Monday, September 11 at 4:30pm to 5:30pm

Kaprielian Hall (KAP), 245

3620 South Vermont Avenue, Los Angeles, CA 90089

**Aaron Mazel-Gee, USC**

Abstract:

In differential geometry, the Chern character is a homomorphism from topological K-theory (the extraordinary cohomology theory of complex vector bundles) to de Rham cohomology. This witnesses de Rham cohomology as an approximation to topological K-theory: the Chern character is an isomorphism after rationalization.

There is a somewhat parallel story in algebraic geometry: it has been known for many years that "topological cyclic homology" (TC) is a close approximation to algebraic K-theory through the "cyclotomic trace" map K --> TC. As a result, the cyclotomic trace is often advertised as an analog of the Chern character. However, this is something of a misnomer: TC is a dramatic refinement of algebraic de Rham cohomology. (However, this discrepancy disappears rationally, and in particular it is invisible in the differential-geometric story.) Indeed, despite the enormous success of so-called "trace methods" in algebraic K-theory computations, the algebro-geometric nature of TC has so far remained mysterious.

I will begin my talk by surveying this state of affairs. Then, I will describe a new construction of TC that affords a precise interpretation of the cyclotomic trace at the level of derived algebraic geometry. This represents joint work with David Ayala and Nick Rozenblyum.