Caltech/USC Joint Algebra and Geometry Seminar

(Joint with Ruofan Jiang and Alexei Oblomkov) The Hilbert scheme of points on planar singularities is an object with rich connections (q,t-Catalan numbers, HOMFLY polynomials, Oblomkov–Rasmussen–Shende conjecture). The Quot scheme of points is a higher rank generalization of the Hilbert scheme of points. As our main result, we prove that for the "torus knot singularity" x^a = y^b with gcd(a,b)=1, the Quot scheme admits a cell decomposition: every Birula-Białynicki stratum is "as nice as possible" despite poor global geometry. The proof uses two key properties of the rectangular‑grid poset: an Ext‑vanishing for certain quiver representations and a structural result on the poset flag variety. Time permitting, I will discuss a conjectured Rogers–Ramanujan type identity, whose sum side is a summation on (nested) a x b Dyck paths and product side has modulus a+b.