Math Colloquium

2020 Fall

2020 Fall Math Colloquia

September 16 Virtual, 4:00pm

Carl Mummert (Marshall University)
A case study in computable graph theory: König’s edge coloring theorem

Abstract Computable graph theory is a subfield of combinatorics and logic. A graph, in this sense, has a set of vertices (points), some of which are connected with edges. Graphs are applied to model many kinds of relationships in numerous disciplines. Computability theory uses a framework originally developed by Alan Turing to study the problems computers are theoretically able to solve. This talk will introduce computable graph theory using a specific theorem. König’s edge coloring theorem gives a way to assign colors to the edges of a graph using the least number of colors possible. I will introduce the theorem and many results about its computability theoretic properties.

No prior familiarity with graph theory or computability is needed, and this talk is aimed at a general audience.

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October 21 Virtual, 4:00pm

Michael Otunuga (Marshall University)
Time Dependent Probability Density Function for Number of Infection in a Stochastic SIS Epidemic Model

Abstract: The closed-form time dependent probability density function of the number of infected individuals at a given time satisfying a stochastic Susceptible-Infected-Susceptible (SIS) epidemic model is derived and analyzed using the Fokker-Planck equation. The mean, median, variance, skewness and kurtosis of the distribution are obtained as a function of time. We study the effect of noise intensity on the distribution and later derive and analyze the effect of changes in the transmission and recovery rates of the disease. The result is applied using published Covid-19 data/parameters.

November 18 Virtual, 4:00pm

Michael Schroeder (Marshall University)
The null space of maximum density ASMs

Abstract: At some point in your life, you have been asked to “set it equal to zero and solve.” This is another such time. A large field of research is dedicated to computing the spectrum of a matrix, which in part asks the question,

If A is a matrix, for what vectors x does A x = 0?

The set of all solutions to this equation is the null space of A. In this talk, we look at a particular family of matrices called alternating sign matrices (or ASMs for short), which are matrices with only 0s, 1s, and -1s in it that satisfy a certain “alternating” condition.

We define and describe some properties of maximum density ASMs, then compute their null spaces. Surprisingly, some elementary graph theory was helpful to solve the problem. This research is joint work with Pauline van den Driessche, from the University of Victoria.

Having some background in linear algebra would be beneficial, but the only requirement is that, at some point in your life, you’ve been asked to set something equal to zero and solve.

2020 Spring

2019 Fall

2019 Fall Math Colloquia

September 18 Smith Hall 516, 4:00pm

Faith Hensley (Marshall University)
Extremal Numbrix Puzzles

Abstract: Numbrix is a puzzle in Parade magazine. The player is given a 9 × 9 grid with some integers between 1 and 81 filled in. The player then needs to fill in the rest of the integers between 1 and 81 so that consecutive integers appear in adjacent cells of the grid. Generalizing this puzzle we consider m × n grids with the entries being the integers between 1 and mn. We say that a set of clues defines a puzzle if there exists a unique solution given those clues. In 2018 Hanson and Nash find the maximum number of clues that fail to define an m × n puzzle for all m and n. I present our work on their conjecture concerning the minimum number of clues necessary to define a puzzle. This research is joint work Ashley Peper conducted as part of the 2019 REU program at Grand Valley State University.

Chloe’ Marcum (Marshall University)
Extended Schur Functions

Abstract: The set of quasisymmetric functions homogeneous of degree n form a polynomial
vector space, QSym, with multiple bases. Many of these bases can be generated
combinatorially using tableaux with various rules. The bases of QSym we are interested
in are the quasisymmetric Schur functions and the extended Schur functions.
Both generalize the symmetric Schur functions. A common question in linear algebra
is how to transition from one basis to another. We have an expansion of
extended Schur functions into quasisymmetric Schur functions for a family of indexing
compositions. We prove this expansion combinatorially by using a tree
whose leaves indicate the polynomials that appear in our expansion.

