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August 3–6, 2016 Abstracts of Papers Presented in Columbus, OH Abstracts of Papers Presented at MathFest 2016 Columbus, OH August 3 – 6, 2016 Published and Distributed by The Mathematical Association of America ii Contents Invited Addresses 1 Earle Raymond Hedrick Lecture Series by Hendrik Lenstra . 1 Lecture 1: The Group Law on Elliptic Curves Thursday, August 4, 10:30–11:20 AM, Regency Ballroom . 1 Lecture 2: The Combinatorial Nullstellensatz Friday, August 5, 9:30–10:20 AM, Regency Ballroom . 1 Lecture 3: Profinite Number Theory Saturday, August 6, 9:30–10:20 AM, Regency Ballroom . 1 AMS-MAA Joint Invited Address . 1 Understanding Geometry (and Arithmetic) through Cutting and Pasting by Ravi Vakil Thursday, August 4, 9:30–10:20 AM, Regency Ballroom . 1 MAA Invited Addresses . 2 Mathematical Sense and Nonsense outside the Classroom: How Well Are We Preparing Our Students to Tell the Difference? Network Science: From the Online World to Cancer Genomics by Robert Megginson Thursday, August 4, 8:30–9:20 AM, Regency Ballroom . 2 Magical Mathematics by Arthur Benjamin Friday, August 5, 10:30–11:20 AM, Regency Ballroom . 2 Immersion in Mathematics via Digital Art by Judy Holdener Saturday, August 6, 10:30–11:20 AM, Regency Ballroom . 2 James R.C. Leitzel Lecture . 2 Inquiry, Encouragement, Home Cooking (And Other Boundary Value Problems) by Annalisa Crannell Saturday, August 6, 8:30–9:20 AM, Regency Ballroom . 2 AWM-MAA Etta Z. Falconer Lecture . 3 Harmonic Analysis and Additive Combinatorics on Fractals by Izabella Laba Friday, August 5, 8:30–9:20 AM, Regency Ballroom . 3 MAA Chan Stanek Lecture for Students . 3 Zombies & Calculus: A Survival Guide by Colin Adams Thursday, August 4, 1:00–1:50 PM, Regency Ballroom . 3 Pi Mu Epsilon J. Sutherland Frame Lecture . 3 Combinatorics - The Mathematics That Counts by Robin Wilson Friday, August 5, 8:00–8:50 PM, Regency Ballroom . 3 NAM David Harold Blackwell Lecture . 3 Urban Analytics: The Case for Smart Parking by Robert Hampshire Friday, August 5, 1:00–1:50 PM, Regency Ballroom . 3 Presentations by Alder Awards Winners 4 Alder Award Session . 4 Do You: How Mathematics + Mentoring + Passion = Opportunities by Dandrielle Lewis Friday, August 5, 2:30–2:50 PM, Hayes . 4 Two Human Faces of Mathematics: Students and Medicine by Jana Gevertz Friday, August 5, 3:00–3:20 PM, Hayes . 4 Modeling Across the Curriculum by Benjamin Galluzzo Friday, August 5, 3:30–3:50 PM, Hayes . 4 Invited Paper Sessions 5 Knot Theory Part A: Thursday, August 4, 8:30–9:50 AM, Fairfield . 5 Part B: Thursday, August 4, 2:00-4:20 PM, Fairfield . 5 The Mathematics of Games Thursday, August 4, 2:00–3:50 PM, Harrison . 6 Mathematics and Magic Friday, August 5, 1:00–3:55 PM, Fairfield . 7 Mathematics and the Life Sciences at MBI Friday, August 5, 1:00–4:10 PM, Harrison . 8 Numbers, Geometries, and Games: A Centenarian of Mathematics Saturday, August 6, 1:00–3:10 PM, Fairfield . 9 Contents iii Undergraduate Research Projects in the Mathematical Sciences Saturday, August 6, 1:00–3:20 PM, Harrison . 10 Themed Contributed Paper Sessions 12 Inviting All Students to Do Mathematics – Engaging Courses, Projects, and Activities for Liberal Arts Students Part A: Thursday, August 4, 8:30–10:05 AM, Union B . 12 Part B: Friday, August 5, 8:30–9:45 AM, Union C . 13 Part C: Friday, August 5, 1:00–6:15 PM, Union C . 14 CAMP: Calculus Applied Mathematics Projects Thursday, August 4, 1:00–3:55 PM, Franklin A . 17 Fostering a Problem-Solving Culture for Students Thursday, August 4, 1:00–4:15 PM,TaftA..................................... 19 My Favorite Math Circle Problem Thursday, August 4, 1:00–4:55 PM, Franklin C . 21 Novel Introductions to Non-Euclidean Geometry Thursday, August 4, 1:00–2:55 PM, Union A . 24 Recreational Mathematics: Puzzles, Card Tricks, Games, Gambling, and Sports Part A: Thursday, August 4, 1:00–3:55 PM,TaftC................................. 25 Part B: Friday, August 5, 1:00–4:15 PM,TaftC .................................. 27 Encouraging Early Career Teaching Innovation Part A: Friday, August 5, 1:00–4:55 PM, Union A . 28 Part B: Saturday, August 6, 9:30–11:45 AM, Union A . 31 Formative Assessment Techniques for Undergraduate Math Courses Part A: Friday, August 5, 1:00–4:35 PM, Union B . 32 Part B: Saturday, August 6, 1:00–3:35 PM, Union B . 35 Programming in Mathematics Classes and Mathematics for Programming Saturday, August 6, 1:00–5:15 PM, Union A . 36 Undergraduate Research Activities in Mathematical and Computational Biology Saturday, August 6, 1:00–2:15 PM,TaftA ..................................... 39 General Contributed Paper Sessions 41 Geometry Thursday, August 4, 8:30–10:25 AM, Union D . 41 Linear & Abstract Algebra Thursday, August 4, 8:30–10:25 AM, Union E . 42 Applied Mathematics Thursday, August 4, 1:00–4:55 PM, Union D . 