Speaker:	Ivan Cherednik (University of North Carolina)
Title:	Motivic superpolynomials and surface singularities
Date (JST):	Tue, May 27, 2025, 13:30 - 15:00
Place:	Seminar Room A
Abstract:	The origin of superpolynomials is Khovanov-Rozanskypolynomials (HOMFLY-PT homology). Conjecturally, they coincide with DAHA superpolynomials (for any colored iterated torus links), and motivic superpolynomials of plane curve singularities (generalizing p-orbital integrals for affine Springer fibers), when these 3 theories overlap. We will begin with an outline of the DAHA construction and its application to the DAHA topological vertex, presumably serving Siefert and lens spaces. The q-Gauss integrals in the DAHA vertex are of independent interest: they can be used to deform Riemann's zeta and the Dirichlet L-functions. The stabilization of knot invariants in families of (colored) knots/links attracted a lot of attention. Punctual Hilbert zeta functions of isolated surface singularities can be obtained as such limits. These zeta functions are Nekrasov-type instanton sums for singularities, closely related to Kapranov's zetas. The extension of this talk will be mostly devoted to the Hilbert zeta function for the “double point”. As q=1, it becomes the generating series of the Euler numbers of Hilbert schemes of the type-A_1 surface singularity, studied (for arbitrary simple singularities) by Toda, Gyenge-Nemethi-Szendroi, Nakajima and others. As t=1, the corresponding Betti numbers are obtained. Potential applications in number theory from the first part of this talk indicate that different instanton sums may be needed here: those extended by ranks and "conductors" (to be discussed).

← Back