Constructibility and continuity in categories of mixed motives
These lectures will be an introduction to the theory of mixed motives, after Voevodsky. Although we will focus on motives over classical schemes, we will present the constructions and proofs in a way which can be adapted to more general settings (algebraic stacks, derived geometry, locally ringed topoi). We will mainly insist on various notions of constructibility, from a geometrical and a categorical points of view. Since we will work with motives locally with respect to the étale topology, this will have rather direct interpretations in terms of both intersection theory and étale cohomology.
Introduction to spectral algebraic geometry
These lectures will consist of an introduction to the theory of spectral algebraic geometry, with the goal of stating (and, time permitting, sketching a proof of) Lurie's generalization of the Artin representability theorem. We will assume familiarity with basic homotopy theory and higher category theory and begin with an account of structured infinity-topoi. We will then go on to define spectral schemes and Deligne-Mumford stacks as locally ringed infinity-topoi satisfying certain conditions, and discuss how they embed into the big etale topos of the sphere via their functor of points. We will then turn to quasicoherent sheaves, the cotangent complex, and deformation theory, leading up to the statement of the representability theorem. The remaining time (if any) will be devoted to applications.