Ken Ono, University of Virginia

Why Does Ramanujan, “The Man Who Knew Infinity,” Matter?

Abstract: Srinivasa Ramanujan, one of the most inspirational figures in the history of mathematics, was an amateur gifted mathematician from lush south India who left behind three notebooks that engineers, mathematicians, and physicists continue to mine today. Born in 1887, Ramanujan was a two-time college dropout. He could have easily been lost to the world, a thought that scientists cannot begin to absorb. He died in 1920. Prof. Ono will explain why Ramanujan matters today and will share several clips from the film, “The Man Who Knew Infinity,” starring Dev Patel and Jeremy Irons. Professor Ono served as an associate producer and mathematical consultant for the film.

Professor Ken Ono is the Thomas Jefferson Professor of Mathematics at the University of Virginia, the Asa Griggs Candler Professor of Mathematics at Emory University and Vice President of the American Mathematical Society. He is considered an expert in number theory. His contributions include several monographs and more than 180 research and popular articles in number theory, combinatorics and algebra. He earned his Ph.D. from UCLA and has received many awards for his research in number theory, including a Guggenheim Fellowship, a Packard Fellowship and a Sloan Research Fellowship. He was awarded a Presidential Career Award for Science and Engineering (PECASE) by Bill Clinton in 2000 and was named a Distinguished Teaching Scholar by the National Science Foundation in 2005. He is also a member of the US National Committee for Mathematics and the National Academy of Sciences. He was an associate producer of the film “The Man Who Knew Infinity” based on the life of Indian mathematician Srinivasa Ramanujan.

Talithia Williams, Harvey Mudd College

Data-Driven Decision Making: Now and Imagined

Abstract: Technology has a history of being a catalyst of change in training and education. We’ve seen it with desktop computers and, more recently, with the emergence of smartphones. But those shifts, substantial as they were, pale in comparison to the next big technological disruption: Data. In this fascinating talk you will discover how the advancing world of data analytics is forever changing the future of learning and work. You will explore the full landscape of data analytics, looking at both the expanding ways in which data is generated, and the advancements in analytics that make that data actionable. You will hear examples of data being used to better understand performance in both education and enterprise, and learn how those insights are being used to inform decision making and transform society.

Statistician Talithia Williams is an innovative, award-winning Harvey Mudd College professor, a co-host of the PBS NOVA series, “NOVA Wonders” and a speaker whose popular TED Talk, “Own Your Body’s Data”, extols the value of statistics in quantifying personal health information. She demystifies the mathematical process in amusing and insightful ways to excite students, parents, educators and the larger community about STEM education and its possibilities. In 2015, she won the Mathematical Association of America’s Henry L. Alder Award for Distinguished Teaching by a Beginning College or University Mathematics Faculty Member, which honors faculty members whose teaching is effective and extraordinary, and extends its influence beyond the classroom. It is this excellence that attracted the attention of online educational company The Great Courses, which selected Williams to produce “Learning Statistics: Concepts and Applications in R,” a series of lectures in which she provides tools to evaluate statistical data and determine if it’s used appropriately. She is the author of “Power in Numbers: The Rebel Women of Mathematics”, a full-color book highlighting the influence of women in the mathematical sciences in the last two millennia.

Williams is a proud graduate of Spelman College (B.A., math), Howard University (M.S., mathematics) and Rice University (M.A., Ph.D., statistics). Her research focus involves developing statistical models that emphasize the spatial and temporal structure of data and applies them to problems in the environment. She’s worked at NASA, the Jet Propulsion Laboratory and the National Security Agency and has partnered with the World Health Organization on research regarding cataract surgical rates in African countries. Faith and family round out a busy life that she shares with a supportive husband and three amazing boys. Through her research and work in the community at large, she is helping change the collective mindset regarding STEM in general and math in particular, rebranding the field of mathematics as anything but a dry, technical or male-dominated but, instead, a logical, productive career path that is crucial to the future of the country.

