MIT-Harvard-MSR Combinatorics Seminar
Wednesday, February 15, 2023 at 4:15pm to 5:15pm
MIT-Math Dept., Room 2-139
Speaker: Minh-Tam Trinh (MIT)
Title: Catalan combinatorics in algebraic geometry
We present three geometric incarnations of the rational q-Catalan numbers, which correspond to three distinct approaches to their combinatorics: namely, rational Dyck paths, noncrossing partitions, and Kreweras decompositions. The first is due to Gorsky–Mazin and Hikita, and specific to type A. The second and third are new, and generalize to Weyl groups. The second is joint work with Galashin, Lam, and Williams, and resolves several conjectures about noncrossing rational Coxeter–Catalan combinatorics from a decade ago. The third generalizes to a Kreweras decomposition of the HOMFLYPT invariant of any positive braid closure, and categorifies to a geometric interpretation of its Khovanov–Rozansky homology involving unipotent matrices. Time permitting, we will explain why we hope the first story also generalizes beyond type A, and why the naive approach already fails in type BC_2.