PME Annual Conference

About

The 2021 Pi Mu Epsilon (PME) National Meeting will be held at the Mathematical Association of America (MAA) virtual MathFest conference on August 4-7, 2021.

Schedule – PME MathFest 2021

Note: All times are in Mountain Daylight Time. (MDT=UTC-6:00)

Wednesday, August 4, 2021

11:00am – 12:00pm	PME Student Session 1
1:00pm – 3:00pm	PME Student Session 2
3:00pm – 3:50pm	J. Sutherland Frame Lecture / About / Previous Lectures
	Speaker: Florian Luca, Arithmetic and Digits Abstract:In our recent paper in the Monthly (October, 2019) with Pante Stănică, we looked at perfect squares which arise when concatenating two consecutive positive integers like $183184 =428^2$ with the smaller number to the left, or $98029801 = 9901^2$ with the larger number to the left. My talk will present variations on this topic with the aim of providing the audience with examples of numbers which are both arithmetically interesting (like perfect squares) while their digital representations obey some regular patterns. The examples will not be limited to perfect squares, but will also include other old friends like Fibonacci numbers and palindromes.

Thursday, August 5, 2021

10:00am – 12:00pm	PME Business Meeting
11:00am – 11:50am	MAA Chan Stanek Lecture for Students
1:00pm – 4:00pm	PME Student Session 3

Friday, August 6, 2021

10:00am – 12:00pm	PME Student Session 4
1:00pm – 2:00pm	PME Student Session 5
3:45pm – 4:00pm	PME Award Session Zoom Link

PME Student Session 1

11:00am – 12:00pm, Wednesday, August 4, 2021


11:00am-11:20am	Ella Wilson: H-Functions and Circle Packing Kenyon College, OH Pi Abstract: Harmonic Measure Distribution Functions ($h$-functions) encode information about the geometry of domains in the plane. Specifically, given a domain and a basepoint within it, for a fixed radius, $r$, $h(r)$ is the probability that a Brownian particle first exits the domain within distance $r$ of the basepoint. There are many open questions about $h$-functions; a central one is whether we can reconstruct a domain from its $h$-function. A challenge is that exact computation is often difficult or impossible, and we need methods to approximate. In this talk, I discuss my explorations into finding a discrete analogue to $h$-functions using the technique of circle packing and connections to open questions about the uniqueness of $h$-functions. Mentored by Dr. Marie Snipes.

11:20am-11:40am	Alyssa Leone: Rittenhouse’s Sums of Sines Youngstown State University, OH Xi Abstract: Next to Benjamin Franklin, Dr. David Rittenhouse was one of the leading American scientists in the eighteenth century. He specialized in multiple areas including astronomy, mathematics, instrumentation, and clock-building. In Spring 1792, Rittenhouse made public a mathematical paper titled “A method of finding the sum of the several powers of the sines”. The purpose of this presentation is to analyze the details of this paper and his ambiguous proofs. This will be accomplished by studying integrals of the form $sin^n(x) $ bounded in the first quadrant and looking for a geometric interpretation. Mentored by Dr. Thomas Wakefield.

11:40am-12:00pm	Ali Hamza Abidi Syed: Optimization Theory in ML University of Houston, TX Theta Abstract: Our goal is the design and analysis of effective numerical schemes for training deep neuronal networks based on optimal control formulations. In particular, we study the methodology presented in [Haber and Ruthotto, Inverse Problems 34 014004, 2018]. In this framework, the forward propagation is interpreted as a dynamical system of an equality-constrained optimization problem. In the first step, we revisited different schemes for forward propagation. In particular, we consider (i) different variants of an explicit Euler discretization of the forward propagation framed as a non-linear ordinary differential equation and (ii) a symplectic Verlet method for solving the forward propagation problem framed as a differential equation inspired by Hamiltonian dynamics. We have replicated the results for the forward propagation presented in the original paper. Deriving the associated optimality conditions and the implementation of an effective scheme for numerical optimization (that potentially will consider second-order derivative information) forms the basis of our current work. Mentored by Dr. Andreas Mang.

