38th Pi Mu Epsilon Conference College of St. Benedict/St. John’s University Friday, April 7th & Saturday, April 8th, 2017

Featured Speaker

Dr. Francis Su Harvey Mudd College and President of the Mathematical Association of America 38th Pi Mu Epsilon Conference Abstracts College of St. Benedict/St. John’s University

Friday, April 7, 2017 — 7:00 pm

PENGL 229 Jennifer Kochaver (Augsburg College) Minority Stressors as Predictors of Sexual Risk Behaviors and HIV Testing in Transgender Individuals Because transgender and gender non‐conforming individuals are at a higher risk for HIV transmission, it is important to understand the factors which put this population at risk. Understanding the psychological factors associated with heightened sexual risk‐taking is a first step towards developing better practices in preventing the transmission of HIV and other sexually transmitted infections in transgender and gender non‐conforming communities. We examined the applicability of Ilan Meyer’s minority stress model to the experiences of transgender and gender non‐conforming individuals in the United States. Meyer modeled internal and external minority stressors as the causes of disproportional adverse health outcomes in minority populations. Using survey data from 300 transgender or gender non‐conforming adults from across the United States, we performed multiple linear and logistic regression adjusting for race, relationship status, level of education, and sex assigned at birth and found that expecting rejection and experiencing discrimination are associated with heightened sexual risk taking in this population, as well as heightened likelihood to get HIV tested. This suggests that the minority stress model may be applicable to transgender and gender non‐conforming individuals with some modifications.

PE 244 Caroline Bang (St. Cloud State University) Inverse Semigroups An inverse semigroup is a set with an associative binary operation where each element has a unique inverse‐like element. This research looks into E‐unitary inverse semigroups, which are a certain type of inverse semigroup with a special condition on the idempotent elements (elements s such that s2 = s). By looking at group actions, we were able to give conditions for when an E‐unitary inverse semigroup has certain properties: specifically, when an E‐unitary inverse semigroup is fundamental and/or combinatorial.

PE 248 Dan Voce (Saint John’s University) A Quantitative Analysis of the Mille Lacs Lake Walleye Population Mille Lacs Lake, located in central Minnesota, has experienced disorder within its natural ecosystem that has led to difficulties maintaining a healthy walleye population balance. Walleye fishing is crucial for the tourist economy so adequate management is essential. We will mathematically model the walleye population present on Mille Lacs Lake. In doing so, we will strive to uncover causes of Mille Lacs’ walleye crash through the historical data of the Lake. Finally, we will use existing patterns to help predict the future of the walleye population and conclude with suggestions based upon mathematical evidence that could help avoid a similar collapse moving forward. (Note that this talk is for 50 minutes)

Friday, April 7, 2017 — 7:30 pm

PE 229 Victoria Bauers (Bethel University) The Nine‐Point Circle In every triangle, there is a set of unique points that exist. Some of these points are the midpoints, the feet of the altitudes, the orthocenter, and the circumcenter. All of these points are linked through one magnificent property: The Nine‐Point Circle. My presentation will look at the history of the Nine‐Point Circle including the discoverers. Next, some preliminary definitions and theorems will be discussed in order to go into the details of the nine points and the proof of their existence. Then Feuerbach’s theorem and proof will be dissected and presented. Lastly, my presentation explores further application of the Nine‐Point Circle, including the Euler line and Feuerbach’s theorem.

PE 244 Joe Zemmels (Winona State University) Buddhabrot: Re‐imagining the Mandelbrot Set Fractals, derived from the Latin frāctus meaning “broken” or “fractured,” are geometric figures characterized by their self‐similarity at any scale. Fractals like the Cantor function existed in theory as far back as the 1880s, but it wasn't until the 1970s and the advent of computers that mathematicians could visualize fractals past simple, manual drawings. One such fractal is known as the Mandelbrot set. The Mandelbrot set is defined as the set of all complex numbers c for which the point 0 does not escape to infinity under iteration of the function f(z) = z2 + c. The classic method of computer generating the Mandelbrot set is well‐documented, yet there have been recent attempts at finding alternative methods of generating and depicting the Mandelbrot set. A particularly interesting method discovered in 1993 produces a depiction of the Mandelbrot set referred to as the “Buddhabrot” because of its visual similarity to Gautama Buddha. The purpose of this project is to discuss the Buddhabrot method and how it relates both mathematically and visually to the classic method of generating the Mandelbrot set.

PE 238 John Incha, Spencer Morrison (University of Wisconsin – River Falls) Road Following Robots in the 21st Century We will describe our experience in the 2017 Mathematical Modeling Contest solving Problem C: Cooperate and Navigate, about improving traffic congestion through the introduction of self‐driving, cooperative automobiles. We used regression analysis and ideas from fluid flow to analyze the effects of different proportions of self‐driving to human operated vehicles. Our analysis was implemented with code written in “R” and “Ruby.”

