One of the most powerful approaches to the study of algebraic K-theory is the use of trace methods: that is, approximations of K-theory by more computable invariants (such as Hochschild homology). In this talk, Jacob Lurie will describe a (conjectural) extension of this methodology to other cohomological invariants in algebraic geometry. For more information on 2025 Simons Collaboration on Perfection in Algebra, Geometry and Topology Annual Meeting, please visit: