Special Year 2024-25: Algebraic and Geometric Combinatorics

Abstract: A Richardson variety R in a cominuscule Grassmannian is defined by a skew diagram of boxes. If this diagram has several connected components, then R is a product of smaller Richardson varieties given by the components. I will show that the...

Abstract: There are two major research trends in the theory of symmetric functions arising from Dyck paths. One is the theory of Catalan symmetric functions and its geometric realization conceived by Chen-Haiman, following the works of Broer and...

Abstract: While mirror symmetry for flag varieties and Grassmannians has been extensively studied, Schubert varieties in the Grassmannian are singular, and hence standard mirror symmetry statements are not well-defined. Nevertheless, in joint work...

Abstract: Springer fibers are subvarieties of the flag variety parameterized by partitions. They are central objects of study in geometric representation theory. Given a partition $λ$, one of the key conclusions of Springer theory is that the top...

Abstract: Suppose $X$ is the affine cone of a projective variety. The Hilbert series of the coordinate ring $C[X]$ is the character of an algebraic torus. More generally, one considers a reductive algebraic group $G$ acting rationally on $X$. When...

Abstract: Chow quotients of projective varieties by affine torus actions provide alternative constructions of interesting geometric objects. For example, the moduli space of stable genus 0 curves with $n$ marked points arises as the Chow quotient of...

Abstract: We will discuss invariants of lattice polytopes and their subdivisions arising from Ehrhart and Hodge theory and introduce their matroid theoretic analogues which are enriched versions of the characteristic and Tutte polynomials.

Abstract: I will discuss some regular subdivisions of the permutahedron, one for each Coxeter element in the symmetric group. These subdivisions are "Bruhat interval" subdivisions, meaning that each face is the convex hull of the permutations in a...

Abstract: The general rule for the interactions between tropical geometry and moduli spaces of course is the following: everything you may wish is going to work like a charm in genus zero, and break down horribly in higher genus. This is the case...

Abstract: In his 2018 paper "Some Schubert shenanigans" Richard Stanley asked for the asymptotic behavior of the maximal principle specialization of a Schubert polynomial. Motivated by this, still open, question we explore the generalization to...