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

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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.


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

List of talks

Date Talk Info
Sep. 14
Speaker: Sebastian Haney
Title: Introduction to Khovanov Homology
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
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.
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.
Oct. 12
Speaker: Patrick Lei
Title: The HOMFLY polynomial and enumerative geometry II
Abstract: Continued
Oct. 19
Speaker: Che Shen
Title: Braid variety and Khovanov-Rozansky homology
Abstract: Given a positive braid word, there is an isomorphism between its Khovanov-Rozansky homology of top degree and the equivariant cohomology of the corresponding braid variety. I will explain some of the ideas behind the proof of this theorem. In the course of the proof, I will introduce Bott-Samelson varieties, which serve as a geometric model for Khovanov-Rozansky homology on the one hand, and gives a good compactification for braid varieties on the other hand.
Oct. 26
Speaker: Sam DeHority
Title: Finite dimensional type A rational Cherednik representations
Abstract: The rational Cherednik algebra of type A admits finite dimensional representations at specific values of its parameter $c$. We will describe these representations using various techniques including the KZ functor and the BGG resolution.
Nov. 2
University holiday
Nov. 9
Speaker 1: Sam DeHority
Continued from Oct. 26
Speaker 2: Cailan Li
Abstract: I will massage what Sam wrote until it becomes a knot, figuratively speaking.
Nov. 16
Speaker: Cailan Li
Title: Torus Knots and the Rational DAHA, II
Abstract: Continuing from last time we will first decompose $L_{m/n}$ as a graded $S_n$ representation into irreducibles. Then, using the representation theory of the Hecke algebra (more accurately just $S_n$) we will prove the main result connecting Cherednik algebras to link invariants, namely that the graded Frobenius character of $L_{m/n}$ coincides with the HOMFLY polynomial of the $(m,n)$ torus knot.
Nov. 23
Speaker: Davis Lazowski
Title: Hilbert schemes, Coulomb branches and DAHA
Abstract: I will introduce affine Grassmannians and Coulomb branches. Then I will explain how studying generalised affine springer theory for Coulomb branches gives a DAHA action on equivariant homology of punctual Hilbert schemes for plane curve singularities.
Nov. 30
Speaker: Alvaro Martinez
Title: The deal with foams
Abstract: We will see how the maps between representations of (quantum) $\mathfrak{sl}_n$ can be drawn as certain trivalent graphs called webs. After expressing the $\mathfrak{sl}_n$ polynomial in terms of webs, we will categorify it using foams. Foams are certain 2-dimensional CW-complexes which can be thought of as cobordisms between webs. We will go over the novel Robert-Wagner foam evaluation, and we will use it to obtain a $\mathfrak{sl}_n$ link homology alla Khovanov homology.
Dec. 7
Speaker: Alvaro Martinez
Title: The deal with foams II
Abstract: We will describe special kinds of webs and foams between them, and see how they give a prerequisite-free interpretation of singular Soergel bimodules, as well as part of their Hochschild homology. After that, we will show how the same machinery can be used to describe the symmetric Khovanov-Rozansky link homology, which categorifies the RT invariant corresponding to symmetric power representations of slN.