HRUMC XXIII

Hudson RiverUndergraduate Mathematics

Conference

Saint Michael’s CollegeColchester Vermont

2016-Apr-02

Schedule overview

8:30–9:50am Registration Dion Center10:00–10:55am Parallel Sessions One St Edmunds and Jeanmarie Halls11:05–12:15pm Welcome, Invited Address McCarthy Arts Center12:25–1:30pm Lunch Dion Center1:40–2:55pm Parallel Sessions Two St Edmunds and Jeanmarie Halls3:00–3:25pm Co�ee and refereshments Dion Center3:30–4:25pm Parallel Sessions Three St Edmunds and Jeanmarie Halls

WifiUse the network “SMC-Guest” with the password “PurpleKnights”.

Abridged program

A brief version of this program, not including the abstracts, is available at http://joshua.smcvt.edu/hrumc/hrumc2016short.pdf.

If you need helpMedical, fire, or police In an emergency call 911. For non-emergency security or safety issues, call

campus security at 802-654-2911, or simply dial 2911 from any campus phone. There is a campusphone in each classroom.

Classroom equipment To help with computers or other equipment there will be student assistants,wearing special tee-shirts, in or near each classroom. You can also call (802) 654-2959, or dial2959 from the campus phone in each classroom.

Rest rooms There are rest rooms on each floor of the academic buildings JEM and STE. There is aunisex rest room on the second floor of JEM, next to room 277.

Welcome everyone!

Welcome to the twenty third annual Hudson River UndergraduateMathematics Conference (HRUMC).Whether you are a student at your first conference or an experienced speaker, we hope that you will findtoday beneficial, rewarding, and inspiring, and that you will make new friends. Our aim is to build anatmosphere that includes the message, “We are glad that you are joining the mathematics community!”

This conference features fifteen minute talks by students and faculty, and a longer plenary address. Eachyear, we invite students and faculty from universities and colleges in New York and New England tosend abstracts for the short talks. These describe research projects, independent study projects, or anyother independent work by students and faculty.

If you are a first time attendee then start by studying the short talks schedule to find some that grab yourinterest. Each of these is marked as Level 1 or Level 2: the Level 1 talks are accessible to everyone whileLevel 2 talks are aimed at faculty and advanced students.

Note that each session has a Chair, who keeps all presentations strictly to the schedule. This means thatyou can easily move from room to room to see talks — you know that each talk that you attend will endon time, and each next one will start when it says it will.

If you are a first time presenter thenwe especially say, “Welcome!” Giving a presentation can be daunting,but is also energizing. The session Chair will be able to help with any questions that you have, includingany technology questions.

The first HRUMC was held at Siena College in 1994, and now it is an annual tradition. For informationabout previous meetings, pictures from this year’s conference, as well as information about next year’sconference you can check out the web site: http://www.skidmore.edu/hrumc.

Presenter or attendee, we hope that you enjoy the HRUMC and that you will you will learn a greatdeal. And, if you can, we hope to see you again, sharing your work, at next year’s conference, hosted byWestfield State University, on Saturday April 8th, 2017.

This conference would not be possible without the generous financial support provided by theO�ce of the Vice President for Academic A�airs at Saint Michael’s College, and by the Depart-ments of Mathematics and Computer Science. Support also comes from the NASA-VT SpaceGrant Consortium, and from the Pi Mu Epsilon national mathematics honor society. We alsothank all of the student and faculty volunteers who contributed their time, talents, energy, andenthusiasm.

HRUMC CommitteeLauren Childs, Williams CollegePaul Friedman, Union CollegeMohammad Javaheri, Siena CollegeJesse Johnson, Westfield State UniversityEmelie Kenney, Siena CollegeJoe Kirtland, Marist CollegeAllison Pacelli, Williams CollegeAlejandro Sarria, Williams CollegeDavid Vella, Skidmore CollegeEdward Welsh, Westfield State UniversityWilliam Zwicker, Union College

Site ArrangementsJim HefferonLloyd Simons

Institutional Greeting and Invited Address

Welcoming Remarks: Dr. Karen Talentino, VPAA, Saint Michael’s CollegeIntroduction of the Speaker: Celsey Lumbra, SMC ’16

