MS Seminar (Mathematics - String Theory)

Speaker: Aaron Mazel-Gee (U of Southern California)
Title: The geometry of the cyclotomic trace
Date: Thu, Mar 22, 2018, 15:30 - 17:00
Place: Seminar Room B
Abstract: Algebraic K-theory — the analog of topological K-theory for varieties and schemes — 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 topological Hochschild homology (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 these talks, I will describe a new construction of TC that affords a precise interpretation of the cyclotomic trace at the level of derived algebraic geometry. In stark contrast with the original construction, this is based on nothing but universal properties (coming from Goodwillie calculus) and the geometry of 1-manifolds (via factorization homology). In terms of TQFT, we find that this arises from certain extra structure on THH governed by covering maps of circles, that in the end reduces to a very down-to-earth observation regarding traces of matrices (which is rationally vacuous).

I will also describe an auxiliary result of independent interest that guarantees the existence of "generalized recollements". Classically, a recollement is a sort of "extension sequence" of triangulated categories; the primordial example is that a closed-open decomposition of a scheme X determines a recollement of its derived category QCoh(X). Generalized recollements accommodate a much broader class of decompositions; in particular, I'll explain how any diagram of closed subschemes of a scheme X determines a generalized recollement of QCoh(X).

This is joint work with David Ayala and Nick Rozenblyum.