# Discrete Math Seminar (DMS)

The **Discrete Math Seminar (DMS)** is intended for Kennesaw State faculty working in the various areas of algebra, number
theory, and discrete mathematics to get together to discuss their current work or
related questions. Seminars often involve advanced mathematical knowledge. However,
the seminars are open to anyone who is interested in attending.

In Fall 2021, all talks will be scheduled from 3:00-4:00pm and will either be held
in person in D-120 Mathematics building, Marietta campus or virtually in Microsoft
Teams.

**Upcoming Events**

#### Friday, November 12, 2021

**SPEAKER:****Dr. Paul Pollack**, University of Georgia**TIME/LOCATION:**3:00-4:00pm online in**Microsoft Teams****TITLE:***"Multiplicative orders mod p: what not everyone knows"***ABSTRACT:**If*a*is an integer coprime to*m*, the multiplicative order of*a*mod*m*is the least positive integer*k*with*a^k==1*mod*m*. I will discuss some lesser-known problems and results about multiplicative orders modulo prime numbers*p*. Particular attention will be paid to the following kind of question: For two fixed integers*a*and*b*, how do the multiplicative orders of*a*mod*p*and*b*mod*p*compare, as*p*varies? This represents joint work with Matt Just and Sergei Konyagin.

## Past Events

#### Friday, September 4, 2020

**SPEAKER: Dr. Andrew Wilson**, Kennesaw State University**TIME/LOCATION:**2:30-3:30pm online in Microsoft Teams**TITLE:***"The combinatorics of harmonic polynomials"***ABSTRACT:**Polynomials in n variables carry an action of the symmetric group on n symbols. If we take the orthogonal complement of the polynomials which are invariant under this action, we get what is called the space of harmonic (or co-invariant) polynomials. In 1942, Emil Artin showed that this harmonic space has dimension n!. Recent work has extended this result to allow the symmetric group to act "diagonally" on multiple sets of variables and to allow some of these variables to anti-commute, motivated by ideas from mathematical physics. I will give an overview of this work, focusing on the combinatorial objects which arise, including permutations, parking functions, and ordered set partitions.

#### Friday, October 2, 2020

**SPEAKER: Dr. Mikhail Lavrov**, Kennesaw State University**TIME/LOCATION:**2:30-3:30pm online in Microsoft Teams**TITLE:***"Monochromatic squares, Hales-Jewett numbers, and why 5 is harder than 4"***ABSTRACT:**I'll begin this talk by talking about a Ramsey problem in the hypercube. This is a distant cousin of the problem for which the record-setting Graham's number was invented, but we'll be less ambitious and accordingly our upper bound will be much smaller: 78. (This result is joint work with John Mackey and Mitchell Lee.)

The proof technique generalizes to solve a different problem. The Hales-Jewett theorem is a Ramsey result for monochromatic lines in a high-dimensional grid; it implies many other results in Ramsey theory, but is very hard to find good upper bounds for. We show how to bound the (two-color) Hales-Jewett number when the grid has side length 4. Finally, we'll see why generalizing to side length 5 or more is hard and needs new ideas.

#### Friday, November 6, 2020

**SPEAKER: Dr. Julianne Vega**, Kennesaw State University**TIME/LOCATION:**2:30-3:30pm online in Microsoft Teams**TITLE:***"Integer Decomposition Property of Schur and Symmetric Grothendieck Polynomials"***ABSTRACT:**In this presentation we will consider Newton polytopes arising from two families of polynomials in algebraic combinatorics: Schur polynomials and inflated symmetric Grothendieck polynomials. In both cases, we show that these polytopes have integer decomposition property. This work is joint with Margaret Bayer, Bennet Goeckner, Su Ji Hong, Tyrrell McAllister, McCabe Olson, Casey Pickney, and Martha Yip.

#### Friday, February 5, 2021

**SPEAKER: Dr. Mikhail Lavrov**, Kennesaw State University**TIME/LOCATION:**2:30-3:30pm online in Microsoft Teams**TITLE:***"Comparisons, edge coloring, and the Erdos-Szekeres theorem"***ABSTRACT: Link to pdf abstract**

#### Friday, February 19, 2021

**SPEAKER: Dr. Erik Westlund**, Kennesaw State University**TIME/LOCATION:**2:30-3:30pm online in Microsoft Teams**TITLE:***"Extendability of k-suitable matchings in Cartesian products"***ABSTRACT: Link to pdf abstract**