Friday, April 7, 2017 — 8:00 pm

PE 229 Jon Blomquist (Saint John’s University) A 3‐Player Game on Cyclic, Dihedral, and Abelian Groups The three‐player game of GEN involves taking turns selecting elements of a group to add to a common pool of elements. Each element can only be selected once, and all players share the pool of elements. The object of the game is to be the player that adds the final element to the pool that will generate the entire group. We will take a closer look at the strategies players must follow in order to win the game when playing with cyclic groups, dihedral groups, and abelian groups.

Friday, April 7, 2017 — 8:00 pm

PE 244 Dayna Jaeger (Bemidji State University) Multiple Approaches to Problem Solving People sometimes say that school mathematics is boring because there is always just one answer. This neglects a fascinating part of mathematics – the possibility of having multiple approaches to a problem result in the same answer. This will be demonstrated with an elementary school level question with various levels of solution processes.

PE 248 Emma Cobian, Jake Minor, Austin Wilcox (University of Wisconsin – River Falls) Anyway You Want It, That's The Way We'll Merge It! In the 2017 Mathematical Contest in Modeling we addressed the problem of merging traffic after a toll. In this talk, we describe how we solved the problem by comparing a horizontal barrier and a staggered barrier, using three metrics. Our methods include non‐deterministic finite cellular automata and a simulation implemented with Java.

PE 238 Brandon Voigt (Saint John’s University) Irregular Graph Labelings For any graph G, label the edges of G using a function f : {1, 2, …, m} → E(G). Define the weight of a vertex v as the sum of every edge incident to v. If each vertex weight is distinct, then f is considered an irregular labeling of G. This talk will discuss irregular labelings on a variety of classes of graphs, particularly trees.

Friday, April 7, 2017 Pellegrene Auditorium, 8:30 PM

Dr. Francis Su Harvey Mudd College and President of the Mathematical Association of America

Fair Division Using Topological Combinatorics

The Brouwer fixed point theorem is a beautiful and well‐known theorem of topology that has a combinatorial analogue known as Sperner's lemma. There are also several other combinatorial analogues of topological theorems. In this talk, I will explain applications to problems of fair division, and trace recent connections and generalizations.

Saturday, April 8, 2017 — 9:00 am

PE 229 Samuel Holen (Minnesota State University – Moorhead) The Collatz Conjecture: An Analysis The Collatz Conjecture, also known as the hailstone sequence, is a mathematical sequence an, where the starting value a0 is a positive integer, defined in the following way.

The conjecture, proposed over 80 years ago by Lothar Collatz, claims that starting from any positive integer a0, repeated iteration will eventually yield the value 1. Simple to state and test, this conjecture starts off as a leisurely stroll in the park, but quickly becomes an Olympic marathon, leaving one drained of all enthusiasm. This talk discusses an analysis of the Collatz Conjecture. We will discuss the basics of the Conjecture along with a brief history and several examples. We will discuss number types that are shown to decrease overall as well as problematic numbers that have not been shown to decrease. We will also derive the "worst case" numbers, numbers that increase the most consecutive times, and where they are headed. Finally, we will examine a few inductive proofs for two infinite sets of numbers that absolutely work.

PE 244 Hannah Davis (University of Minnesota – Twin Cities) Algebraic Varieties of Gerechte Designs A gerechte design is an n‐by‐ n Latin square and a partition of the square into n regions, each containing n distinct symbols. The popular Sudoku puzzle is a gerechte design with a partition consisting of nine 3‐ by‐ 3 squares. Representations of gerechte designs have been studied in many areas of mathematics to determine the number and relationships between the solutions. We consider gerechte designs as the solution set of a system of polynomial equations over a finite field, and present the relationship between the resulting geometric object, the restrictions which generate it, and the automorphisms acting upon it.

PE 248 Oliver O’Keefe (University of Saint Thomas) Subatomic Knots Glueballs are subatomic particles which are hypothesized to take the shape of tightened knotted and linked uniform‐radius tubes. These model glueballs can decay through quantum tunneling events, where the tube representing the particle passes through itself at a random self‐contact point. These events can result in a change in knot type of the model glueball. The goal of this research is to determine all the possible knot type changes that occur from quantum tunneling events in model glueballs. (A background in physics or knot theory is not needed)

Saturday, April 8, 2017 — 9:30 am

PE 229 Michael Holmblad (Winona State University) Elliptic Curve Cryptography Elliptic Curve Cryptography is one of two major forms of public key cryptography. Between it and RSA, Elliptic Curve is much faster and more secure. It also has a trapdoor that as of right now is not vulnerable to sub‐exponential attacks. Understanding Elliptic Curve Cryptography is no trivial matter. The learning curve is very steep and sharp. However, by using concepts from Abstract Algebra, basic Number Theory, and Calculus we may be able to understand Elliptic Curve Cryptography from a mathematical standpoint. Elliptic curves themselves are not trivial, but together we will try to understand how they work and how to use them in cryptography.

