The philosophy of "categorification" seeks to lift various concepts involving vector spaces to those involving triangulated categories so that old structures are recovered by passing to the Grothendieck groups. The conjectural concept of perverse schobers can be seen as a categorification, in this sense, of the classical concept of perverse sheaves. A class of examples allowing a precise formulation, is provided by the birational geometry of flops.
The role of passing to the dual vector space is played, in the categorified picture, by Toen's Morita duality for dg-categories. The talk will analyze, from this point of view, schobers associated to flops of relative dimension 1. In this case, the familiar Poincare-Verdier duality between hypercohomology of perverse sheaves and hypercohomology with compact support, lifts to a statement involving the coherent and perfect derived categories of a singular variety. The corresponding statement for a "web of flops" associated with the Grothendieck resolution, remains conjectural.
Joint work with A. Bondal and V. Schechtman.
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.
We will give an informal introduction to derived algebraic geometry and to symplectic and Poisson geometry inside derived algebraic geometry, and concentrate on examples that will be useful in the remaining two talks. Then we will explain the problem of defining a pointed formal neighbourhood in algebraic geometry, give one solution, explain how to get from this formal gluing results of (pseudo-)perfect complexes and G-bundles, given a closed substack (or a flag of these) in a fixed stack, and speculate about possible relations with the Geometric Langlands Program for surfaces.
In the final lecture we will consider moduli spaces of (possibly irregular) connections on higher dimensional varieties, study their derived Poisson structures and construct their symplectic leaves by means of a derived analog of quasi-hamiltonian reduction construction.