Keynote address: The P vs. NP ProblemScott Aaronson

MIT, University of Texas at Austin

AbstractI’ll discuss the status of the famous P ?= NP problem in 2016, o�ering a personal perspective on whatit’s about, why it’s important, why many experts conjecture that P != NP is both true and provable,why proving P != NP is so hard, the landscape of related problems, and crucially, what progress hasbeen made in the last half-century. I’ll say something about diagonalization and circuit lower bounds;the relativization, algebrization, and natural proofs barriers; and the recent works of Ryan Williams andKetanMulmuley, which (in di�erent ways) hint at a duality between impossibility proofs and algorithms.

BiographyProfessor Aaronson holds a Ph.D. in Computer Science from University of California, Berkeley. Heis currently Associate Professor of Electrical Engineering and Computer Science at the MassachussetsInstitute of Technology. Starting in July he will be the David J. Bruton Jr. Centennial Professor ofComputer Science at the University of Texas at Austin. He has an international reputation as an expertin the Theory of Computation and Complexity Theory.

Scott studies the fundamental limits on what can be e�ciently computed in the physical world. This en-tails studying quantum computing, the most powerful model of computation that we have. His work hasincluded limitations of quantum algorithms in the black-box model; the learnability of quantum states;quantum proofs and advice; the power of postselected quantum computing and quantum computingwith closed timelike curves; and linear-optical quantum computing.

PARALLEL SESSIONS ONE

Abstract Algebra I JEM 380Chair: Blair Madore

10:00-10:15 The Amazing Connections Between Groups and Symmetric Subgroups (Level 2) HENRYYOUNG (WHEATON COLLEGE) The history of group theory started with the development of per-mutation groups. However, our mathematical predecessors did not know they were dealing withsuch an essential algebraic structure. Arthur Caley proved that all groups are isomorphic to a sub-group of a permutation group. Thus, permutations groups contain the structure of all of the fa-mous groups that we know and love. By looking at these isomorphisms and their applications tothe theory of groups, we may as well rethink what it is meant to be a group.

10:20-10:35 How Commutative are Dihedral Groups? (Level 1) AMANDA PETERSON (SUNY POTSDAM)We will show the probability of any two randomly selected elements of a dihedral group com-muting through the use of Clifton, Guichard, and Keef’s work. We will explain how many andwhich elements in a dihedral group commute and use that information to compute the probabilityof any two elements commuting in a product of dihedral groups.

10:40-10:55 Subnormal Subgroups of M-Groups (Level 2) JOHN MCHUGH (UNIVERSITY OF VERMONT) Awealth of structural information about a finite group can be obtained by studying its irreduciblecharacters. Of particular interest are monomial characters — those induced from a linear char-acter of some subgroup — since Brauer has shown that any irreducible character of a group canbe written as an integral linear combination of monomial characters. Our primary focus is theclass of M-groups, those groups all of whose irreducible characters are monomial. A classical the-orem of Taketa asserts that an M-group is necessarily solvable, and Dade proved that every solv-able group can be embedded as a subgroup of an M-group. After discussing results related to M-groups, we will construct explicit families of solvable groups that cannot be embedded as subnor-mal subgroups of any M-group.

Analysis JEM 389Chair: Andrew McIntyre

10:00-10:15 The Equality in Young’s Inequality (Level 1) JAMES VEES (HAMILTON COLLEGE) Young’sInequality is an important result in functional analysis that relates the areas enclosed by a functionand its inverse. This presentation will o�er two di�erent calculus-based proofs of the conditionsthat imply equality in Young’s Inequality. The talk will also give an example of an integral thatbecomes easy to compute if we use this result.

