# Knot Homology and DAHA Seminar Fall 2021

** Last Updated:**

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 himwich@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.

### List of talks

Date | Talk Info |
---|---|

Tuesday Sep. 14 |
Speaker: Sebastian Haney Title: Introduction to Khovanov Homology Notes:pdf |

Tuesday Sep. 21 |
Speaker: Álvaro Martínez Title: HOMFLY-PT homology Abstract:We will briefly motivate the HOMFLY polynomial, a powerful invariant of links. Then we will motivate and construct (from scratch) HHH, the categorification of this notion by Khovanov and Rozansky. There will be explicit computations and a brief overview of the surprising conjectures connecting HHH with the geometry of Hilbert schemes. Notes: pdf |

Tuesday Sep. 28 |
Speaker: Avi Zeff Title: Hilbert Schemes Abstract: We’ll introduce Hilbert schemes, look at some examples, and prove (or, time depending, state) some properties such as representability and smoothness under certain conditions. Then we’ll specialize to the cases of the plane and plane curve singularities and explore connections to representation theory and knot invariants. |

Tuesday Oct. 5 |
Speaker: Patrick Lei Title: The HOMFLY polynomial and enumerative geometry I Abstract: We will state (again) the Oblomkov–Shende conjecture and introduce an important element in its proof: stable pairs enumerative invariants. We will define the objects involved, compute an example, and then (time-permitting) start the proof of Oblomkov–Shende, following Maulik. |

Tuesday Oct. 12 |
Speaker: Patrick Lei Title: The HOMFLY polynomial and enumerative geometry II Abstract: Continued |

Tuesday Oct. 19 |
Speaker: Che Shen Title: TBA Abstract: TBA |