Knot Homology and DAHA Seminar Fall 2021

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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)

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:

References on knot homology and DAHA:

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