10:20-10:35 Limiting Distributions for Topological Markov Chains with Holes (Level 2) MARK F. DEMERS(FAIRFIELD UNIVERSITY), CHRIS IANZANO (STONY BROOK UNIVERSITY), PHILIP MAYER (FAIRFIELDUNIVERSITY), PETER MORFE (COOPER UNION), ELIZABETH C. YOO (COLUMBIA UNIVERSITY) Opendynamical systems are models of physical systems in which mass or energy is allowed to escapefrom the system. Central questions involve the existence of conditional equilibria (measures thatare invariant under the dynamics conditioned on non-escape) which can be realized as limitingdistributions under the dynamics of the open system. We study this problem in the context oftopological Markov chains, a class of symbolic dynamical systems with a wide variety of appli-cations. Under a combinatorial condition on the Markov chain, we study transfer operators as-sociated with positive recurrent potentials and prove the existence of a spectral gap on a naturalfunction space. This implies the existence (and uniqueness in a certain class) of limiting distribu-tions which represent conditional equilibria for the open system. We also prove a relation betweenthe escape rate from the system and the entropy on the survivor set (the set of points that neverenters the hole).

10:40-10:55 Measure theory on the real line (Level 2) JEFF JAUREGUI (UNION COLLEGE) If you have abounded subset of the real line, how do you define its “size”, or “measure”? For an interval [𝑎, 𝑏],the measure ought to be just the length of the interval, 𝑏 − 𝑎. But what about more complicatedsets that have lots of gaps, like the subset of rational numbers in [𝑎, 𝑏], or the famous Cantor set?In this talk I will give an introduction to the subject of measure theory, an important branch ofanalysis.

Applied Mathematics Ia JEM 378Chair: Amy Wehe

10:00-10:15 Origami Square Twist: Possible to Fold it Rigidly? (Level 1) AUTUMN PHANEUF (HOLYOKECOMMUNITY COLLEGE) The ancient art of origami is now being integrated into technological ad-vances. A challenge that many artists, mathematicians, and engineers face is to figure out whetheror not a structure can be folded rigidly. “Rigid folding” means that one can replace the paper witha sheet of metal with hinges and still be able to fold the origami model. This presentation will an-alyze the origami square twist. We know that we can fold a square twist with paper. Is it possibleto rigidly fold one?

10:20-10:35 Applications of Inverse Problems to Image Processing (Level 2) HANNAH SOPER (NORWICHUNIVERSITY) The goal of this study is to form a basic understanding of inverse problems and theirapplications, culminating with solving a specific problem relating to image processing: deblur-ring an image. We do this with the matrix form of convolution used for blurring an image. Bycalculating the singular value decomposition of this matrix, we can invert the process of blurringan image. When Tikhonov regularization is then imposed on the deblurring process, the problembecomes well-posed and an exact solution can be found. We confirm the viability of this processby performing it on a picture including blurred text, which then becomes readable.

10:40-10:55 Describing Continuous Symmetries with Lie Groups (Level 2) KENNETH RATLIFF (HAMILTONCOLLEGE) Continuous symmetries play an important role in modern physics. The study of thesesymmetries invokes matrix groups; for example, given a system that is invariant about an axis, theset of rotations that leaves the system unchanged is the group 𝑆𝑂(2). In this talk we will explorethe mathematics of Lie groups, their actions and representations as matrix groups, and how theyencode the symmetries of a system.

Applied Mathematics Ib JEM 166Chair: Joseph Kirtland

10:00-10:15 Portfolio optimization: maximizing return of an investor’s portfolio of assets for a given level ofrisk (Level 2) NAMINI DE SILVA (SUNY PLATTSBURGH) Modern Portfolio Theory in finance at-tempts to optimize a portfolio of assets, i.e., maximize return for a given level of risk. The theorystates that when determining the di�erent proportions of wealth an investor should invest in eachasset, those assets should not be considered in isolation, but rather relative to every other asset inthe portfolio. We will examine how to determine these proportions (called portfolio weights) forany given number of assets.

10:20-10:35 Transportation Problem (Level 2) REBECCA LERMA (FITCHBURG STATE UNIVERSITY) Trans-portation problems are problems involving transporting products from several sources to severaldestinations. To solve these problems a type of linear programming problem used, which is thesimplex method. In this talk we will take an example of a transportation problem of transportingproduce from farms to retail location. We seek to find the best solution to the constraints of bothfrom what farms can produce and what the retail locations desire. The aim in this problem wouldbe to minimize the total cost of transportation and satisfy the necessary demands.