Lisa Holden, Northern Kentucky University

26 Years of Teaching: A Series of Fortunate Events

Abstract: In a career shaped largely by serendipity, I’ve had the good fortune to be affiliated with institutions that are particularly devoted to their undergraduates. Indeed, creating a welcoming community, engaging students in mathematical experiences, and providing research opportunities to upper level students is the norm at NKU. And it’s within this context that I’ve had the opportunity to explore ideas ranging from modeling star formation to finding optimal fingerings for a cellist. In this talk, I’ll take you on a meandering path of fortunate events and highlight some of the work I’ve done with my colleagues and our students over the past few years.

Lisa Holden joined joined Northern Kentucky University in 2002 and became associate professor in 2013. She has directed a number of undergraduates in research projects with backing that includes the MAA/SIAM PIC Math program.

She graduated magna cum laude from Boston College, then completed her M.S. and Ph.D. from Northwestern University in applied mathematics. Before coming to Northern Kentucky University she taught at institutions like Kalamazoo College and Wesleyan College.

Jeffrey Heath, Centre College

A Random Walk Through Sports Analytics

Abstract: Sports Analytics is a growing field in which people use mathematics, statistics, and computation to better understand the world of sports, often with the goal of providing a competitive advantage to a teamâ€™s performance or operations. In this talk I will discuss some of my sports analytics projects, such as modeling baseball with Markov chains, win probability-based ratings, and optimal decision-making in football. In addition, I will weave in some lessons learned through working with student researchers on these projects, and how this work has shaped my career both in and out of the classroom.

Jeffrey Heath joined Centre’s faculty in 2007, and became associate professor of mathematics in 2013. He was named a Centre Scholar in 2011, and received the Kirk Award for excellence in teaching in 2019. His research interests include sports analytics and applied statistics in pharmacology.

Heath graduated summa cum laude from Georgetown College with a B.S. in mathematics. He earned M.S. and Ph.D. degrees in applied mathematics and scientific computation from the University of Maryland, where he served as a teaching fellow in the mathematics department.

Shreeya L. Arora (h), Gatton Academy of Math and Science, Western Kentucky University

Detecting Gerrymandering with Computational Algorithms

Partisan gerrymandering is the manipulation of district boundaries in order to establish an unfair political advantage for a particular party or group. The goal of this research project is to develop code to evaluate the current districting of a state using computational mathematics. We developed a random walk model using Wolfram Mathematica that generates thousands of district plans. The new districts continue to meet criteria regarding population bounds and precinct adjacencies. These plans are then analyzed to find normal distributions. In order to validate the algorithm, a model state, “Pentucky,” made up of 6 districts and 100 precincts, was used. Two disparate initial district plans were used as inputs for the algorithm; a consistent winning-party average resulted from both district plans. This concept is being further applied with the 3,692 precincts within Kentucky. Using registration statistics, the algorithm will generate a distribution of districting plans with hopes to guarantee equal voter representation.

Tibor E. Burdette (f), Nelson County High School

Jacobsthal’s Ladder

Euler’s totient function ϕ(`n`) is defined as the number of integers less than `n` which are coprime to `n`. Carmichael’s Totient Conjecture states that for each positive integer `n` there exists a different integer `m` such that ϕ(`m`) = ϕ(`n`). While investigating a condition posed by Pomerance as sufficient to disprove the conjecture, we discovered bounds for Dirichlet’s theorem of arithmetic progressions given by the Jacobsthal function by reducing its calculation to a deceptively difficult game of modular whack-a-mole. Also, the conjecture would imply that any integer satisfying Pomerance’s condition must be divisible by the square of every prime, hence Jacobsthal’s Ladder.

Courtney T. Chaffin (u), Morehead State University

Statistics to Spare

Using data on 44 high school bowlers from the Eastern Cincinnati Conference, we examine statistical properties of 10-pin bowling scores. We look for a relationship between average scores and standard deviation by using the Wilcoxon Rank-Sum Test and the Rank Correlation Test. Additionally, we determine if bowling scores are normally distributed using Skew-Kurtosis Tests. Finally, we examine Handicap Systems and their effectiveness and impact on high school bowling.

Doug Chatham (f), Morehead State University

Social Distancing on the Chessboard

On an `n` × `n` chessboard, every two squares are at most two queen moves (or rook moves) from each other. We ask how many pawns need to be placed on the board to increase the maximum possible distance from 2 to some desired number `d`. We produce an algorithm to answer the question for given values of `n` and `d`. For some small values of `d`, we extend the results produced by that algorithm to all `n`.

