Alexei Oblomkov

The Hilbert scheme of a plane curve singularity and the HOMFLY polynomial of its link

The intersection of a complex plane curve with a small three -sphere surrounding one of its singularities is a non-trivial link. The refined punct ual Hilbert schemes of the singularity parameterize subschemes supported at the singular point of fix ed length and whose defining ideals have a fixed number of generators. We conjecture that the generati g function of Euler characteristics of refined punctual Hilbert schemes is the HOMFLY polynomial of the link. The conjecture is verified for irreducible singularitiesy = x, whose links are thek, n torus knots, and for the singularity y = x − x + 4xy + 2xy, whose link is the 2,13 cable of the trefoil.

Torus knots and the rational DAHA

Author(s): Gorsky, E; Oblomkov, A; Rasmussen, J; Shende, V | Abstract:

Quantum cohomology of the Hilbert scheme of points on An-resolutions

We determine the two-point invariants of the equivariant quantum cohomology of the Hilbert scheme of points of surface resolutions associated to type A_n singularities. The operators encoding these invariants are expressed in terms of the action of the affine Lie algebra \hat{gl}(n+1) on its basic representation. Assuming a certain nondegeneracy conjecture, these operators determine the full structure of the quantum cohomology ring. A relationship is proven between the quantum cohomology and Gromov-Witten/Donaldson-Thomas theories of A_n x P^1. We close with a discussion of the monodromy properties of the associated quantum differential equation and a generalization to singularities of type D and E.

On stable Khovanov homology of torus knots.

We conjecture that the stable Khovanov homology of torus knots can be described as the Koszul homology of an explicit irregular sequence of quadratic polynomials. The corresponding Poincaré series turns out to be related to the Rogers–Ramanujan identity.

Double affine Hecke algebras and Calogero-Moser spaces

In this paper we prove that the spherical subalgebra

Donaldson-Thomas theory of An × P 1

eH_{1,\tau}e

Generalized Lamé operators

of the double affine Hecke algebra

Generalized double affine Hecke algebras of higher rank

H_{1,\tau}

Knot homology and sheaves on the Hilbert scheme of points on the plane

is an integral Cohen-Macaulay algebra isomorphic to the center

Two-dimensional algebro-geometric difference operators

Z