# Algebra and Algebraic Geometry Seminar Fall 2024

The seminar normally meets 2:30-3:30pm on Fridays, in the room **Van Vleck** **B131**.

## Algebra and Algebraic Geometry Mailing List

- Please join the AGS mailing list by sending an email to ags+subscribe@g-groups.wisc.edu to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).

## Fall 2024 Schedule

date | speaker | title | host/link to talk |
---|---|---|---|

September 27 | Joshua Mundinger (Madison) | Hochschild homology and the HKR spectral sequence | local |

October 4 | Dima Arinkin (Madison) | Derived category of the stacky compactified Jacobian | local |

November 15 | Yunfeng Jiang (Kansas) | TBA | Andrei/Ruobing |

## Abstracts

### Joshua Mundinger

**Hochschild homology and the HKR spectral sequence**

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.

### Dima Arinkin

**Derived category of the stacky compactified Jacobian**

Abstract: The Jacobian of a smooth projective curve is an abelian variety which is identified with its own dual. This implies that its derived category carries a non-trivial auto-equivalence - the Fourier-Mukai transform. When the curve has planar singularities, the Jacobian is no longer compact (and, in particular, not an abelian variety), but it turns out that the Fourier-Mukai transform still exists, provided we compactify the Jacobian. The transform can be viewed as the `classical limit' of the geometric Langlands correspondence.

In this talk, I will explore what happens when the curve becomes reducible. From the point of view of the geometric Langlands conjecture, it is important to work with the compactified Jacobian viewed as a stack (rather than the corresponding moduli space). In my talk I will show that this also leads to certain issues, and in fact that the most general version of the statement is inconsistent, while more conservative versions are true.