José N. Contreras (f), Ball State University

Experiencing the Thrill of Formulating and Solving Problems with GeoGebra

In this presentation, I will illustrate how my students and I have used a problem-posing framework and GeoGebra to pose and solve Varignon problems using four main strategies: Specializing, generalizing, extending, and reversing. To enrich the students’ experience, I start the investigation with the following version of the Varignon’s problem:
Let `E`, `F`, `G`, and `H` be the midpoints of the consecutive sides of a parallelogram `ABCD`. What type of quadrilateral is `EFGH`?

Molly W. Dunkum (f) and Robert G. Donnelly (f), Western Kentucky University

How to do representation theory by putting numbers in boxes

Representation theory is the idea that we can learn something new about an algebraic structure, say a group, by realizing its elements as something more concrete, such as matrices or symmetries. Combinatorial representation theory is the idea that objects with interesting discrete characteristics often accompany concrete representations of algebraic structures. Our goal is to exhibit a really nice example of these ideas that we discovered recently in our work with representations of symmetric groups and certain matrix algebras — this is where the numbers-in-boxes stuff will happen.

Claus Ernst (f), Western Kentucky University

Energies of a knot

Knots can be considered as thick-knotted tubes or a stiff thin curves. These considerations influence the shape the knots will have. We discuss the mathematics behind these ideas.

Allison Fitisone (g), University of Kentucky

The Normalized Measure of Simplicial Solid Angles

We view the solid angle as a cone spanned by extreme rays. The measure of the solid angle is the proportion of space that the cone occupies. Of particular interest is the normalized measure of a simplicial solid angle. It is well known that in dimension two, the angle between two vectors can be computed using the dot product. In dimension three, the normalized measure of a simplicial solid angle can be obtained using the scalar triple product, as noted by Oosterom and Strackee (1983). In 2006, Jason Ribando discovered that the normalized measure of a simplicial solid angle in higher dimensions can be obtained via a multivariable hypergeometric series, when the series converges. In this talk, I will showcase Python code that I developed in collaboration with Dr. Yuan Zhou, which uses a truncated form of Ribando’s formula to approximate the normalized measure of solid angles.

Nicholas M. Gaubatz (u), Murray State University

An Examination of Dn Networked-Numbers Games

We will explore aspects of the networked-numbers game—a game played on a graph whose moves consist of “firing” weighted nodes. The networked-numbers game is relatively simple to understand, but its structure is rich with results in combinatorics, root systems, and Lie Algebra. First, we will look at the construction of this game. Next, we will examine the game played on a general Dn-type graph. Lastly, we will give an overview of the significance of this game in algebraic combinatorics.

Ryan Gipson (f), Campbellsville University

A Characterization of Puiseux Algebras with Bounded Factorization

We will investigate the bounded factorization property of the monoid domain `F`[`X`;`M`], where `M` is an additive submonoid of the nonnegative rational numbers. We accomplish this task by examining sufficient and necessary factorization properties of the associated monoid `M` and, in particular, its irreducibles. We divide our work into two sections: we first present a useful characterization of all bounded factorization monoid domains by precisely determining the necessary factorization properties of its associated Puiseux monoid; then, we consider those bounded factorization domains with the finite factorization property.

Harrison T. Gover (u), Western Kentucky University

How Much Curvature is in a Knot?

We describe several ways of how a mathematical knot can curve when drawn on paper, including braid diagrams and spiral diagrams. Using these, we can define knots as “curly,” or as having a braid index greater than its spiral index. From there, groups and families of curly knots can be described and proven to be curly.

Peyton Hennessy (u) and Hanna Schmitt (u), Northern Kentucky University

Surface Art in Mathematica

Inspired by recent year’s Joint Mathematics Meetings’ Mathematical Art Galleries, our research seeks to produce esthetically pleasing surfaces using Mathematica as the medium. In our work, we used minimal surfaces, surfaces of least area, and tube surfaces, surfaces built around curves. We used code to modify these sets of equations, which were then combined with built-in functions generating a range of results. Our results included 3D prints, movies, and graphics in Mathematica. The 3D prints helped us visually display the position, in space, of the surfaces we created on a real-life scale. We used digitally generated movies to show how a minimal surface can morph into another surface by altering a parameter value. Finally, we altered parametric equations to produce an array of surfaces and tubes to show how minor changes in these equations affect the appearance of each graphic.