10:40-10:55 Phase control of quantum walks (Level 2) JONATHAN VANDERMAUSE, DANIEL ROCKMORE(DARTMOUTH COLLEGE) It has recently been shown that node-to-node transport in a continuous-time quantum walk can be slowed down or sped up by introducing suitably chosen complexphases to the adjacency matrix of the network. With the one-shot hitting time as our measure

of transport between nodes, we frame our study as a control problem, with the goal of enhanc-ing or suppressing transport between an initial node and a target node by optimizing the complexphases of certain links in the network. We first study walks over simple graphs and show that theextent to which transport is enhanced or suppressed depends on the topology of the network, theposition of the link in the network, and the values of the chosen phases. In particular, we showthat transport can be totally suppressed along the diagonal of an 𝑁-by-𝑁 lattice and that transportcan be enhanced in graphs containing diverging paths. Finally, we propose a more general pro-cedure for optimizing phases in a complex network. A greedy algorithm is used to reduce the hit-ting time between the most distant nodes in an Erdős-Rényi network, a preferential attachmentnetwork, and a Watts-Strogatz small-world network.

Combinatorics I JEM 364Chair: Lauren Heller

10:00-10:15 Counting MV Assignments of the Origami Snake Tessellation (Level 1) KRISTEN KENNEY(WESTERN NEW ENGLAND UNIVERSITY) The origami snake tessellation is a corrugation-patternedfold that, unlike the Miura-ori or the square twist tessellation, contains vertices of degree four andsix. This makes it challenging to determine the number of valid mountain-valley (MV) assign-ments of the snake tessellation, or the number of ways which the tessellation can be folded flat.We present upper bounds on 𝑆𝑚,𝑛, which denotes the number of valid MV assignments for an𝑚 × 𝑛 snake tessellation. We also describe our e�orts for finding a closed formula for 𝑆(𝑚,𝑛).This work was supported by the NSF ODISSEI grant EFRI-1240441 and advised by Dr. ThomasHull.

10:20-10:35 A Mathematician’s View of Scheduling (Level 1) SIMONA BOYADZHIYSKA (WELLESLEY COL-LEGE) Consider the schedule of events happening at a conference; we can say that one event is“less than” another if the former ends before the latter begins, but if they overlap in time, then theevents are called “incomparable.” The events together with this relation form a partial order. Par-tial orders arising from schedules in this way are called interval orders. In this talk, we will addressquestions such as the following: Which partial orders come from schedules? Which come fromschedules in which all events have the same length? Which come from schedules in which eachevent is at least one hour and at most two hours long?

10:40-10:55 A Partially Ordered World (Level 1) ALAN SHUCHAT, RANDY SHULL, ANN TRENK (WELLES-LEY COLLEGE) In this talk we introduce partial orders and in particular, the class of interval orders.Interval orders, which can be used to model scheduling problems, can be represented by sets ofhorizontal lines. We will discuss properties that characterize interval orders and recent research inwhich we characterize a related class. Our results include an e�cient algorithm to construct theintervals in a representation.

Di�erential Equations I JEM 362Chair: Lucy Spardy

10:00-10:15 Comparison of KDV and RLW PDE models of shallow-water waves (Level 1) DAMARIS ZA-CHOS, GIANMARCO MOLINO (UNIVERSITY OF HARTFORD) We study traveling wave solutions of theKorteweg-de Vries Equation (KDV): 𝑢𝑡 + 𝑎𝑢𝑢𝑥 + 𝑢𝑥𝑥𝑥 = 0 and Regularized Long-Wave Equa-tion (RLW): 𝑢𝑡 + 𝑢𝑥 + 𝑎𝑢𝑢𝑥 + 𝑢𝑥𝑥𝑡 = 0. Even though the equations di�er from each other,both have been used to describe shallow-water waves because their traveling wave solutions aresimilar to each other for certain parameters. The objective of our research is to compare numer-ical properties of the two in terms of speed and stability. We support our results with numericalsimulations.