Saturday, April 8, 2017 — 9:30 am

PE 244 Jake Park (Saint John’s University) Nim on Groups Nim is a simple mathematical game that has existed for years. It is believed that the game Nim originated in China and is one of the oldest games in the world. Nim has been played using three distinct piles of anything from stones, matchsticks, tokens, or any other counter. Each pile contains some amount of stones, but there may be different numbers of stones among the piles. In its basic form, two players compete to be the one to take the last stone of the game. Players alternate turns until there is a winner. On each turn, a player chooses a single pile to remove stones. The player may remove as many stones as desired, as long as it is at least one, from the single pile chosen. Then the other player takes a turn. Nim continues in this fashion until a player takes the last stone, and that player wins since there are no more stones to eliminate.

PE 248 Emily Vecchia (University of Saint Thomas) To what extent are measurements of open knotting consistent with traditional knot theory? In traditional knot theory, researchers primarily study closed knotted loops. However, important biological objects, such as DNA and proteins, have structures that can be entangled but form open curves. To understand the knotting in these systems, a rigorous mathematical definition of knotting for open curves is needed. In working towards such a definition, scientists have devised methods to measure knotting in open curves by closing the curves in some fashion. In this talk, we analyze to what extent two prominent closure methods are consistent with traditional knot theory. In particular, given an open subarc of a closed polygonal knot, we analyze to what extent each closure method’s classification of the knotting in the subarc coincides with the knot type of the closed knot as a function of length of the subarc.

Saturday, April 8, 2017 — 10:00 am

PE 229 Le Tang (Winona State University) Theorems on p‐adic Continued Fractions One of the major results for continued fractions is the Stern‐Stolz Theorem that relates the convergence of the continued fraction to the divergence of the associated series. The standard proof of the Stern‐Stolz Theorem uses a collection of recurrence relations on the numerator and denominator of the convergents. In this project we study the geometric approach as described in the research paper “The Seidel, Stern, Stolz and Van Vleck Theorems on continued fractions” by Beardon and Short. They reconsidered the Stern‐Stolz Theorem (and other important convergence theorems for continued fractions) by treating continued fractions as a sequence of Möbius transformations on the complex plane. In our study, we modify the methods to prove the analogous theorem in the complete p‐adic field Cp and in this talk we present our result.

Saturday, April 8, 2017 — 10:00 am

PE 244 Nicholas Meyer (Winona State University) On The Algebra of Rotations in R3 The need to represent rotations of objects in 3‐D Euclidean space arises daily in many fields: animation, computer vision, and physics, to name a few. Ever since Euler first described his eponymous angles, without giving a general method for constructing them, mathematicians have longed for a better system to describe rotations. In 1843, William Rowan Hamilton had an epiphany whilst walking across Brougham Bridge in Dublin with his wife. Therein he inscribed the laws defining the quaternions, forever changing the face of rotations. The quaternions, when limited to having unit norm, form a group under multiplication which is isomorphic to SU(2). This paper will discuss the interplay between these two groups and will clarify the use of quaternions to represent rotations. We will delve into the relationship between SU(2) and SO(3).

PE 248 Evan Camrud (Concordia College – Moorhead) A Tractable Numerical Model for Exploring Nonadiabatic Quantum Dynamics Simple quantum systems may be described in full by elegant solutions to eigenvalue problems. Sadly, the physical world is anything but simple, and quantum systems must be approximated by numerical means. A quantum phenomenon known as a “conical intersection” arises in molecules containing potential energy states degenerate at a single point. The dynamics induced by nonadiabatic coupling cannot be solved outside of approximation. By utilizing discrete variable representation, a versatile one‐ dimensional model that is numerically tractable and illustrates fundamental aspects of nonadiabatic quantum dynamics was developed. The model demonstrates how the local topography of an unavoided crossing can affect a photochemical quantum yield.

Saturday, April 8, 2017

Pellegrene Auditorium, 10:30 AM

Dr. Francis Su Harvey Mudd College and President of the Mathematical Association of America

The Geometry of Cubes

Cubes are one of the simplest geometric objects. Or are they? I will ask some basic questions that show how cubes are connected to many other mathematical ideas. Including pi? Some are recent discoveries by undergraduates.

Notes:

MATHEMATICS DEPARTMENT

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