Back to Algebraic Geometry Seminar: Spring 2022
Algebraic Geometry Seminar
Date: Tuesday, Apr 26, 2022
Time: 3:30PM - 4:30PM
Location: See individual entries
Virtual
Hunter Dinkins
UNC Chapel Hill
Title |
Exotic Quantum Difference Equations and Integral Solutions |
Abstract |
From their beginning, Nakajima quiver varieties have bridged the gap between algebraic geometry and representation theory. Maulik and Okounkov pioneered new such directions in 2012, as they constructed a Hopf algebra acting in the cohomology of quiver varieties. One of their main results is the identification of the quantum differential equation governing the enumerative geometry of quiver varieties with operators in this Hopf algebra. A similar identification was obtained later by Okounkov and Smirnov for the K-theoretic difference equations. Their results suggest a natural generalization of the difference equations, which will be the focus of this talk. I will define these new difference equations and demonstrate how their solutions can be related back to enumerative geometry. In the case that the quiver variety has finitely many torus fixed points, these results provide integral solutions for these difference equations and are related to the algebraic Bethe Ansatz. I will give examples of all results for the Hilbert scheme of points in the plane. |
Algebraic Geometry