October 16 Smith Hall 516, 4:00pm

Logan Rose (Marshall University)
Modeling Malaria with Controls

Abstract: Malaria is a deadly parasitic disease that has been a major threat to public health for centuries, particularly in Sub Saharan Africa and South Asia. Although there are still 200 million cases reported yearly according to the CDC, there have been renewed efforts in recent decades to combat this illness. Mathematicians have created various models using differential equations to simulate the spread of malaria and to determine which control strategies are most effective in reducing the number of new infections. Many of these consider the interactions between humans and the Anopheles mosquito, the primary vector of the disease. This project considers the effectiveness of two popular malaria control strategies: using bednets and releasing sterile mosquitoes. We used an eight-equation model that combines an SEIR (Susceptible-Exposed-Infected-Recovered) model for the human population, an SEI (Susceptible-Exposed-Infected) model for the vector population, and an additional equation for the sterile mosquito population. In addition, we applied the Next Generation Matrix method to our model to derive a formula for the basic reproduction number for infectious disease. Finally, we apply optimal control theory to find the optimal releasing strategies, and numerical simulations are presented for various cases.

Michael Waldeck (Marshall University)
Generalizing the Classical Construction for Complete Sets of Mutually Orthogonal Latin Squares

Abstract: Latin squares are a well established topic of study in Mathematics. A latin square of size n is an n by n square where each number from 1 to n appears exactly once in each row and in each
column. Two latin squares are orthogonal mates if, when the squares are superimposed, every
possible ordered pair of entries appears in the superimposed square. In the 18th century, Leonhard
Euler studied the question of finding a complete set of mutually orthogonal latin squares (MOLS)
of a given size. Euler proved that for each n, a set of n by n MOLS can have no more than n – 1
squares. It has also been proved that, where n is a prime power, this upper bound is achieved
through the classical construction. The classical construction for a complete set of MOLS of size
n by n can be given by a formula M(i, j) = i + cj, where c is a nonzero element in the finite field of order n. We study the number of orthogonal mates and the graph of the mate relationship between a family of latin squares based on a generalization of that formula. That is, when q is prime, we study the set of latin squares of size q by q that can be given by a formula S(s, c)(i, j) = s(i) + cj,
where s mapping the finite field of order q to itself is a permutation of the field, c is a nonzero element in the finite field of order q, and the addition and multiplication are carried out in the finite field.

November 20 Smith Hall 530, 4:00pm

Carl Mummert (Marshall University)
How to Multiply Big Natural Numbers

Abstract: Big natural numbers are all around. They keep your data private on the internet, and scientists use them for high precision simulations. Adding big numbers is not very hard, but multiplying them is much more challenging. The problem of multiplying numbers efficiently is at the border between mathematics and computer science.

I will introduce several multiplication algorithms currently used in state of the art software. Then I will describe a breakthrough announced by two researchers in March 2019. They produced an algorithm that can multiply two natural numbers – of any size – with an efficiency we think is optimal. This talk is aimed at anyone with a background of college algebra or higher.

2019 Spring

2018 Fall

2018 Fall Math Colloquia

September 19
Elizabeth Niese (Marshall University)
The RSK algorithm and applications
Smith Hall 516, 4:00pm

Abstract: The Robinson-Schensted-Knuth (RSK) algorithm is a classical algorithm in algebraic combinatorics. It is a bijection between words and pairs of tableaux which has many interesting combinatorial properties. There are a number of distinct constructions equivalent to this algorithm, including the jeu-de-taquin and Viennot’s shadow lines construction. We will look at several of these constructions and their use in proofs of algebraic formulas.

October 17
Chloé Marcum (Marshall University)
Using polynomials to study knots
Smith Hall 516, 4:00pm

Abstract: A knot is any closed loop in space. Two knots may be the same knot even though they appear to be different. A local move is a change to a small portion of a knot and the rest of the knot is assumed to stay the same. A local move may or may not change the knot. An invariant is a characteristic of a knot that help us tell the difference between knots. In our research, we studied certain polynomials that are knot invariants. We studied the effect of certain local moves on Homflypt and Kauffman polynomials. As a consequence, we discovered some new properties of these invariants.

This research was completed in summer 2018 as part of a Research Experience for Undergraduates (REU) at St Mary’s College of Maryland. At the end of the talk, I will share general information about REUs as well as my experience at St. Mary’s.

Knot theory is an area of math that is understandable without a lot of math background. All students interested in math are encouraged to attend.

November 28
Raid Al-Aqtash (Marshall University)
Market Basket Analysis
Smith Hall 516, 4:00pm

Abstract: Market basket analysis is a data mining technique based upon using association rules to uncover the purchasing trends in large transaction datasets. The key factor here is to determine possible lists of items that are frequently sold together. In this talk, I will speak about association rules and how they can be used by large retail companies in market basket analysis. A real dataset will be provided in an application of association rules.