10:20-10:35 Soliton escape velocity after scattering for a nonlinear Klein-Gordon partial di�erential equation(Level 2) SALEM MOGES, REID BASSETTE (UNIVERSITY OF HARTFORD) We study the scattering oftraveling wave solutions (kink and anti-kink) of a nonlinear Klein-Gordon equation: 𝑢𝑡𝑡 + 𝑏𝑢𝑡 =𝑎𝑢𝑥𝑥 − 𝑢+ 𝑢3. Waves traveling in opposite directions collide and the number of collisions dependson the initial velocity. After colliding a number of times, the waves escape from each other with a

constant escape velocity. The objective of our research is to relate the initial and escape velocitiesand the number of collisions.

10:40-10:55 Self-trapping Behavior of One Dimensional Solitons with a Sign-changing Nonlinearity Coef-ficient (Level 2) SAMUEL MCLAREN (WESTERN NEW ENGLAND UNIVERSITY) In the present work,we consider the Nonlinear Schrödinger (NLS) equation in one dimension with a sign-changingnonlinearity coe�cient. The emphasis is on the study of the dynamics of solitons, with either Hy-perbolic Secant or Gaussian profiles, as they transition from the defocusing to focusing NLS. Thiswork is motivated by advancements in the study of Bose Einstein Condensates (BECs). We exam-ine the behavior of these profiles by discretizing the system and varying the control parameters forboth the nonlinearity function and the initial profile. We find that for certain parameter regimeswe typically get either dispersion or a combination of self-trapping solitons and dispersion, pastthe sign-change of the nonlinearity coe�cient.

Geometry I JEM 281Chair: Eva Goedhart

10:00-10:15 Hyperbolic Crocheting (Level 1) BETHANY RAMRATH (SAINT MICHAEL’S COLLEGE) Hyper-bolic geometry is the least known of the geometries, and the best way to explore this topic is withthe crochet coral reef project. First one may ask “why crochet?” Then perhaps “why coral reefs?”Crochet is the most pliable three dimensional representation of a hyperbolic curve a person cancreate. As for the coral reefs they are the most naturally occurring hyperbolic structures that canbe found in the world. The crochet coral reef project was originally developed to bring aware-ness to global warming and its e�ects on the Great Barrier Reef. Australian twins Margaret andChristine Wertheim brought this project to life in 2005. Not only did this project bring aware-ness to the Great Barrier Reef, but it brought awareness to math. Through this project we will beable to explore parallel lines, areas, and angles of triangle on hyperbolic planes. We will calculatedistances distortions and curvature. There is so much to explore with hyperbolic geometry andcrochet gives the best representation to start our exploration.

10:20-10:35 Noether’s Theorem: Symmetry and Conservation (Level 1) TRISTAN JOHNSON (UNION COL-LEGE) A common problem is to find an input that optimizes (maximizes or minimizes) a function.An extension of this problem is to find a function that optimizes an expression depending on thefunction. We will look at how small (di�erentiable) variations of functions give us more infor-mation about expressions dependent on these functions. Specifically, Noether’s Theorem statesthat in a system of functions, each di�erential symmetry – or small variation where the system isinvariant– constructs a conserved quantity for the system. We will apply this to physical examplesunder the Least Action Principle such as the wave equation, Schrödinger’s equation, or electro-magnetism.

10:40-10:55 Archimedean Solids (Level 1) CELSEY LUMBRA, MACKENZIE EDMONDSON (SAINT MICHAEL’SCOLLEGE) This paper will examine the mathematics behind Archimedean solids with a primaryfocus on the truncated icosahedron. We will begin with a history of polyedra and the di�erencebetween the nature of Platonic and Archimedean solids. After a brief introduction of the polyhe-dra, we will traverse through the mathematics behind the truncated icosahedron while analyzingits properties. We will look at the object through several lenses, primarily algebraic ones, in whichwe examine symmetry groups and how they apply to the object. Subsequent sections will look atorbits and stabilizers as well as stellations. The final section will examine our process of render-ing a 3D version of a truncated icosahedron and the mathematics utilized to get us there via thecomputer algebra system Maple.

