## Advanced Research Initiative (ARI)

The Advanced Research Initiative (ARI) of the Department of Mathematics and College of Science invites distinguished researchers to Marshall University to speak with both general and specialized audiences and interact with graduate and undergraduate students. For upcoming ARI presentations please see the Department of Mathematics Events Calendar.

### Spring 2017 – ARI Mathematics Speaker

**Dr. Damir Dzhafarov**

Assistant Professor of Mathematics

University of Connecticut

**“Problems, Problems, Problems”**

Monday, April 10, 2016 • Corbly Hall 105 • 4:00pm • PDF Flyer

Here is a problem just about everyone has seen in a math class at some point: Given a function, find its range. For example, if we are given the function f (x) = x^{ 2}, defined for x = 0, 1, 2, 3, 4, …, we know that its range is 0^{2} = 0, 1^{2} = 1, 2^{2} = 4, 3^{2} = 9, 4^{2} = 16, etc. Easy enough, it seems. Here is another problem just about everyone has faced: Your computer is showing you the dreaded spinning beach ball, and you forgot to save the document you have been working on—do you try to wait it out, or reboot now and start over? That seems like a much harder task… But surprisingly, these two problems—the range problem, and the spinning beach ball problem—are actually one and the same, in a certain precise sense. I will explain why this is the case, and mention other related problems, some more familiar than others. Some of the key insights here go back to Turing, and the advent of modern computing, which I will discuss along the way.

**“Mathematics, Backwards and Forwards”**

Tuesday, April 11, 2016 • Smith Hall 530 • 4:00pm • PDF Flyer

Abstract: Mathematics today benefits from having “firm foundations”, by which we usually mean a system of axioms sufficient to prove the theorems we care about. But given a particular theorem, can we specify precisely which axioms are needed to derive it? This is a natural question, and also an ancient one: over 2000 years ago, the Greek mathematicians were asking it about Euclid’s geometry. Reverse mathematics is a program in mathematical logic that provides a modern approach to this kind of question. A striking fact repeatedly demonstrated in this research is that the vast majority of mathematical propositions can be classified into just five main types, according to which axioms are needed to carry out their proofs. I will discuss these five categories, and what they reveal about relationships between different areas of mathematics. I will also mention a few notable examples of theorems that do not fit inside these five categories.

### Fall 2016 – ARI Mathematics Speaker

**Dr. Richard Ehrenborg**

Professor of Mathematics

University of Kentucky

**“Mathematical Magic”**

Monday, October 10, 2016 • Corbly Hall 104 • 4:00pm • PDF Flyer

We will demonstrate a few “magic” tricks, all based upon mathematics. We will reveal the tricks including the mathematics behind them. Among other topics this will include parity, recursions and Hall’s Marriage Theorem.

**“Counting Permutations with Calculus”**

Tuesday, October 11, 2016 • Smith Hall 516 • 4:00pm • PDF Flyer

A permutation π = π_{1} π_{2} … π_{n} is a list of the elements 1 through n in some order. A permutation is alternating if the elements zig-zag, that is, π_{1}< π_{2} > π_{3} < … . We prove a classic formula for the number of alternating permutations using the interplay of geometry and calculus. Using the same techniques we also study the number permutations with no double ascents, that is, there is no index i such that π_{i} < π_{i+1} < π_{i+2}. Unfortunately, this last case becomes messy and leads to open questions where else these techniques can be applied.

### Spring 2016 – ARI Mathematics Speaker

**Dr. Shubhabrata (Joey) Mukherjee**

Research Assistant Professor of Mathematics

Division of General Internal Medicine,

Department of Medicine, University of Washington

**“Introduction to Genetic Epidemiology in GWAS era”**

Monday, March 7, 2016 • Science Building 376 • 4:00pm • PDF Flyer

Genetic epidemiology holds great potential for personalized medicine and improved biological knowledge of disease pathogenesis. In this talk, I will introduce some basic concepts needed to understand, analyze, and interpret Genome-wide Association Studies (GWAS) data with late-onset Alzheimer’s disease (LOAD) as an example. I will briefly talk about extending these results to gene-wide and network/pathway-based analysis, which are complementary to GWASs.

