Title: Protection Statistics in Rooted Trees

Abstract: For the first part of the talk, we will look at different classes of unlabeled simply generated rooted trees and study the asymptotic behavior of statistics which relate to the distance of vertices to the fringe: rank, height, and balance. I will focus on some general results on balanced trees using analytic techniques. Time permitting, I will also give a combinatorial proof of a recent result by Bona: the number of neighbors of leaves in increasing trees of size n is equal to the number of permutations of [n] with no fixed points. This talk is based on my doctoral thesis.

Title: Boundary Algebras of Positroids

Abstract: Positroid varieties are subvarieties of the Grassmannian defined by requiring the vanishing and nonvanishing of certain maximal minors. Positroid varieties are known to have a cluster structure which is categorified by the Gorenstein-projective module category over the completed boundary algebra of the associated dimer model. I will talk about positroid combinatorics and describe the boundary algebras which arise from arbitrary connected positroids.

Abstract: Over 50 years ago, Victor Kac and Robert Moody introduced Kac-Moody algebras as a natural extension of the already classified semisimple Lie algebras. There are three types of Kac-Moody algebras: finite, affine, and indefinite. Both finite and affine have had all root multiplicities calculated. Some partial results have been obtained in the indefinite case, but the root multiplicities are not completely known. In this work, we realize the indefinite Kac-Moody algebras HE_7^(1)and HE_8^(1) as minimal graded Lie algebras whose local part is V\oplus gl(n;C)\oplus V' where V and V' are suitably chosen gl(n)-modules. This will allow us to use the combinatorial Kang's multiplicity formula to compute the root multiplicities to level 7 in the case of  HE_7^(1) and level 9 in the case of HE_8^(1). Additionally, we verify the counterexample to Frenkel's conjecture for HE_8^(1) found by Kac, Moody, and Wakimoto by computing the relevant root multiplicity and we provide a root whose multiplicity is a counterexample to Frenkel's conjecture for HE_7^(1) , showing that Frenkel's conjecture does not hold for HE_7^(1) .

This is a joint work with Kailash Misra.

Title: Equitable Facility Location

Abstract: Facility location is one of the most common applications of combinatorial optimization. Models that minimize the mean distance between a “customer” and their assigned facility behave well computationally and make sense from a modeling perspective in many contexts. However, if equity is a concern, the mean is not an ideal metric. Many equitable facility location models have been developed in the literature, but they tend not to scale well computationally because nonlinear optimization models with integer variables are very hard to solve. Historically applied to incomes, equally distributed equivalents (EDEs) provide more accurate measures of the experience of a population than the population mean by penalizing values on the “bad” end of the distribution; i.e., very low incomes pull the EDE below the mean.  We develop a computationally scalable facility location model that minimizes the Kolm-Pollak EDE, a metric that is applied in the Environmental Justice literature to compare exposure to environmental harms, such as air pollution, across demographic groups. We apply our methods to food deserts and election polling locations, demonstrating that optimizing over the Kolm-Pollak EDE, rather than the mean, can lead to big gains in equity while still resulting in near-optimal average distances.

DOCTORAL DEFENSE

Speaker:  Benjamin Reese

Title:  Geometry of pipe dream complexes

DOCTORAL DEFENSE

Speaker:  Benjamin Reese

Title:  Geometry of pipe dream complexes

DOCTORAL DEFENSE

Speaker:  Benjamin Reese

Title:  Geometry of pipe dream complexes