Ryan N. Hinson (u), Morehead State University

An Exploration in the Tools of Options Pricing

Stock options can be a useful tool in any investor’s portfolio. They allow the skilled investor to increase their leverage and possibility for a higher payout with less risk. However, they are only useful if the investor knows how to use them, and if they accurately reflect the price of the option. This presentation hopes to offer some insight in the binomial method of options pricing along with new adjustments to the model in hopes of reflecting a more accurate options price. This, along with the Monte Carlo simulation presented, will hopefully offer some insight into the mathematical investor’s toolbox.

Wilson D. Horner (g), Western Kentucky University

Analysis of Boundary Observability of Strongly Coupled One-Dimensional Wave Equations with Mixed Boundary Conditions

In control theory, the time it takes to receive a signal after it is sent is referred to as the observation time. For certain types of materials, the observation time to receive a wave signal differs depending on a variety of factors, such as material density, flexibility, etc. Suppose we have a piezoelectric beam and want to find the observation time of a given signal. To represent this mathematically, we can discretize the spatial variable in a system of coupled wave equations via finite differences. We find that for this particular discretization, the observability time approaches infinity which does not mimic the continuous case, and is unsuitable in terms of controlling the system. To bypass this issue, a filtering technique known as direct filtering is applied, and a (non-optimal) observation time is obtained.

Jacob D. Kretzer (u), Morehead State University

A Statistical Analysis of COVID-19 Deaths Using Multi-Regression

COVID-19 is the virus that is responsible for the pandemic that currently effects the entire globe, causing mass shutdowns and economic disruption. In the United States, over 515,000 people have died due to COVID-19. This analysis began with first analyzing the state with the highest death toll and working to build several models using multi-regression to predict future deaths in this state. After these models were complete, work began on building models for the entire United State in order to predict deaths for the nation as a whole rather than just a singular state using what was learned previously.

Alexis Krumpelman (a) and Joshua Qualls (f), Morehead State University

Hackers Byte: Developing a Course on Cryptography and Python

Cryptography plays an essential role in daily communication through technology; programming is a valuable skill because of its numerous applications in STEM. Our goal is to develop a course to forge student proficiency in both cryptography and the basics of the Python programming language. In this presentation, we will elaborate on the objectives of the course and how they are to be achieved. Student learner outcomes include proficiency in Python programming, expounding on security properties in cryptographic theories, analyzing and applying cryptography in Python, and considering the corresponding limitations and applicability. The presentation will conclude with an update on the current implementation of the curriculum.

Elizabeth G. LaBreche (u), Morehead State University

Developing a Simplistic Model for COVID-19

This presentation examines the dynamics of COVID-19 and the factors which affect outcomes within a population. A Susceptible-Exposed-Infected-Recovered compartmental model was developed to represent the various disease-states of a population. Euler’s Method was used to find approximate solutions of the first-order differential equations governing the compartmental model. The model was applied to a population approximately the size of Louisville, KY, in order to simulate a possible outcome of events for the largest metropolitan area in the state. Three scenarios were addressed using the developed model: one resembling the historical course of the outbreak, one in which a mask mandate was applied early in the course of the outbreak but individuals still chose to gather during the holiday season, and one in which a mask mandate was applied early in the course of the outbreak but individual refrained from gathering. Outcomes suggest that significant morbidity and mortality is expected except when masking is applied early on in the outbreak and individuals refrained from gathering during holidays.

