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
Please email firstname.lastname@example.org if you would like to be on the mailing list.
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.
This seminar will be hybrid. Details TBA If you are interested in attending virtually, please let us know.
List of talks
|Speaker: Sebastian Haney
Title: Introduction to Khovanov Homology
|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.
|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.
|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.
|Speaker: Patrick Lei
Title: The HOMFLY polynomial and enumerative geometry II
|Speaker: Che Shen