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

Masters Exam

Speaker:  Chloe Napier

Title:  TBA

Abstract:

Speaker:  William Dugan, U Mass Amherst

Title:        Faces of generalized Pitman-Stanley polytopes

Abstract:

The Pitman-Stanley polytope is a polytope whose integer
lattice points biject onto the set of plane partitions of a certain
shape with entries in {0 ,1}. In their original paper, Pitman and
Stanley further suggest a generalization of their construction depending
on $m \in {\mathbb N}$ whose integer lattice points biject onto the set
of plane partitions of the same shape having entries in  $\{ 0 , 1, ...
, m \}$. In this talk, we give further details of this
generalized Pitman-Stanley polytope, $PS_n^m(\vec{a})$,
demonstrating that it can be realized as the flow polytope of a certain
graph. We then use the theory of flow polytopes to describe the faces of
these polytopes and produce a recurrence for their f-vectors.

William Dugan is a student of Alejandro Morales who is funding this visit.

Masters Exam

Speaker:  Chloé Napier, University of Kentucky

Title:     New Interpretations of the Two Higher Stasheff-Tamari Orders

Abstract:

In 1996, Edelman and Reiner defined the two higher Stasheff-Tamari orders on triangulations of cyclic polytopes and conjectured that they are equal. In 2021, Nicholas Williams defined new combinatorial interpretations of these two orders to make the definitions more similar. He builds upon the work by Oppermann and Thomas in the even dimensional case of giving an algebraic analog to these orders using higher Auslander-Reiten Theory. He then gives a completely new result for the odd dimensional case. In this talk, we will discuss the combinatorial interpretations of the even dimensional case and motivate the odd dimensional case and algebraic analog by example. 

See schedule

https://www.ms.uky.edu/~readdy/KOI/

See schedule https://www.ms.uky.edu/~readdy/KOI/

KOI Combinatorics Lectures

Speaker:  Richard Ehrenborg, University of Kentucky

Title:         Sharing pizza in n dimensions

Abstract:

We introduce and prove the n-dimensional Pizza Theorem. Let H be a real n-dimensional hyperplane arrangement. If K is a convex set of finite volume, the pizza quantity of K is the alternating sum of the volumes of the regions obtained by intersecting K with the arrangement H. We prove that if H is a Coxeter arrangement different from A₁ⁿ such that the group of isometries W generated by the reflections in the hyperplanes of H contains the negative of the identity map, and if K is a translate of a convex set that is stable under W and contains the origin, then the pizza quantity of K is equal to zero. Our main tool is an induction formula for the pizza quantity involving a subarrangement of the restricted arrangement on hyperplanes of H that we call the even restricted arrangement. We get stronger results in the case of balls. We prove that the pizza quantity of a ball containing the origin vanishes for a Coxeter arrangement H with |H|-n an even positive integer.

This is joint work with Sophie Morel and Margaret Readdy.

https://www.ms.uky.edu/~readdy/KOI/