Friday February 09, 2018 1:30 PM  2:30 PM P131
 Aaron MazelGee, USC
The geometry of the cyclotomic traceAlgebraic $K$theory  the analog of topological $K$theory for varieties and schemes  is a deep and farreaching 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 socalled ``trace methods'' in $K$theory computations, the algebrogeometric 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 1manifolds (via factorization homology). This is joint work with David Ayala and Nick Rozenblyum.