Linear Algebra JEM 377Chair: William Zwicker

3:30-3:45 Computing stability of neural activity (or, can neurons compute the 𝐿2 norm?) (Level 1) REBECCAWARZER (BENNINGTON COLLEGE) Recent research has found that the cortical activity correspond-ing to a conscious percept can be described as a stable attractor in state space. The stability of the

neural activity associated with stimulus perception is computed by taking the norm of the sum ofnormed vectors, where the vectors are either MEG sensor data or firing rates of neurons in a net-work. If neural activity is more stable when stimuli are perceived, then it may be important for aneuron or network of neurons to be able to compute the stability of neural activity such that thestimulus representation can move on to later processing stages. I propose a mathematical model ofa neuron that computes the stability and directional variance of a network by implementing thenormalization function in a biologically plausible way.

3:50-4:05 A Speculation on “Spectral Analysis of the Supreme Court” (Level 2) SCOTT OGLE (WHEATONCOLLEGE) Lawson, Orrison, and Uminsky use techniques from Voting Theory and Linear Al-gebra to analyze case splits from the Rehnquist Supreme Court from 1994 to 1998. From theorthogonal decomposition of vectors corresponding to voting splits, we can identify the degreeto which a single Supreme Court Justice or minority sub-group a�ected the outcome of a givencourt case. In this talk, we will discuss the techniques used in the paper and apply them to datafrom the Roberts Supreme Court from 2010 to 2015.

4:10-4:25 The Hadamard Product, Inverse, and Square Root for Matrices (Level 2) ROBERT REAMS (SUNYPLATTSBURGH) Instead of the usual matrix product, inverse, and square root of matrices, we willdefine the Hadamard (or entry-wise) product, inverse, and square root for matrices. We will showthat there are some nice results that come from these alternative definitions.

Number Theory I JEM 168Chair: Paul Friedman

10:00-10:15 Primality Tests Based on Eisenstein Integers (Level 1) MIAOQING JIA (UNION COLLEGE) Ac-cording to a theorem of Berrizbeitia, a highly e�cient method for certifying the primality of aninteger 𝑁 ≡ 1 (mod 3) can be created based on pseudocubes in the ordinary integers Z. Williamsand Woodings move this method into the Eisenstein integers Z[𝑤] and define a new term, Eisen-stein pseudocubes. They create a new algorithm in this context to prove primality of integers𝑁 ≡ 1 (mod 3) in a shorter period of time. We will look at the Eisenstein integers and analyzehow to use the technique of congruential sieving to compute a table of Eisenstein pseudocubese�ciently.

10:20-10:35 Applications of Elliptic Curves in Cryptography and Number Theory (Level 1) KELLY ISHAM(SKIDMORE COLLEGE) Elliptic curves are algebraic curves of the form 𝑦2 = 𝑥3 + 𝑎𝑥 + 𝑏 over afield 𝐾. These curves form a group over 𝐾 under an addition operator that is defined using thecurves’ special structure. This talk will focus on elliptic curves over finite fields. Over such curves,we can create secure cryptosystems that work e�ciently and require smaller keys than cryptosys-tems based on factoring, such as RSA. We can also create analogs to classic number theoretic algo-rithms such as Pollard 𝑝− 1.

10:40-10:55 Further Studies on the Diophantine Equation (𝑎2𝑐𝑋𝑘 − 1)(𝑏2𝑐𝑌 𝑘 − 1) = (𝑎𝑏𝑐𝑍𝑘 − 1)2 for𝑘 = 6 (Level 2) CATRICE CHONG, ELIZABETH MCGRADY, GAYEE PARK (SMITH COLLEGE) In thispresentation, we examine the Diophantine equation (𝑎2𝑐𝑋𝑘 − 1)(𝑏2𝑐𝑌 𝑘 − 1) = (𝑎𝑏𝑐𝑍𝑘 − 1)2

for 𝑘 = 6. In the previous study by Goedhart and Grundman, it has been proven that there areno positive integer solutions for 𝑘 ≥ 7. Our research is a continuation of this study, where weattempt to narrow the possible solution sets for 𝑘 = 6.

Paradoxes JEM 375Chair: Daniel Velleman

10:00-10:15 The Braess Paradox (Level 1) NICHOLAS HARDING (SUNY PLATTSBURGH) The topic thatis going to be covered is the Braess Paradox. The paradox, discovered by Dietrich Braess in 1968,deals with the surprising and counterintuitive result of adding a new pathway to a congested sys-tem.

