ICED Summer Learning Seminar

Organization:

Time: Wednesdays at 1:30pm ET, June, July, August, 2022.

Location: Zoom. Anyone who is interested in attending is welcome; email one of the organizers and we will send you the link.

Organizers: Andrew Harder (anh318@lehigh.edu), Joe Kramer-Miller (jjk221@lehigh.edu), Yury Ustinovskiy (yuu221@lehigh.edu).

Seminar topics:

• Twisted (co)homology,

• Irregular connections and irregular Hodge structures,

• Exponential differential equations.

Roughly, our goal is to understand a little bit about the Hodge theory of irregular connections, following papers of Yu, Esnault--Sabbah--Yu, and subsequent work of Sabbah--Yu. We also plan to explore related topics in arithmetic geometry and mathematical physics.

Schedule:

• June 8, Andrew Harder:

Introduction to various cohomology theories.

We introduced the de Rham, log de Rham complex, and the twisted de Rham complex. We introduced the stupid filtration on the de Rham complex and the Hodge filtration in cohomology. (Notes)

• June 15, Andrew Harder

Degeneration of the Hodge-to-de Rham spectral sequence.

We explain a bit about the Hodge-to-de Rham spectral sequence for the holomorphic de Rham complex and describe, in broad terms, why it degenerates at the E1 term, including an overview of the Hodge theorem, Dolbeault cohomology. (Notes)

• June 22, Andrew Harder:

Irregular Hodge filtration and examples.

I'll introduce the irregular Hodge filtration and we will work through a few examples (Notes).

• June 29, Andrew Harder

E1 degeneration and the irregular filtration

I discussed proof of the E1 degeneration of the spectral sequence attached to the irregular filtration in the case of curves, and in the case where the polar divisor of f has only poles of order 1 (Notes).

• July 6, Cancelled.

• July 13, Joe Kramer-Miller

Twisted cohomology with compact support and Poincare duality, Part I

• July 20, Joe Kramer-Miller

Twisted cohomology with compact support and Poincare duality, Part II

• July 27, Joe Kramer-Miller

Twisted cohomology with compact support and Poincare duality, Part III

• August 3, Cancelled.

• August 10, Andrew Harder

The Brieskorn lattice, Part I: Definitions (Notes)

• August 17, Andrew Harder

The Brieskorn lattice, Part II: Examples and V-filtration (Notes)

• August 26 (NEW DAY), Andrew Harder

The Brieskorn lattice, Part III: Mixed Hodge structure on the vanishing cohomology