Special Topology Seminar: The geometry of the cyclotomic trace

Seminar | May 4 | 2-4 p.m. | 736 Evans Hall

 Aaron Mazel-Gee, Oklahoma State Univ.

 Department of Mathematics

Algebraic K-theory is a deep and far-reaching invariant, but it is notoriously difficult to compute. To date, the primary means of understanding K-theory is through its "cyclotomic trace" map K — > TC to topological cyclic homology. This map is usually advertised as an analog of the Chern character, but this is something of a misnomer: TC is a further refinement of any flavor of de Rham cohomology (even "topological", i.e. built from THH), though this discrepancy disappears rationally. However, despite the enormous success of so-called "trace methods" in K-theory computations, the algebro-geometric nature of TC has remained mysterious.

In this talk, I will present a new construction of TC that affords a precise interpretation of the cyclotomic trace at the level of derived algebraic geometry, which is based on nothing but universal properties (coming from Goodwillie calculus) and the geometry of 1-manifolds (via factorization homology). This is joint work with David Ayala and Nick Rozenblyum. It is inspired by forthcoming work of Thomas Nikolaus and Peter Scholze, though it is more general in a few ways; in particular, the full connection with derived algebraic geometry is only visible in our broader context.