#### Friday, April 2, 2021

**SPEAKER: Dr. Thomas McConville**, Kennesaw State University**TIME/LOCATION:**2:30-3:30pm online in Microsoft Teams**TITLE:**"Determinantal formulas with major indices"**ABSTRACT: Link to pdf abstract**

#### Friday, April 16, 2021

**SPEAKER: Dr. Richard Ehrenborg**, University of Kentucky**TIME/LOCATION:**2:30-3:30pm online in Microsoft Teams**TITLE:***"Sharing pizza in n dimensions"***ABSTRACT: Link to pdf abstract**

#### Friday, September 3, 2021

**SPEAKER: Dr. Thomas McConville**, Kennesaw State University**TIME/LOCATION:**3:00-4:00pm in D-120 Mathematics building, Marietta campus**TITLE:***"Enumerative aspects of bubbling and shuffling"***ABSTRACT:**The shuffle lattice is a partial order on words determined by two common types of genetic mutation: insertion and deletion. Curtis Greene discovered many remarkable enumerative properties of this lattice that are inexplicably connected to Jacobi polynomials. In this talk, I will introduce an alternate poset called the bubble lattice. This poset is obtained from the shuffle lattice by incorporating a third type of mutation, namely transposition. Using the structural relationship between bubbling and shuffling, we provide insight into Greene’s enumerative results. This talk is based on joint work with Henri Mülle.

#### Friday, September 24, 2021

**SPEAKER:****Dr. Andrés Vindas Meléndez**, University of California, Berkeley and MSRI**TIME/LOCATION:**3:00-4:00pm in D-120 Mathematics building, Marietta campus and streaming online in Microsoft Teams**TITLE:***"Ehrhart Theory of Paving Matroid Polytopes"***ABSTRACT:**Ehrhart theory is a topic in geometric combinatorics which involves the enumeration of lattice points in integral dilates of polytopes. We introduce pan-handle matroids, which can be understood as a certain lattice-path matroid. We provide a formula for the Ehrhart polynomial of the matroid polytope of pan-handle matroids. This Ehrhart polynomial allows us to obtain a formula for matroid polytopes of paving matroids. No prior knowledge will be assumed and the talk will aim to be accessible to non-experts. This is joint work with Mohsen Aliabadi, Derek Hanely, Jeremy L. Martin, Daniel McGinnis, Dane Miyata, George D. Nasr, and Mei Yin.

#### Friday, October 15, 2021

**SPEAKER:**Dr. Zhu Cao, Kennesaw State University**TIME/LOCATION:**3:00-4:00pm in D-120**Mathematics building, Marietta campus**and online in Microsoft Teams**TITLE:***"Exact covering systems, quadratic forms, and identities for the Rogers-Ramanujan functions"***ABSTRACT:**The Rogers-Ramanujan functions satisfy the famous Rogers-Ramanujan identities as well as Ramanujan's continued fraction. These identities have combinatorial interpretations and combinatorial proofs. As an example of applications, Rogers-Ramanujan identities appeared in Baxter's solution of the hard hexagon model in statistical mechanics. Ramanujan established a list of forty identities involving the Rogers-Ramanujan functions. In the past, people have developed various tools to attack the forty identities.

From the point of view of exact covering systems (ECS), we can extend the definition of the equivalence of quadratic forms. Furthermore, we show that there are natural connections among ECS, quadratic forms, and product identities for theta functions. In the past several years, I have unified most of the previous approaches and provided nice explanations for most of the identities among the forty identities. The applications of our main result are not just restricted to identities for the Rogers-Ramanujan functions. We can also work on n-ary quadratic forms (especially ternary quadratic forms) and present a list of new identities.

#### Friday, October 29, 2021

**SPEAKER:****Dr. Anthony Bonato**, Ryerson University**TIME/LOCATION:**3:00-4:00pm online in Microsoft Teams**TITLE:***"Pursuit-evasion games on graphs"***ABSTRACT:**In pursuit-evasion games, a set of pursuers attempts to locate, eliminate, or contain an evader in a network. The rules, specified from the outset, greatly determine the difficulty of the questions posed above. For example, the evader may be visible, but the pursuers may have limited movement speed, only moving to nearby vertices adjacent to them.

Central to pursuit-evasion games is the idea of optimizing certain parameters, whether they are the search number, burning number, or localization number, for example. We report on progress in several pursuit-evasion games on graphs and conjectures arising from their analysis. Finding the values, bounds, and algorithms to compute these graph parameters leads to topics intersecting graph theory, the probabilistic method, and geometry.