**“Genetic analyses of late-onset Alzheimer’s Disease”**

Tuesday, March 8, 2016 • Smith Hall 529 • 4:00pm • PDF Flyer

Alzheimer’s disease is the most common form of dementia. Genetics of late-onset Alzheimer’s disease (LOAD) is very complex. Genome-wide Association Studies (GWAS) are an important first step. Gene-based and network/pathways based analyses also contribute to the understanding of genetic determinants of LOAD. Thus, a three-component approach — GWAS, gene-based, and network-based analyses — is likely to best illuminate genetic determinants of a disease. I will explore the three component approach in this talk.

### Fall 2015 – ARI Mathematics Speaker

**Dr. Nick Loehr**

Associate Professor of Mathematics

Virginia Tech

**“Rook Theory 101”**

Monday, September 28, 2015 • Location TBA • 4:00pm • PDF Flyer

Combinatorics is the mathematical theory of counting. In this talk, we introduce combinatorics through the subject of rook theory — which has nothing to do with chess! The first goal of rook theory is to count configurations of rooks on generalized chessboards so that no two rooks attack each other. The k’th rook number of a board counts the number of such ways to place k rooks on the board. It often happens that two different boards share the same rook numbers. This talk investigates when and why this occurs by using geometric and algebraic manipulations of boards and rook placements. We present theorems of Foata/Schutzenberger and Goldman/Joichi/White characterizing rook-equivalent boards in various ways. The techniques used illustrate three branches of modern combinatorics — enumerative combinatorics, bijective combinatorics, and algebraic combinatorics. No prior knowledge of counting (or chess) is required for this talk.

**“Sweep Maps and Bounce paths”**

Tuesday, September 29, 2015 • Location TBA • 4:00pm • PDF Flyer

Mathematics is filled with open (unsolved) problems, ranging from deep foundational issues of physics, computation, geometry, and number theory to highly specialized research questions. This talk describes an open problem in algebraic combinatorics that can be stated and investigated with virtually no mathematical background, although the problem appears to be fiendishly challenging to solve. We define a family of maps on words, called “sweep maps.” A sweep map assigns a level to each letter in a word according to a simple rule, then sorts the letters according to their level. Surprisingly, although sweep maps act by sorting, they appear to be invertible: i.e., different input words are always sent to different output words by any given sweep map. The open problem is to prove the invertibility of all sweep maps, preferably by explicitly describing the inverse functions. We explain some known special cases of this problem using a model in which words are visualized using lattice paths. In some cases, we can pass from a lattice path to an associated “bounce path,” which provides the additional data needed to invert the sweep map. These bounce paths originally arose in the study of objects called q,t-Catalan polynomials. Many algorithms that have appeared in the q,t-Catalan literature over the last 20 years turn out to be particular instances of the sweep maps or their inverses. The sweep maps thus provide a simple unifying framework for understanding all of these algorithms.

### Spring 2015 – ARI Mathematics Speaker

**Dr. Judy Day**

Assistant Professor of Mathematics

University of Tennessee

**“Modeling the host response to inhalation anthrax to uncover the mechanisms driving risk of disease”**

Tuesday, April 14, 2015 • Science Building 276 • 4:00pm • PDF Flyer

*Bacillus anthracis*, the causative agent of anthrax, can exist in the form of highly robust spores, making it a potential bio-terror threat. Once inhaled, the spores can germinate into vegetative bacteria capable of quick replication, leading to progressive disease and death. There is a critical need to better quantify the risk of disease from different inhalation exposure scenarios. Key to this effort is the use of mathematical and computational modeling to uncover the mechanisms driving risk. To this end, this presentation will discuss ongoing work on the development of models and methods that explore the host response to inhalation anthrax and provide insight into the mechanisms that drive the risk of disease. This is joint work is joint University of Tennessee graduate student Buddhi Pantha, and the NIMBioS Working Group on Modeling Anthrax Exposure.