43 Teaching Calculus Thursday, August 4, 1:00–4:55 PM, Union E . 46 Graph Theory and Other Topics Friday, August 5, 8:30–11:40 AM, Union E . 49 Teaching Introductory Level Mathematics and Assessment Friday, August 5, 8:30–11:40 AM, Union D . 52 Teaching Advanced Level Mathematics Friday, August 5, 1:00–5:10 PM, Union E . 54 Teaching Introductory Level Mathematics Friday, August 5, 1:00–4:55 PM, Union D . 58 History of Mathematics Saturday, August 6, 8:30–11:40 AM, Union E . 61 Number Theory Saturday, August 6, 8:30–11:40 AM, Union D . 63 Outreach and Other Topics Saturday, August 6, 1:00–4:55 PM, Union D . 65 Probability, Statistics and Calculus Saturday, August 6, 1:00–5:00 PM, Union E . 68 iv Contents Graduate Student Session 72 Great Talks for a General Audience: Coached Presentations by Graduate Students Saturday, August 6, 1:00–5:00 PM, Madison . 72 PosterFest 2016 73 PosterFest 2016 Friday, August 5, 3:30–5:00 PM, Exhibit Hall . 73 Index of Speakers 76 Invited Addresses 1 Invited Addresses Earle Raymond Hedrick Lecture Series Hendrik Lenstra Universiteit Leiden Lecture 1: The Group Law on Elliptic Curves Thursday, August 4, 10:30–11:20 AM, Regency Ballroom The theory of elliptic curves is a showpiece of modern mathematics. Its implications are felt from the brightest parts of number theory, where it supplied the key to Fermat’s Last Theorem, to the darkest corners of cyberspace, where it provides the workhorses of secret communication. How much of the theory can with lucidity and rigor be developed in an undergraduate algebra course? The lecture outlines an approach to at least establishing the group law, using no other tools than what such a course ordinarily already covers. Lecture 2: The Combinatorial Nullstellensatz Friday, August 5, 9:30–10:20 AM, Regency Ballroom Noga Alon’s combinatorial Nullstellensatz (1999) is a quantitative sharpening of the familiar fact that a non-zero polynomial in several vari- ables over an infinite field defines a function that does not vanish everywhere. It has an impressive number of consequences of a combinatorial nature, and forms an excellent example of what algebra can do for the non-algebraist. As the lecture shows, the combinatorial Nullstellensatz also belongs to the algebraist’s own toolbox. A characteristic application is an elegant theorem about matrices that is just one step away from one of the loveliest theorems of Galois theory: the existence of a normal basis in any finite Galois extension. Lecture 3: Profinite Number Theory Saturday, August 6, 9:30–10:20 AM, Regency Ballroom What is a number? Surprisingly, one of the fundamental notions of mathematics is never given a rigorous definition. Usage determines the meaning of the word. When one refers to the typical element of some ring as a “number” , one expresses a certain familiarity with that ring. The algebraist who defines a new ring creates a new species of number, and may feel master of the universe. Profinite numbers are elements of the ring of profinite integers, an important instrument in both infinite Galois theory and arithmetic geometry. In the lecture, the world of profinite numbers with its wealth of wonders is explored for its own sake. AMS-MAA Joint Invited Address Thursday, August 4, 9:30–10:20 AM, Regency Ballroom Ravi Vakil Stanford University Understanding Geometry (and Arithmetic) through Cutting and Pasting Euler’s famous formula tells us that (with appropriate caveats), a map on the sphere with ff countries (faces), e borders (edges), and v border- ends (vertices) will satisfy v − e + f = 2. And more generally, for a map on a surface with g holes, v − e + f = 2 − 2g. Thus we can figure out the genus of a surface by cutting it into pieces (faces, edges, vertices), and just counting the pieces appropriately. This is an example of the topological maxim “think globally, act locally” . A starting point for modern algebraic geometry can be understood as the realization that when geometric objects are actually algebraic, then cutting and pasting tells you far more than it does in “usual” geometry. I will describe some easy-to-understand statements (with hard-to-understand proofs), as well as easy-to-understand conjectures (some with very clever counterexamples, by M. Larsen, V. Lunts, L. Borisov, and others). I may also discuss some joint work with Melanie Matchett Wood. 2 Invited Addresses MAA Invited Addresses Thursday, August 4, 8:30–9:20 AM, Regency Ballroom Robert Megginson University of Michigan Mathematical Sense and Nonsense outside the Classroom: How Well Are We Preparing Our Students to Tell the Differ- ence? Network Science: From the Online World to Cancer Genomics “Mathematics will always be a key element of liberal education, since it promotes logical reasoning.” You have likely heard this claim, or perhaps made it yourself.
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