Abstracts
Martijn Kool
Intersection theory on Hilbert schemes on Calabi-Yau 4-folds
I discuss intersection theory on Hilbert schemes of subschemes of Calabi-Yau 4-folds (reviewing lower dimensions along the way). For Hilbert schemes of points on affine 4-space, this is related to Nekrasov’s Magnificent Four formula (joint with Rennemo). For Hilbert schemes of surfaces on compact Calabi-Yau 4-folds, this is related to the variational Hodge conjecture (joint with Bae-Park).
Georg Oberdieck
Quantum cohomology of the Hilbert scheme of points on an elliptic surface
Quantum cohomology is a deformation of the classical cup product that encodes the 3-pointed Gromov-Witten invariants of a space. In this talk I will explain the proof of a formula for the quantum product with divisor classes on the Hilbert scheme of points on an elliptic surface S with respect to fiber curve classes. By the work of Hu-Li-Qin this determines the full quantum product with divisor classes whenever p_g(S)>0. Joint work with Aaron Pixton.
Andrei Negut
Quot schemes of curves and Yangians
In the '90s, Nakajima and Grojnowski showed how to realize the cohomology of Hilbert schemes of surfaces by having a family of commuting operators act on a single "vacuum" vector. In this project, we will do the same for Quot schemes of curves, and the analogous family of operators will live inside the Yangian of sl_2. Joint work with Alina Marian.
Eugene Gorsky
Generic curves and non-coprime Catalans
We compute the Poincaré polynomials of the compactified Jacobians for plane curve singularities with Puiseaux exponents (nd,md,md+1), and relate them to the combinatorics of q,t-Catalan numbers in the non-coprime case. This is a joint work with Mikhail Mazin and Alexei Oblomkov.
Dimitri Wyss
Non-archimedean integration on quotients and BPS-invariants
Non-archimedean integration on smooth Deligne-Mumford stacks provides an efficient tool to encode their orbifold cohomolgy. It can for example be used to give a short proof of Göttsche's formula for the Hilbert scheme of points on a surface.
Motivated by Donaldson-Thomas theory for del Pezzo surfaces, we generalize this theory to moduli spaces of smooth Artin stacks. As an application we obtain a new expression for BPS-invariants of CY1 categories and recover Maulik-Shen’s $\chi$-independence result for BPS-invariants on del Pezzo surfaces. This is joint work with Michael Groechenig and Paul Ziegler.
Alexei Oblomkov
Title: Interpolating between the Alexander polynomial and Igusa Zeta function.
Joachim Jelisiejew
Limits of Gorenstein algebras: oriented Iarrobino scheme
We introduce the oriented Iarrobino scheme. This is a fine moduli space with a map to the Hilbert scheme of points. From the global perspective, its construction is a step in understanding the singularities of Hilbert schemes of points on higher-dimensional varieties, especially threefolds. From the local perspective, it geometrizes Iarrobino's symmetric decomposition of the Hilbert function: the torus limit of a Gorenstein algebra in the Iarrobino scheme of affine space yields the symmetric decomposition. I will speculate a bit on the nonoriented case. This is a work in progress, parts of this work are joint with Alessio Sammartano and Ritvik Ramkumar.
Tony Iarrobino
Ilaria Rossinelli
Motivic integration on Hilbert schemes: curvilinear Hilbert schemes and applications to plane curve singularities
Alessio Sammartano
Hilbert schemes of points in affine 3-space and their tangent spaces
Several basic questions about the geometry of Hilbert schemes of points remain open. I will discuss some of these problems, and some recent progress, with particular emphasis on the 3-dimensional case. This talk is based on joint works with and Joachim Jelisiejew and Ritvik Ramkumar.
Richard Rimányi
3d mirror symmetry of stable envelopes
In this joint work with T. M. Botta we study the elliptic characteristic classes called stable envelopes introduced by M. Aganagic and A. Okounkov. Stable envelopes measure singularities, they geometrize quantum group representations, and they can be interpreted as monodromy matrices of certain differential or difference equations. We prove that for a rich class of holomorphic symplectic varieties---called Cherkis bow varieties---their elliptic stable envelopes display a duality inspired by mirror symmetry in d=3,N=4 quantum field theories. In the key step of our proof, we ``resolve'' large charge branes to a number of smaller charge branes. This phenomenon turns out to be the geometric counterpart of the algebraic fusion procedure. Along the way we discover more about the rich geometry of bow varieties, such as their Bruhat order and the elliptic Hall algebra structure on their stable envelopes.
Lev Rosansky
Title: B-type 2-categories associated with Nakajima quiver varieties and their applications.
Abstract: This is a review of my joint work with A. Oblomkov and R. Rimanyi. We describe a 2-category of holomoprhic Lagrangian subvarieties associated with a quiver variety and demonstrate two applications of this construction: the categorical (affine) braid group action leading to link homology and a quiver-in-quiver construction of varieties associated with superlagebras gl(m|n).
Alexander Braverman