Spring 2023

- The Combinatorics Seminar meets on
**Fridays, 3-4pm**, in**Snow 302**. (We may sometimes use Zoom instead.) - Organizers: Mark Denker and Jeremy Martin.
- Good general seminar-attending advice, especially for graduate students: The "Three Things" Exercise for getting things out of talks by Ravi Vakil
- Also consider attending the Graduate Online Combinatorics Colloquium.

**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