PME Student Session 2

1:00pm – 3:00pm, Wednesday, August 4, 2021


1:00pm-1:20pm	Jackson Krebsbach: Mapping Vegetation Density Hope College, MI Delta Abstract: Active coastal sand dune complexes are dynamic environments characterized by interactions between plant populations, topography, and physical processes. The above-ground portion of dune plants reduce sand transport by reducing wind speeds at the surface, and the roots anchor the sediment. Dune activity also affects the plants, and some dune species have adaptations that allow them to thrive in depositional environments. Our goal is to create maps of dune vegetation and topography so that we can better understand these interactions. With two different scales of photography, both at ground level and from the air using a drone, our group developed a method to map vegetation density at high resolution across an entire dune complex. First, as calibration, a machine learning algorithm was used to classify pixels in ground-based photographs into different classes. Next, the Normalized Difference Vegetation Index (NDVI) was calculated in the drone imagery over the same areas. Finally, a regression model was developed using NDVI values to predict vegetation density. This model was applied to an NDVI orthomosaic map, resulting in a vegetation density map of the entire dune complex. Mentored by Dr. Brian Yurk.

1:20pm-1:40pm	Blake Harlow: Modeling Dune Topography Hope College, MI Delta Abstract: Lake Michigan dune complexes evolve as winds and waves erode the sand, causing major topographic changes over time. These dynamics are not fully understood or modeled. Drone photography is an efficient method for collecting precise multispectral imagery for these areas. The Hope College Dune Group has been using this remote sensing data to model various aspects of the dune, although much remains to be understood about sand transport mechanisms and how they lead to dune topography changes over time. We have used various ground point classification and interpolation algorithms in Python and R to create digital terrain models (DTMs) which map the bare ground surface of the dune. One of our overall aims is to use machine learning along with DTMs created at different points in time to model changes in surface topography. In this talk, we report on our method of constructing DTMs for this unique terrain and also discuss our progress in understanding dune surface dynamics. Modeling changes of the dune surface in this way will provide useful information for protecting and maintaining healthy dune ecosystems. Mentored by Dr. Darin Stephenson.

1:40pm-2:00pm	Katie Yan: Modeling the Plague in Eyam Skidmore College, NY Alpha Theta Abstract: In the 17th century, the bubonic plague spread across much of Europe. After reaching London, the Great Plague killed 25% of the population. The plague arrived in Eyam, England by the way of a bale of cloth from London. During the epidemic, 76% of all Eyam residents died from the plague, leaving only 83 surviving villagers. In my talk, I explore what makes the story of Eyam such a compelling case study for mathematical disease models. The first model I considered was a simple, compartmental S-I-R model. I then considered more complex systems of differential equations and their numerical solutions. Additionally, I worked with an agent-based modeling software, NetLogo, to help explore my research question, “was the quarantine effective?” Mentored by Dr. Rachel Roe-Dale.

2:00pm-2:20pm	Bandita Karki: Virus Suppression with DIPs University of Idaho, ID Alpha Abstract: Defective Interfering Particles (DIPs) are viruses that arise from deletion mutations during viral replication within a host cell. All viruses produce some defective variants. DIPs cannot replicate themselves independently, but require co-infection with non-defective versions of the virus. Such co-infected cells produce almost exclusively DIP progeny, leading to some suppression of the original virus through predator-prey-like dynamics. We use simple differential equation models to study the dynamics of DIPs and wild-type viruses in the presence of adaptive and innate immunity. Our goal is to understand whether therapeutic administration of DIPs will augment or interfere with the immune response and, in the former case, we seek to provide guidance on how virus suppression is affected by infection and clearance parameters, as well as by the timing of DIP introduction. We find that DIPs can create a significant impact on the total and maximum viral abundance and that timing matters. While the application of DIPs to overcome viruses is still in preclinical stages, mathematical predictions can guide potential therapeutic uses. Mentored by Dr. Stephen M Krone.