10:20-10:35 The Banach-Tarski Paradox (Level 2) ADAM DANIERE (HAMILTON COLLEGE ) This presen-tation will demonstrate the Banach-Tarski paradox, in which one topological ball becomes two,using mathematics generally accessible to undergraduates. Specifically, I will define free groups on

two generators and use group actions and homomorphisms to sketch how assuming the axiom ofchoice leads to this surprising result.

10:40-10:55 A Poker Paradox (Level 1) GREGORY QUENELL (SUNY PLATTSBURGH) We rank pokerhands according to their frequencies: a rarer hand always beats a more common one. A straightflush (the rarest) beats four of a kind, which beats a full house, and so on. When we introduce oneor more wild cards, the frequencies change so that, for example, two pair becomes a rarer handthan three of a kind. So should two pair beat three of a kind? If it does, then three of a kind sud-denly becomes the rarer hand, and so it should beat two pair. With wild cards in the deck, there isno ranking of the usual poker hands that is consistent with their frequencies.

Statistics Ia STE 104Chair: Phil Yates

10:00-10:15 A Sports Analytics Consulting Course: Part I (Level 1) MICHAEL SCHUCKERS (ST. LAWRENCEUNIVERSITY) In this talk, I describe a course taught at St. Lawrence University during the Fall2015 semester. As part of that course thirteen students were statistical consultants for five St.Lawrence University sports teams: men’s soccer, women’s soccer, track, volleyball and women’shockey over the course of the semester. This talk will discuss the motivation for the course, thestructure of the course, the outcomes of the course and lessons learned from running the coursefrom a faculty perspective.

10:20-10:35 A Sports Analytics Consulting Course: Part II (Level 1) KELSEY A. WEST, CHRISTOPHER J.ROMAGNA, BAILEY J. O’KEEFFE, SHAUNA BULGER, MATTHEW MONHART, MICHAEL A. THEOBALD,SYDNEY A. BELL, JOSIAH BARTLETT, MICHAEL L. LENGIEZA, ALEXANDRIA J. HAEHL, JOHN H.TANK III, CURTIS J. HURLBUT, JULIA H. SIMOES, COLTON F. RANSOM (ST. LAWRENCE UNIVER-SITY) In this talk, we describe student projects done as part of a course taught at St. LawrenceUniversity during the Fall 2015 semester. As part of that course thirteen students were statisticalconsultants for five St. Lawrence University sports teams: men’s soccer, women’s soccer, track,volleyball and women’s hockey over the course of the semester. Throughout the semester, the stu-dent collected and analyzed data for each sport. A summary of the work done for each team willbe presented. Further, we present some of the student final projects that were part of this course aswell as discuss the challenges and rewards of this course.

10:40-10:55 Predicting the NCAA Men’s Postseason Basketball Poll More Accurately (Level 2) JOHN A.TRONO (SAINT MICHAEL’S COLLEGE) A previous study investigated how well a linear model couldpredict where teams would be ranked in the final NCAA coaches’ poll (for men’s basketball)which is announced right after the post season, single elimination, championship tournament(known as March Madness) has concluded. Monte Carlo techniques were able to improve uponthose results, which were obtained via a weighted, linear regression model. This Monte Carlo ap-proach produced a model whose Spearman correlation coe�cients were roughly equal to 0.85 forthe top 15, top 25 and top 35 teams, respectively, with regards to said final poll. This article willdescribe a non-linear model that is approximately 10% more accurate than the previous model,and incorporates Zipf’s law — and a quantity known as the Tournament Selection Ratio.

Statistics Ib STE 102Chair: Jessica Mao

10:00-10:15 Statistical Examination of Female Representation in Oscar Best Picture Nominated Films(Level 1) BAILEY O’KEEFFE (ST. LAWRENCE UNIVERSITY) Due to criticism about the gender dis-crepancy in Oscar-nominated films, we analyzed the screen time of male and female leads in filmsnominated for Best Picture from 2006-2014 using R and Minitab. We discuss the di�erencesfound between lead screen times according to year and director gender, the relationship betweenhow long a female lead is on screen and how likely it is that a film will win the Oscar, a logisticregression model that we built to predict Oscar Winners, and also the lack of racial diversity inthese films.

