Two-page Summary of Research Statement

The Springer variety SX is defined to be the set of flags stabilized by a nilpotent operator X. Springer varieties can be generalized to a two-parameter family of varieties called Hessenberg varieties H(X, h), defined by a nilpotent operator X and a certain step function h (or equivalently, a Dyck path). In 1992, Garsia-Procesi gave a presentation of the cohomology ring H(SX) as the polynomial quotient ring Z[x1, . . . , xn]/I, where I is the so-called Tanisaki ideal [5]. They also give a monomial basis BX for this ring. My first paper generalized this work of Garsia-Procesi when X is regular nilpotent [6]. In it, I conjectured that H∗(H(X, h)) is isomorphic to a polynomial quotient ring Z[x1, . . . , xn]/Jh, identified a basis of monomials BX,h, and proved that this basis recovers the Betti numbers of H(X, h). In my second paper, Julianna Tymoczko and I define a new family of ideals Ih which generalizes Tanisaki ideals [7]. To do this, we modified methods from Biagioli-Faridi-Rosa’s work on finding reduced generating sets for the Tanisaki ideals [1]. We prove that this new presentation Ih coincides with Jh. Moreover, upon reading a preprint of our paper, a Jerzy Weyman student Federico Galetto proved that our conjectural minimal generating sets for these generalized Tanisaki ideals is indeed minimal. Interestingly enough, it was Weyman himself who first conjectured a minimal generating set for the Tanisaki ideal in 1989 [12] to which the Biagioli-Faridi-Rosas paper gave a counterexample [1]. However, their methods helped us develop techniques to find minimal generating sets for our generalized Tanisaki ideals. Perhaps a combination of the two methods can be used to settle the still open problem of finding minimal generating sets for the Tanisaki ideals. After hearing a talk I gave on these generalized Tanisaki ideals, Aaron Lauve of Loyola University Chicago (who independently has been thinking of minimalized generating sets for the classic Tanisaki ideals) expressed interest in pursuing this project with me.