Keywords and phrases: data mining, basket analysis, beer and diapers, association rules, support, confidence, lift.

This will be a general audience talk, using a bit of information about proportions and percentages.

2018 Spring

2018 Spring Math Colloquia

February 21
Michael Otunuga (Marshall University)
Global stability for a (2n+1)-dimensional HIV/AIDS epidemic model with treatments
Smith Hall 518, 4:00pm

Abstract: In this work, we derive and analyze a (2n+1)-dimensional deterministic differential equation modeling the transmission and treatment of HIV (Human Immunodeficiency Virus) disease. The model is extended to a stochastic differential equation by introducing noise in the transmission rate of the disease. A theoretical treatment strategy of regular HIV testing and immediate treatment with Antiretroviral Therapy (ART) is investigated in the presence and absence of noise. By defining R(0,n), R(t,n) and R(t,n) as the deterministic basic reproduction number in the absence of ART treatments, deterministic basic reproduction number in the presence of ART treatments and stochastic reproduction number in the presence of ART treatment, respectively, we discuss the stability of the infection-free and endemic equilibrium in the presence and absence of treatments by first deriving the closed form expression for R(0,n), R(t,n) and R(t,n). We show that there is enough treatment to avoid persistence of infection in the endemic equilibrium state if R(t,n)=1. We further show by studying the effect of noise in the transmission rate of the disease that transient epidemic invasion can still occur even if R(t,n)<1. This happens due to the presence of noise (with high intensity) in the transmission rate, causing R(t,n)>1. A threshold criterion for epidemic invasion in the presence and absence of noise is derived. Numerical simulation is presented for validation.

April 19
Carl Mummert (Marshall University)
The number TREE(3), and counting down in base infinity
Smith Hall 518, 4:00pm

Abstract: The motivation of this talk is a peculiar situation from computer science. In some cases, we know that a program will eventually stop, but we have no way to concretely describe or even bound the number of steps the program will take. For one such program, the number of steps is a number TREE(3) so large that there is no concrete way to describe it or bound it from above.

This talk will introduce TREE(3) and the related result known as Kruskal’s theorem. We will look at some simpler versions of the theorem, leading us to a “base infinity” number system. This system is like base ten, but each digit can be arbitrarily large. We will see that counting down to 1 from a base infinity number is not as easy as it sounds.

The work on base infinity numbers is joint research with mathematics major Samantha Colbert.

2017 Fall

2017 Fall Math Colloquia

September 20
Matt Davis (Muskingum University)
Non-transitive dice: Constructions, Complications, and Questions
Smith Hall 509, 4:00pm

Abstract: Non-transitive dice have been a source of fascination for mathematicians for over 50 years. We are given a set of dice which are numbered in strange ways. Each player chooses a die, rolls it, and the higher roll wins. Our intuition suggests that in any set of dice, one is the “best”. However, it turns out that it is relatively easy to construct a set of dice which are non-transitive – where most dice are strong against some opponents and weak against others. In this talk we will look at lots of examples of these fascinating objects, aiming for a goal of a single construction that allows us to create a set of dice in any desired configuration. We will also talk briefly about the much harder problem of finding the most efficient way to create such a set of dice.

October 18
Skye Smith (Service Pump & Supply, Huntington WV)
Three Things I Wish I Had Known When I Was a Math Major
Smith Hall 509, 4:00pm

Abstract: Since graduating from Marshall University with an applied mathematics degree in 2014, I have used my degree in several various business roles. Each position has provided a new way to use my mathematics degree in a business setting and each role brought new lessons I wished I had considered throughout my time as an undergraduate student. In this presentation, I will discuss the three things I wish I had known while I was a mathematics major at Marshall University. Addressing these three observations will help guide mathematics students who are hoping to use their skill set in a business setting at a time when math minds are more important than ever to companies undergoing digital transformations and embracing the era of big data.

November 15
Avishek Mallick (Marshall University)
Statistical Modeling of Discrete/Count Data

Abstract: In this talk, I will introduce the idea of Statistical modeling, especially in context of count data. We will look at different facets of data fitting like estimation techniques and criterion for assessing goodness-of-fit. A substantial part of the talk will be about modeling inflated count data. We will be looking at lots of real world examples. This talk is intended for a general audience and thus should be appropriate for Mathematics undergraduate and graduate students.

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