Peter L. Lefkovitz (u), Northern Kentucky University

Melodies As Curves

The curvature function of a 2D curve describes the shape of the curve regardless of how it is moved or rotated. In music, a motif is string of notes that appears throughout a piece, although sometimes moved up or down. The measure of curvature is a number, and we can assign each note in a motif a numerical value that describes its distance from some fundamental frequency. Because each curve has a unique curvature function, we are able to create a unique song based on a curve, and vice versa. However, this does not easily allow for rhythm. A 3D curve has curvature and torsion, which measure how much it “bends” and how much it “twists”, respectively. We can make curvature rhythm, and torsion pitch, and get a unique song based on any space curve, and vice versa.

Allen Lin (h), Gatton Academy at Western Kentucky University

Values of Dirichlet Series

We generalize the results of the alternating Dirichlet series first examined by Junesung Choi. Specifically, we examine its analytical continuation and its special values. We also connect values of the Choi Dirichlet series to values of another Dirichlet series. Finally, we connect this last class of Dirichlet series to Dirichlet L-functions, ultimately obtaining values of the alternating Dirichlet series.

Andy Martin (f), Kentucky State University (retired)

All Math Teachers should visit 3Blue1Brown

In this talk I will announce my favorite YouTube channels which focus on math topics. The title betrays my #1 .

Emma Moore (u), Western Kentucky University

Application of Direct Fourier Filtering for the Exact Observability of a Wave Equation

In control theory, a system is exactly observable if, when you measure a certain system property for a finite time, you are able to distinguish any two different initial states on a string. With partial differential equations (PDEs), like the one-dimensional wave equation, there are infinitely many eigenvalues, which makes the PDE infinite-dimensional, so exact observability of vibrations on a string is an infinite-dimensional problem. In practice, however, system properties are observed by the sensors, and sensors can only observe finitely many vibrational modes. Therefore, sensor design for the observed property has to be done for a numerical approximation of the PDE, which makes it a finite dimensional system. However, exact observability does not hold for well-known numerical approximation techniques like Finite Difference Method (FDM) and Finite Element Method (FEM). The major issue we encounter in trying to show the exact observability of the approximated system is that the uniform gap property of eigenvalues is lost, which is not the case for the infinite-dimensional system. This uniform gap ensures exact vibrational observations by the sensors. Without it, sensors cannot distinguish one vibrational mode from another. This happens because the deficiencies of the numerical techniques cause artificial, high-frequency eigenvalues, which results in the uniform gap approaching zero as the mesh parameter goes to zero. This talk will focus on the lack of observability for the FEM space-approximation of the wave equation and a remedy, the so-called “direct filtering technique,” which enables us to prove exact observability. The proofs are based on the so-called “discrete multipliers technique.”

Tom Richmond (f), Western Kentucky University

The Lower-Rational-Limit Upper-Irrational-Limit Topology

In the Euclidean topology on ℝ, a basic neighborhood of `x` has form (`x`−`a`, `x`+`a`). In the lower-limit topology, a basic neighborhood of `x` has form [`x`, `x`+`a`), so the “neighboring points” are close to `x` without going under `x`. The upper-limit topology, defined dually, has basic neighborhoods of form (`x`−`a`, `x`], so the neighboring points to `x` are close to `x` without going over `x`. We consider the topology on ℝ having lower-limit type neighborhoods at rational points and upper-limit type neighborhoods at irrational points.

Jay L. Schiffman (f), Rowan University

Explorations With The Tribonacci Sequence 1, 1, 1, 3, 5, 9, ....

This presentation explores divisibility, periodicity and prime outputs in the Tribonacci sequence in addition to palatable number tricks associated with the sequence. In contrast to the companion Fibonacci sequence which embodies a rich history of productive research, the OEIS (The On Line Encyclopedia of Integer Sequences) indicates the study of the Tribonacci sequence is ripe for discovery. For example, only fourteen prime entries were known as of 2000. Recently I secured the fifteenth and sixteenth prime outputs using Mathematica 12.0 which I will share and was able to completely factor the initial three hundred twenty-five terms. While every integer `n` enters the Fibonacci sequence no later than the `n`² term, no power of three beyond nine serves as a factor of any term in our sequence. The Pisano periods of the Tribonacci sequence are messy at best in contrast to the Fibonacci sequence, but illustrate the power of modular arithmetic. This presentation is accessible for undergraduate students who enjoy topics in number theory, discrete mathematics and computer science as well as recreational mathematics enthusiasts.