Open Collections

Description

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 $K \to TC$ 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 describe 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.