Date: Sunday, January 12, 2025
Location: Eckhart Hall, Room 202, University of Chicago
Support: This workshop is supported by NSF grant DMS 2152311 FRG: Higher Categorical Structures in Algebraic Geometry.
Pretalk on Hochschild homology (after work of Moulinos-Robalo-Toën and Raksit)
Hochschild homology and the HKR spectral sequence
Abstract: Hochschild homology of an algebraic variety carries the Hochschild-Konstant-Rosenberg (HKR) filtration. In characteristic zero, this filtration is split, yielding the HKR decomposition of Hochschild homology. In characteristic p, this filtration does not split, giving rise to the HKR spectral sequence. We describe the first nonzero differential of this spectral sequence. Our description is related to the Atiyah class and is based on the filtered circle introduced by Moulinos-Robalo-Toën and Raksit.
Dieudonné theory and cohomology of K(G,n)
Abstract: I will talk about joint work in progress with S. Li and, partially, S. Mondal where, given a finite commutative locally free group scheme G, we give a uniform structural result for cohomology of K(G,n) for a huge class (e.g., singular, de Rham, prismatic, ...) of cohomology theories: namely, it is isomorphic to the free derived divided power algebra on the corresponding "Dieudonné module" D(G)[-n] as En-1-algebra. A posteriori, this also gives a way to define the Dieudonné module purely in terms of cohomology of BG: namely, it turns out that the homology of BG has a natural animated ring structure and D(G) is given by the fiber of its tangent complex at the augmentation (up to a shift).
Quantization of Lagrangians in mixed characteristic
Abstract: Let X be a smooth algebraic variety and let L be a smooth Lagrangian sub variety inside the cotangent bundle of X. In this talk we will discuss various ways of attaching to L a D-module on X. The answer we present goes through some novel aspects of the theory of D-modules in mixed characteristic, which we will discuss at length. Time permitting, we will also discuss applications to the theory of the Weyl algebra, and to rigid connections on the affine line.