2:20pm-2:40pm	Hannah Scanlon: Disease on an Adaptive Network Wake Forest University, NC Lambda Abstract: The spread of infectious diseases depends on both properties of the disease and actions of the infected population. This study investigates a unique adaptive network model which replicates human tendency to avoid disease while also maintaining preexisting, interpersonal relationships. Specifically, our adaptation applies a probability that individuals will pause connections to people while they are infected and another probability that they will reconnect those same connections once neither party is infected. This adaptive network model is used on an SIR disease type. Based on this model design, we used a moment closure approximation to create a closed system of 14 ordinary differential equations describing the node and edge densities of our model for further system analysis. While most infectious disease models focus on basic infection and recovery rates to assess the severity of an outbreak, our model also considered network structure and human response. By analyzing this model, we were able to verify the influence of average node degree and pausing probability on the severity of an infectious event. Mentored by Dr. John Gemmer.

2:40pm-3:00pm	Phuong Pham: Covid-19 Analysis University of Massachusetts Lowell, MA Delta Abstract: The outbreak of COVID 19 has created a significant impact in the world since the end of 2019, and the number of new cases continues increasing. Therefore, analyzing and predicting COVID 19 data to understand more about the virus is essential. Our research uses a raw dataset from open access databases from Github, and it aims to find the relationship between the fourteen features and number of total cases and total deaths using linear and logistic regression analysis. The results indicated that even though total cases and deaths per million is related to stringency index, median age , gdp per capita, cardiovasc death rate, female smokers, handwashing facilities, hospital beds per thousand, life expectancy , human development index, only stringency index (how strict the country’s policy is) factor has significant relationship with the total cases per million, and only hospital beds per thousand has significant relationship with total deaths per million when fitting the multiple linear regression. For logistic regression, all the variables appeared significant to the responses: total cases per population, total deaths per population, and total deaths per total cases respectively. Mentored by Dr. JongSoo Lee.

PME Student Session 3

1:00pm – 4:00pm, Thursday, August 5, 2021


1:00pm-1:20pm	Hanna Noelle Griesbach: Isomorphic Polynomials Elon University, NC Nu Abstract: Let $\mathbb{Q}[x]$ be the polynomial ring over the rational numbers, and let $f,g\in \mathbb{Q}[x]$ be two irreducible polynomials of the same degree. Call $f$ and $g$ isomorphic if there exist $a,b\in\mathbb{Q}[x]$ such that $f(a(x)) = b(x)\cdot g(x)$. Equivalently, $f$ and $g$ are isomorphic if the respective field extensions they define are isomorphic. In this talk, we answer the following: Given an arbitrary irreducible polynomial $f(x)\in\mathbb{Q}[x]$, when does there exist an irreducible polynomial $g(x^2)\in\mathbb{Q}[x]$ such that $f(x)$ and $g(x^2)$ are isomorphic, and how do we compute such a polynomial $g$ when it is known to exist? We also show that our approach can be generalized, under certain assumptions, to the case where $g(x^2)$ is replaced by $g(x^k)$ for positive integers $k$. Mentored by Dr. Chad Awtrey.

1:20pm-1:40pm	Micah Phillips-Gary: Five Lemma for Category Theory The College of Wooster, OH Phi Abstract: (This presentation is based on interdisciplinary research in mathematics and philosophy.) We will give the Five Lemma as a theorem for groups and give definitions for the analogue category theory concepts necessary to state the Five Lemma as it applies to any abelian category. In particular, we will define “isomorphism,” “kernel,” “image” and “exact sequence” as applicable to any abelian category. We will then briefly discuss the philosophical implications of this abstraction. Namely, it shows the difference between generalization and abstraction as conceptual procedures and suggests a relativization of the notion of material conceptual content so as to make multiple levels of abstraction possible. Mentored by Dr. Rob Kelvey.

1:40pm-2:00pm	Ayush Kumar: ME of Inverse Hulls University of Texas-Tyler, TX Phi Abstract: Given a one-sided Markov shift X on finite alphabets, we study Morita Equivalence of the inverse semigroup associated with the shift. In particular we relate Morita Equivalence of inverse semigroup to flow equivalence of shift spaces. Mentored by Dr. David Milan.

