Verlinde series are generating functions of Euler characteristics of line bundles on the Hilbert schemes of points on a surface. Formulas for Verlinde series were determined for surfaces with $K=0$ by Ellingsrud, Göttsche, and Lehn. More recently, Göttsche and Mellit determined Verlinde series for surfaces with $K^2=0$, and gave a conjectural formula in the general case. In this talk, I will give a formula for the Euler characteristics of line bundles on the Hilbert schemes of points on $\mathbb{P}^1 \times \mathbb{P}^1$, and a combinatorial (but less explicit) formula for ample line bundles on the Hilbert schemes of points on Hirzebruch surfaces. By structural results of Ellingsrud, Göttsche, and Lehn, this determines the Verlinde series for all surfaces. The proof is based on a new combinatorial description of the equivariant Verlinde series for the affine plane.
For more information see https://personal.math.ubc.ca/~jbryan/Zoominar-UBC-ETH/