**“Determining the what, when, and how of therapeutic intervention strategies for controlling complex immune responses”**

Wednesday, April 15, 2015 • 376 Science Building • 4:00pm • PDF Flyer

Ideally, when challenged with a bacterial insult, a host orchestrates an immune response that, not only eliminates the offending pathogen, but also restores the host to homeostasis. However, due to the complex nature of the response, this is not always possible, especially in critically ill patients. Clearly, intervention is needed; however, determining the types of intervention that should be given, when they should be given, and in what amount remains a challenge. Computational modelling and control methodologies can provide fresh insight into this challenging biomedical problem and potentially offer techniques to answer these difficult questions. This talk will discuss computational control methodologies that are being explored to determine the what, when, and how of therapeutic intervention strategies for controlling complex immune responses.

### Fall 2014 – ARI Mathematics Speaker

**Dr. Richard Brualdi**

UWF Beckwith Bascom Professor of Mathematics (Emeritus)

University of Wisconsin – Madison

**“The Gale-Berlekamp Light-Switching Problem and a Permutation Variation”**

Wednesday, November 5, 2014 • Smith Hall 154 • 4:00pm • PDF Flyer

Consider an *n* by *n* array of light bulbs each controlled by a switch. Suppose there are also 2*n* other switches which allow one to simultaneously switch all the light bulbs in a row or all the light bulbs in a column. Now use the individual switches and turn some of the light bulbs on. With the row and column switches only, can one get all the lights in the off position? If not, how few on-lights are possible? This problem, its connections to coding theory, and a permutation variation is the subject of this talk.

**“All Things Bruhat: Matrix Bruhat Decomposition,Complete Flags, Bruhat Order of Permutations, (0,1) and Integral Matrices, and Tournaments”**

Thursday, November 6, 2014 • Science Building 375 • 4:00pm • PDF Flyer

The title is a bit of an exaggeration, but we will discuss the topics it contains and various connections between them.

### Spring 2014 – ARI Mathematics Speaker

**Dr. Jeffry L. Hirst**

Professor of Mathematics

Appalachian State University

**“Alan and Ada’s Theoretical Machines”**

Monday, March 10, 2014 • Shawkey Room, MSC • 4:00pm • PDF Flyer

Ada Lovelace is often called “the first computer programmer,” though she died a century before the first general purpose computers were built. This claim is based on the notes she appended to her translation of Menabrea’s 1842 paper, *Notions sur la Machine Analytique de M. Charles Babbage*. Alan Turing is often called “the father of theoretical computer science” on the basis of his 1936 article *On computable numbers, with an application to the Entscheidungsproblem*. Lovelace’s notes describe Babbage’s Difference Engine and Babbage’s Analytical Engine, and Turing’s paper proves the existence of a Universal Turing Machine. This talk will compare the designs and capabilities of these early theoretical computing machines.

**“Reverse Mathematics, Graphs, and Matchings”**

Tuesday, March 11, 2014 • Science 465 • 4:00pm • PDF Flyer

How can we tell if two theorems are essentially the same? If we can prove that they are equivalent, then they are in some sense interchangeable. If our equivalence proof relies on a particularly small set of assumptions, then our claim of similarity is even stronger. This is the fundamental motivation of reverse mathematics, a program in the foundations of mathematics initiated by Harvey Friedman and Stephen Simpson. This talk will illustrate some results and techniques of the program.