2:00pm-2:20pm	Rebecca Odom: Self-Conjugate Partitions University of Texas at Tyler, TX Phi Abstract: A $\textit{partition}$ of a positive integer $n$ is defined as a non-increasing sequence $P=\left[y_0,y_1,\dots,y_m\right]$ of positive integers which sum to $n$, where the $y_i$ are called the $\textit{parts}$ of the partition. A Young diagram is a visual representation of a partition, where each row of boxes corresponds to a part. A conjugate partition is similar to a transpose of a matrix; we switch the rows with columns, or the index of a part with the part itself. Self-conjugate partitions are partitions that are equal to their conjugate; previously, the only known way to verify self-conjugate partitions was through the use of a Young diagram. In this research, by proving preliminary lemmas and theorems about easily identifiable shapes which are symmetric, we come to the main result: by simply adding the multiplicities of parts appropriately, we can show whether or not a partition is self-conjugate. Mentored by Dr. Madeline Dawsey.

2:20pm-2:40pm	Tyler Russell: Partition Polynomials University of Texas at Tyler, TX Phi Abstract: $\textit{Partitions}$ are a common topic of study in number theory. They express the different ways that a positive integer may be written as a sum of other positive integers. In this research, we explore the analytic properties of a polynomial $f_\lambda(x)$ that we call the $\textit{partition polynomial}$ for the partition $\lambda$, with the hope of learning new properties of partitions. We prove a recursive formula for the derivative of $f_\lambda(x)$ involving Stirling numbers of the second kind, show that the set of integrals of $f_\lambda(x)$ is dense in $[0,1/2]$, and pose a conjecture relating the integral to the number of parts of the partition. Mentored by Dr. Madeline Dawsey.

2:40pm-3:00pm	Ben Gobler: Listing the Rationals Worcester Polytechnic Institute, MA Alpha Abstract: It is well known that the rational numbers are countable. From a list which includes each rational number exactly once, we may ask, “What is the 200th rational in the list?” or “Where does 22/7 appear in the list?” To answer these questions, we will explore an original method which uses continued fractions to evaluate and locate terms in the Calkin-Wilf sequence, one such list of the rationals. Mentored by Dr. Brigitte Servatius.

3:00pm-3:20pm	Nicholas Adduci: Visual/Geometric Rep of Primes Youngstown State University, OH Xi Abstract: This paper will reconceptualize the natural numbers by creating a novel geometric basis for their representation. Once this is achieved, the paper will go on to describe two different visual representations of the natural numbers against a grid. The first representation will organize the natural numbers in accordance with the novel geometric basis that has been established, whereas the second representation will display the natural numbers in terms of their coprimality relationships. Both visual representations contained herein will elucidate patterns that relate to the distributions of the prime numbers. New terminology and specialized functions will be developed in order to formalize the perceived patterns in the visual arrangements. Finally, the specialized functions contained herein will be used to establish mathematical relationships with respect to the structure of the prime numbers. Mentored by Dr. Thomas Wakefield.

3:20pm-3:40pm	Johnathan Koch: Defining the Cycle Youngstown State University, OH Xi Abstract: In this presentation, I will present and justify an alternate criterion that is necessary and sufficient for a cycle within the permutation group. The justification is built off the classical definition of a cycle. It is then proved that disjoint cycles commute to prove the relation between the order of a product of disjoint cycles and the order of the cycles themselves. Finally, a note on the irreducibility of a cycle is needed before proving: $\sigma \in S_n$ is an $m$-cycle $\Longleftrightarrow X_\sigma = \{ \sigma(a)^i \| 0 \leq i < m \} \forall a \in X_\sigma $ Mentored by Dr. Thomas Madsen.

PME Student Session 4

10:00am – 12:00pm, Friday, August 6, 2021


10:00am-10:20am	Luke Hetzel: Prisoner Dilemma & Cooperation Youngstown State University, OH Xi Abstract: The repeated prisoner’s dilemma is an aspect of Game Theory which has been used by evolutionary biologists to model the development of cooperation in a biological system. While a solitary game of the prisoner’s dilemma has a strictly dominant strategy to always defect, repeating the game adds complexity and allows for strategies to incentivize cooperation for a higher total benefit. Previous work has shown that the strategy “Tit-for-Tat” which incentivizes cooperation through reciprocity is a highly effective strategy. This paper analyzes the initial viability of the “Tit-for-Tat” strategy in a continually defecting system. Mentored by Dr. Padraic Taylor.

