraic K-theory or derived algebraic geometry will be assumed.
In the second talk, I will explain our theory of stratified noncommutative geometry. This provides a means of decomposing and reconstructing categories; it generalizes the theory of recollements, which e.g. from a closed-open decomposition of a scheme (or of a topological space) gives a decomposition of its category of quasicoherent (resp. constructible) sheaves. It encompasses a version of adelic reconstruction (after Beilinson, Parshin, and Tate), which is itself an elaboration of the classical "arithmetic fracture square" that reconstructs an abelian group from its p-completions and its rationalization. It also applies to chain complexes or spectra with "genuine" G-action, which are the relevant objects for equivariant Poincare duality. We use this theory to deduce a universal mapping-in property for TC (generalizing recent work of Nikolaus--Scholze), which gives rise to the cyclotomic trace from algebraic K-theory.