Knot Homology and DAHA Seminar Fall 2021

The goal of this seminar is to understand part of the triangle of connections between Khovanov-Rozansky homology, the double affine Hecke algebra and Hilbert schemes.

Time and location information

Organisers: Sam DeHority, Zoe Himwich, Davis Lazowski
Time/date: Tuesdays 5pm EST
Location: TBA

Please email lazowski@math.columbia.edu if you would like to be on the mailing list.

Plan/references

First, we will cover necessary background on Hilbert schemes, DAHA and Khovanov–Rozansky homology, following (for instance)

• Punctual Hilbert Schemes, Iarrobinio
• Double Affine Hecke Algebras, Cherednik
• Lecture Notes on Cherednik Algebras, Etingof–Ma
• On Khovanov’s categorifcation of the Jones polynomial, Bar-Natan
• Lectures on knot homology, Nawata–Oblomkov

Then, depending on attendee interest, we will hone in on either the connection between DAHA and knot homology or the connection between Hilbert schemes and knot homology. (We do not expect to have time during the seminar to treat both connections adequately).

References on knot homology and Hilbert schemes:

• Refined knot invariants and Hilbert schemes, Gorsky–Negut
• Stable pairs and the HOMFLY polynomial, Maulik
• The Hilbert scheme of a plane curve singularity and the HOMFLY polynomial of its link, Oblomkov–Shende

References on knot homology and DAHA:

• Torus knots and the rational DAHA, Gorsky–Oblomkov–Rasmussen–Shende

We hope that talks will illustrate any connections made with explicit computations where possible.

Virtual attendance

This seminar will be hybrid. Details TBA If you are interested in attending virtually, please let us know.

Notes

There are live-TeXed notes of this seminar here graciously provided by Patrick Lei.

List of talks