10:20am-10:40am	Nathan LeRoy: Survey Labyrinth Probabilities St. Norbert College, WI Delta Abstract: Labyrinth is a board game that was manufactured by Ravensburger in 1986. It is for 2-4 players with an age range of 7-99. The goal of the game is to slide the moveable columns and rows with an extra piece to maneuver your character to the treasure icons scattered about the board. We will be exploring mathematical elements such as graph theory, game theory, and probability in relation to the configurations of the standard 7×7 board for this game. We will be starting our research investigating a 3×3 board with similar ratios of pieces to the actual board. We will derive computations on the 3×3 board and then scale up our findings to the larger game boards as well as a board of arbitrary size. Mentored by Dr. Lindsey Bosko-Dunbar.

10:40am-11:00am	Lydia Mader: Strategies in Labyrinth Game St. Norbert College, WI Delta Abstract: Labyrinth is a board game for two to four players by Ravensburger in 1986. The object of the game is to collect all of your treasures and return to your starting corner before your opponents. Turns are sequential and consist of two parts–1) shift the board by inserting an extra game tile into a moveable row or column to change the configuration of the current board and 2) take the option of either moving your playing token along an open path made by connecting the game tiles or staying at the current coordinate on the board. Turns are completed in sequential order by players in a clockwise direction until one player obtains all their respective treasures and returns to their starting corner. The board consists of three basic types of game pieces: T, L, and I. The standard gameboard is a 7×7 square, with even rows and columns movable. We explore several factors that would optimize success during gameplay. Some of these factors considered are the configuration of the board, the location of the intended target, and the least amount of moves to a target. The analysis builds from a 3×3 reduced version to a 5×5 reduced version to analyze possible strategies for the full 7×7 board. Mentored by Dr. Lindsey Bosko-Dunbar.

11:00am-11:20am	Chase Reiter: Trigonometry and Spirographs Youngstown State University, OH Xi Abstract: Spirographs are fun and interesting mathematical graphs that can be drawn using a programming interface (such as MATLAB) and an elementary understanding of the unit circle and parametrizations of equations of a circle. The mathematical spirograph is based on the common child’s drawing toy which relies on spinning plastic gears of differing numbers of teeth with a pencil to draw a beautiful curve. This talk will expound upon a brief derivation of the parametric equations that allow one to draw a spirograph. Mentored by Dr. Thomas Madsen.

11:20am-11:40am	Francesca Amato: Bias in our Courts Adelphi University, NY Xi Abstract: Text mining is a tool used by many disciples to extract meaningful information from documents. The Cook Partisan Voting Index (PVI) is a non-partisan measure of how strongly each state leans towards a political party as compared to the nation. In this study, we perform a textual analysis on state Supreme Court cases from 2000 to 2020. The Cook PVI is used to select the top five Democratic states and the top five Republican states for analysis with the purpose of exploring social justice issues that may be present. Statistical models are then used to analyze these results and test their significance. We make important observations on these data and state meaningful conclusions based on these observations. The goal of this research is to determine if bias is present in our court system, and if so, if there is a difference between Democratic leaning states and Republican leaning states. This type of study would be beneficial for the public and can help our court system with their decision-making process. Mentored by Dr. Carl Giuffre.

11:40am-12:00pm	Jonathan Homan: Strong Fusion Pretzel Links Andrews University, MI Gamma Abstract: A link is a collection of circles embedded into 3-dimensional space. The fusion of a link fuses together two components of the link via a band, whereas strong fusion fuses two components of the link and adds an unknotted circle about the band. Such moves naturally arise in the study of cobordisms and have been shown to model biological phenomena. Determining links obtained by fusion has been a well-studied problem, but much less work has focused on strong fusion. Here we present a complete and original classification of those pretzel links which can be obtained by strong fusion. Pretzel links are an important family of links which comprises those links that fit a general form that includes many of the most common links. The primary tools we depend on are linking number and a dichromatic resolution of the link in which we conceive of the link as being colored with two colors and resolve crossings in such a way that respects those colors. Solving the classification problem in a number of subcases gives the general result. Mentored by Dr. Anthony Bosman.

