Problems

Background

• Hopkins, "Upho Lattices I: : examples and non-examples of cores" (link)

• Hopkins-Lewis, "Upho lattices II: ways of realizing a core" (link)

• Hopkins, "A Note on Möbius Functions of Upho Posets" (link)

Background on Schubert calculus

• Manivel, “Symmetric Functions, Schubert Polynomials and Degeneracy Loci”

• Macdonald, “Notes on Schubert Polynomials” 

• Billey–Gao–Pawlowski, “Introduction to the cohomology of the flag variety” (link)

Background on the project

• Billey–Haiman, “Schubert Polynomials for the Classical Groups” (link)

• Lam—Lee—Shimozono “Back Stable Schubert Calculus" (link)

• Assaf–Searles, “Schubert polynomials, slide polynomials, Stanley symmetric functions and quasi-Yamanouchi pipe dreams” (link)

• Nadeau–Tewari, “P-partitions with flags and back stable quasisymmetric functions” (link)

• Hardt–Wallach, “When do Schubert polynomial products stabilize?” (link)

Abstract: 

This project studies the graded Varchenko--Gelfand ring also called the Cordovil algebra, of a real hyperplane arrangement, which is a finite-dimensional algebra built from Heaviside functions on the chambers of the arrangement. We will present this algebra as a quotient of a polynomial ring and study its minimal free resolution, Betti table, and Koszul homology algebra. We will be particularly interested in understanding the Cordovil algebra associated to the uniform matroid and the type A braid arrangement. The project is computational and experimental, but the goal is to extract precise combinatorial conjectures from the data.

A more detailed description of the problem and some background can be found here.