KU Combinatorics Seminar
Spring 2023

Thursday 1/19
Faculty candidate talk (4-5pm)

Friday 1/20
Organizational meeting

Tuesday 1/24
Faculty candidate talk (4-5pm)

Thursday 1/26
Faculty candidate talk (4-5pm)

Friday 1/27
No seminar

Tuesday 1/31
Faculty candidate talk (4-5pm)

Friday 2/3
No seminar

Friday 2/10
Marge Bayer
Magic and Antimagic Graphs and Hypergraphs

Abstract: Label the edges of a graph with nonnegative integers. The labeling is magic if each vertex has the same sum of the labels on the edges incident to it. The labeling is antimagic if the sums at the vertices are all different. It is not known if every graph (besides \(K_2\)) has an antimagic labeling. Fifty years ago Stanley showed that the number of magic labelings with sum \(r\) is a quasipolynomial in \(r\). This talk will review Stanley's result, discuss the work of Beck and Fahramand on antimagic labelings, and present our attempts to extend results to hypergraphs. This is work from a GRWC project with Amanda Burcroff, Tyrrell McAllister and Leilani Pai.

Friday 2/17
Jeremy Martin
Cluster algebras: an introduction

Abstract: I will try to explain what a cluster algebra is and why there are important, using Lauren Williams' article "Cluster algebras: an introduction" [Bull. Amer. Math. Soc. 2014] as my main source. I will not assume any background with cluster algebras, particularly since I have none myself.

Friday 2/24
No seminar

Friday 3/3
No seminar

Friday 3/10
No seminar (Spring Break)

Friday 3/17
No seminar (Spring Break)

Friday 3/24
Online talk [Zoom link] (passcode 1430)
Watch party with cookies in Snow 302

Shira Zerbib (Iowa State University)
KKM-type theorems and their applications

Abstract: The KKM theorem, due to Knaster, Kuratowski and Mazurkiewicz in 1929, is a topological lemma reminiscent of Sperner's lemma and Brouwer's fixed point theorem. It has numerous applications in combinatorics, discrete geometry, economics, game theory and other areas. Generalizations of this lemma, in several different directions, were proved over the years (e.g., by Shapley, Gale, Komiya, Soberon) and have been widely applied as well. We will discuss a recent common generalization of all these theorems. We will also show two very different applications of KKM-type theorems: one is a proof of a conjecture of Eckhoff from 1993 on the line piercing numbers in certain families of convex sets in the plane, and the other is a theorem on fair division of multiple cakes among players with subjective preferences.

Friday 3/31
Kyle Maddox
Katzman and Ehrhart: A tale of two cities

Abstract: In this talk, we will discuss some delightful connections between the Hilbert polynomial of a class of toric rings and the volume of a related polytope. In particular, we will try to demystify a paper of Moty Katzman, "The Hilbert series of algebras of Veronese type," which is often cited in the realm of Ehrhart theory. No background beyond elementary algebra should be required.

Friday 4/7
Mark Denker
The Hopf Monoid on Edge Colored Digraphs

Abstract: Generalized chromatic functions were introduced by Aliniaeifard, Li, and van Willigenburg in a recent preprint to unify the theory of chromatic symmetric functions, chromatic quasisymmetric functions, and P-partitions. Generalized chromatic functions are defined from vertex-colorings of edge-colored digraphs and can be thought of as the image of a Hopf morphism from a Hopf monoid on edge-colored digraphs to the Hopf algebra on quasisymmetric functions. In this talk we introduce the Hopf monoid on edge-colored digraphs, its connections to the Hopf monoids on graphs and directed graphs, and give a cancellation-free antipode formula.

Friday 4/14
Arian Ashourvan (KU Department of Psychology)
Untangling the Brain's Web: Insights from Graph Theory and Network Neuroscience

Abstract: In this talk, I will explore the applications of graph theory in understanding large-scale brain dynamics in network neuroscience. I will begin by introducing the basics of graph theory and its relevance to studying the brain. Next, I will discuss how graph theory is used to examine structural and functional brain networks and can reveal their organizational principles, such as small-worldness, modularity, and rich-club organization. Finally, I will showcase the use of generative models to predict the functional dynamics of brain networks and highlight current challenges and future directions in this exciting area of research.

Friday 4/21
Natasha Rozhkovskaya (Kansas State University)
Applications of symmetric functions for solving soliton equations

Abstract: Symmetric polynomials are polynomials of several variables that do not change their value under any permutations of variables. Their applications in modern mathematics are ubiquitous. In this talk we will give a brief summary of the role of symmetric functions in the theory of soliton equations.

Soliton equations are non-linear partial differential equations that posses remarkable solutions in the form of stable traveling waves. The modern study of these equations is situated at the crossroads of various areas of mathematics.

It turns out, that an infinite collection of soliton equations with the name KP-hierarchy has common solutions that can be obtained from a family of symmetric functions called Schur symmetric functions, which are already famous for their important meaning in representation theory.

We plan to overview main definitions and properties of symmetric functions, examples of solitons, explain the meaning of boson-fermion correspondence, and give general outline of steps that lead to interpretation of Schur symmetric functions as solutions of the KP-hierarchy.

Friday 4/28
José Bastidas (Université de Québec à Montréal)
The Primitive Eulerian polynomial (and type B shenanigans)

Abstract: The coefficients of the Eulerian polynomial count permutations with a given number of excedances. The Primitive Eulerian polynomial keeps track of this statistic for "irreducible" permutations. We will explain what we mean by "irreducible" and state the corresponding result/definition in the case of signed permutations.

We will then define the Primitive Eulerian polynomial \(P_{\mathcal{A}}(z)\) for any central hyperplane arrangement \(\mathcal{A}\) as a reparametrization of its cocharacteristic polynomial. Previous work on the polytope algebra of deformations of a zonotope (2021) implicitly showed that this polynomial has nonnegative coefficients whenever \(\mathcal{A}\) is a simplicial arrangement. However, in that work, we only found a combinatorial interpretation of its coefficients when \(\mathcal{A}\) is the reflection arrangements of type A or B (permutations and signed permutations).

We also present a geometric/combinatorial interpretation for the coefficients of \(P_{\mathcal{A}}(z)\) for all simplicial arrangements \(\mathcal{A}\), along with some real-rootedness results and conjectures. Time permitting, I will discuss some relations between the type B Primitive Eulerian polynomial and other polynomials previously defined in the literature. Based on joint work with Christophe Hohlweg and Franco Saliola.

Friday 5/5 (Stop Day)
Melody Yu
Title TBA

For seminars from previous semesters, please see the KU Combinatorics Group page.

Last updated Fri 4/21/23