PME Student Session 5

1:00pm – 2:00pm, Friday, August 6, 2021


1:00pm-1:20pm	Yifan Zhang: Subsums of Random Numbers University of Illinois at Urbana-Champaign, IL Alpha Abstract: If you pick $n$ random numbers in $[0,1]$, what is the probability that their sum also falls into the interval $[0,1]$? The answer turns out to be $1/n!$, which can be seen using symmetry arguments or multidimensional integrals. We consider more generally the probability, $p(n,k)$, that, given $n$ random numbers in $[0,1]$, there are $k$ of these numbers whose sum falls into the interval $[0,1]$. We show that these probabilities satisfy a generalized Pascal triangle type recurrence and we give an explicit formula for $p(n,k)$. In the special case $k=n-1$, these probabilities can be expressed in terms of Stirling numbers of the second kind and also in terms of harmonic numbers. Mentored by Dr. AJ Hildebrand.

1:20pm-1:40pm	Connor Lehmacher: The Generalized Matcher Game Vanderbilt University, TN Zeta Abstract: Recently the matcher game was introduced. In this game, two players create a maximal matching on a graph by one player repeatedly choosing a vertex and the other player choosing a path of length 1 containing that vertex. One player tries to minimize the result and the other to maximize the result. In this talk, I’ll discuss a generalization of this game where the path of length 1 is replaced by a path of length 2. I’ll provide some general results and bounds for the game, and calculate the outcome for specific classes of graphs. Mentored by Dr. Wayne Goddard.

PME 2021 Speaker Award Winners & Pictures

Name	Institution	Chapter	Talk Title
Nicholas Adduci	Youngstown State University	OH Xi	An Investigation into Visual and Geometric Representations of Prime Numbers
Ben Gobler	Worcester Polytechnic Institute	MA Alpha	Listing the Rationals using Continued Fractions
Hanna Noelle Griesbach	Elon University	NC Nu	When is a Polynomial Isomorphic to an Even Polynomial?
Luke Hetzel	Youngstown State University	OH Xi	Using Agent Based Modeling in NetLogo to Visualize Game Theory
Jonathan Homan	Andrews University	MI Gamma	Classifying Pretzel Links Obtained by Strong Fusion
Bandita Karki	University of Idaho	ID Alpha	Modeling the therapeutic potential of defective interfering particles
Johnathan Koch	Youngstown State University	OH Xi	Defining the cycle within the permutation group
Nathan LeRoy	St. Norbert College	WI Delta	Probabilities of the Game of Labyrinth
Rebecca Odom	University of Texas at Tyler	TX Phi	Identifying Self-Conjugate Partitions
Chase Reiter	Youngstown State University	OH Xi	Using Trigonometry to Make Spirographs with Parametrizations
Tyler Russell	University of Texas at Tyler	TX Phi	Polynomials Associated to Integer Partitions
Hannah Scanlon	Wake Forest University	NC Lambda	Modeling the Spread of Infectious Diseases on an Adaptive Network
Ella Wilson	Kenyon College	OH Pi	Using Circle Packings to Approximate Harmonic Measure Distribution Functions
Katie Yan	Skidmore College	NY Alpha Theta	Modeling the Plague in Eyam
Yifan Zhang	University of Illinois at Urbana-Champaign	IL Alpha	Subsums of Random Numbers

Donors and Supporters

Pi Mu Epsilon is grateful for its strong working relationship with the Mathematical Association of America (MAA). The MAA regularly cost-shares various MathFest activities with PME, helps secure spaces for student talks and meetings, and handles all audio-visual needs. Moreover, the MAA is indispensable in assisting with the PME online registration system.

For many years, the National Security Agency (NSA) has provided generous support to help defray costs for Pi Mu Epsilon students who attend MathFest and represent their chapters. Through a series of consecutive grants over a quarter of a century, NSA has provided more than $250,000 in support. PME appreciates the NSA’s recognition that attending a national conference can have a lasting impact on students and expose them to the mathematics community outside their home institutions.

In addition, Pi Mu Epsilon would like to express its sincere appreciation to the following organizations for their generous sponsorships of the Awards for Outstanding Presentations: The American Mathematical Society, the American Statistical Association, Budapest Semesters in Mathematics, the Council on Undergraduate Research, and Bio-SIGMAA.