Abstracts D-modules-2024

The radius of convergence of a $p$-adic differential equation on a p-adic disc has interesting variational properties. It turns out that its $\ell$-adic analog, the Swan conductor of an $\ell$-adic local system $\mathcal{F}$ on a closed rigid unit disc $D$, has similar properties: using the ramification theory of Abbes and Saito, we show that the Swan conductor function ${\rm sw}_{\mathcal{F}}:\mathbb{Q}\to\mathbb{Q}$ (of variable the radius of a sub-disc of $D$) is continuous, convex and piecewise linear; its slopes are integers that can be computed in terms of local characteristic cycles of $\mathcal{F}$ (analogs of the characteristic cycle of a holonomic $\mathcal{D}$-module).

Roman Bezrukavnikov (MIT)

Crystalline differential operators on a smooth variety X in positive characteristic form an Azumaya algebra over the Frobenius twist of its cotangent bundle. The natural extension to the case of a smooth algebraic stack X has derived nature: it involves a gerbe over the cotangent of X which is a derived stack. I will discuss an attempt to define a similar derived enhancement for divided power differential operators on a stack, focusing on the example where X is the classifying stack of a maximal unipotent subgroup in a reductive algebraic group G; the construction is motivated by the study of G-modules via derived localization. Time permitting I will explain related structures in the topology of the affine Grassmannian, connected to the above by the geometric Satake equivalence. The results are partly joint with Pablo Boixeda Alvarez.

Ekaterina Bogdanova (Harvard)