### Fall 2013 – ARI Mathematics Speaker

**Dr. Thomas Mathew**

Presidential Research Professor

University of Maryland, Baltimore County

**“The Assessment of Bioequivalence: A Statistical Overview”**

Monday, November 18, 2013 • MSC BE5 • 4:00pm • PDF Flyer

The topic of bioequivalence deals with procedures for testing the equivalence of two drug products: typically, a generic drug and a brand name drug on the market. Bioequivalence testing consists of showing that the concentration of the active drug ingredient that enters the blood is similar for the two drugs. Area under the time-concentration curve, or the AUC, is usually used for this purpose, and the data are obtained based on cross-over designs. In the talk, the bioequivalence problem will be introduced, its history will be discussed, and examples will be provided. Statistical criteria that are used for bioequivalence testing, especially the criterion of average bioequivalence, will be discussed. Methodology for testing the hypotheses of average bioequivalence will be addressed. The emerging area of equivalence testing in the context of biosimilars will be briefly touched upon.

**“Methodology and Some Applications”**

Tuesday, November 19, 2013 • CH 105 • 4:00pm • PDF Flyer

Standard likelihood based methods that are usually used to analyze data arising from a parametric model are typically accurate to the first order. Higher order inference procedures provide major improvements in accuracy, and are available for discrete as well as for continuous data. In the talk, two applications of higher order inference will be described. Both the applications deal with the computation of an upper tolerance limit: a limit that is expected to capture a specified proportion or more of a population with a given confidence level. The limit is constructed using a random sample, and the confidence level refers to the resulting sampling variability.

The first application that will be discussed is on the computation of tolerance limits under the logistic regression model for binary data. The data consist of binary responses, and upper tolerance limits are to be constructed for the number of positive responses in future trials corresponding to a fixed level of the covariates. The problem has been motivated by an application of interest to the U.S. Army, dealing with the testing of ballistic armor plates for protecting soldiers from projectiles and shrapnel, where the probability of penetration of the armor plate depends on covariates such as the projectile velocity, size of the armor plate, etc. The second application is on the computation of upper tolerance limits under a general mixed effects model with balanced or unbalanced data. Higher order inference procedures will be used to obtain accurate solutions in both the applications. Numerical results, examples and data analysis will also be reported.

### Spring 2013 – ARI Mathematics Speaker

**Dr. Howie Wiess**

Professor of Mathematics

Georgia Institute of Technology

**“Sir Ronald Ross, the SIR transmission model and the Foundations of Public Health”**

Wednesday, March 6, 2013 • Drinko 402 • 4:00pm

After some brief comments about the nature of mathematical modeling in biology and medicine, we will formulate and analyze Sir Ronald Ross’s SIR infectious disease transmission model. The model is a system of three non-linear differential equations that does not admit a formula solution. However, we can apply methods of calculus to understand a great deal about the nature of solutions. Along the way we will use this model to develop a theoretical foundation for public health, and we will observe how the model yields several fundamental insights (e.g., threshold for infection, herd immunity, etc.) that could not be obtained any other way. At the end of the talk we will compare the model predictions with data from actual outbreaks.

**“Perspectives on multiple waves during flu pandemics”**

Thursday, March 7, 2013 • SH 511 • 4:00pm

A striking characteristic of the past four influenza pandemic outbreaks in the United States has been the multiple waves (peaks) of infections. However, the mechanisms responsible for the multiple waves are uncertain. We use mathematical models to exhibit mechanisms each of which can generate multiple waves.

The first two mechanisms capture changes in virus transmissibility and behavioral changes. The third mechanism involves population heterogeneity (e.g., demography, geography, etc.); each wave spreads through one sub-population. The fourth mechanism is virus mutation which causes delayed susceptibility of individuals. The fifth mechanism is waning immunity. Four of the models reproduce the two waves of the 2009 H1N1 pandemic in the United States, both qualitatively and quantitatively. One model reproduces the two waves only qualitatively.

We use the models to study the effects of border control and vaccination strategies on the outbreak, and we propose a hypothesis about why China only experienced a single wave of infections during the 2009 flu pandemic.