10:20-10:35 Giving Real People Access to Big Data — Analyzing Bike Rentals in NYC (Level 1) XUE-HANG PAN (ST. LAWRENCE UNIVERSITY) We are now in an era of “Big Data” but challenged tofind ways for people to extract meaning from the data e�ectively. We discuss the process of datascraping, building a database, and giving a user tools to investigate the data. We build these toolsusing R and provide a user interface with Shiny apps. We illustrate these ideas using data from theCiti Bike service in New York City that covers 330 stations and millions of rentals in 2015.

Topology JEM 373Chair: Adam Lowrance

10:00-10:15 Non-Orientable Surfaces (Level 1) MARCELLA DALEY (ST. MICHAEL’S COLLEGE) In thistalk, we will define non-orientable surfaces and discuss their essential properties. Three exam-ples of non-orientable surfaces are the Möbius strip, Klein bottle, and Boy’s surface. Möbius stripsare non-orientable surfaces with boundary, meaning that they only have one side and one edge.Half-twisting a strip of paper once, then attaching the ends together, makes the simplest Möbiusstrip. Klein bottles are non-orientable, closed manifold, one-sided surfaces. Attaching two Möbiusstrips together creates a Klein bottle, although unlike a Möbius strip, they cannot be embeddedin R3. Boy’s surfaces are real projective planes immersed in three dimensions. A Boy’s surface iscreated by manipulating one cap of a sphere. They also cannot be embedded in R3 without self-intersection.

10:20-10:35 The Influence of Distinct Topological Spaces on the Complex Analytic Functions that may AriseUpon Them (Level 2) EVAN GALL (BENNINGTON COLLEGE) With two poles of a galvanic batteryin hand we ask, how do di�erent topological spaces a�ect the types of steady steamings (electricfield lines) that may arise upon them? Drawing Upon Felix Klein’s On Riemann’s Theory of Alge-braic Functions and their Integrals, we explore how complex analytic functions and their integrals arethe most general class of functions that describe electrostatics on surfaces, how the constructionof these functions changes depending on the surface they are applied to, and how these surfacesa�ect the properties of these functions.

10:40-10:55 An Interesting Invariant for Generic Smooth Closed Planar Curves (Level 1) PATRICK DRAGON(BARD COLLEGE AT SIMON’S ROCK) Fabricius-Bjerre’s publication in 1962 established the existenceof an invariant for generic, smooth, closed curves in the plane. Since then, the result has seen ex-tensions to curves with cusps, as well as to curves in other topological spaces, including 𝑆2, RP2,and R3. This presentation will include the statement and a proof of the original result for planarcurves.Let 𝛾 be a generic, smooth, closed curve in the plane. Let 𝐶 be the number of crossings in 𝛾. Let𝐼 be the number of inflections in 𝛾. We sort the doubly-tangent lines to 𝛾 into two types; thenumber of type-one doubly-tangent lines denoted by 𝑇 and the number of type-two doublytangent lines denoted by 𝑆. A doubly-tangent line 𝐿 is called type one if both its nearby neigh-borhoods in 𝛾 occur on the same side of 𝐿. A doubly-tangent line 𝐿 is called type two if its nearbyneighborhoods in 𝛾 occur on opposite sides of 𝐿. Our invariant is given by the surprisingly sim-ple formula 𝑇 − 𝑆 = 𝐶 + 1

2𝐼 .

PARALLEL SESSIONS TWO

Abstract Algebra II JEM 373Chair: John McHugh

1:40-1:55 Subnormal Subgroups of M-Groups (Level 2) JOHN MCHUGH (UNIVERSITY OF VERMONT) Awealth of structural information about a finite group can be obtained by studying its irreduciblecharacters. Of particular interest are monomial characters — those induced from a linear char-acter of some subgroup — since Brauer has shown that any irreducible character of a group canbe written as an integral linear combination of monomial characters. Our primary focus is theclass of M-groups, those groups all of whose irreducible characters are monomial. A classical the-orem of Taketa asserts that an M-group is necessarily solvable, and Dade proved that every solv-able group can be embedded as a subgroup of an M-group. After discussing results related to M-groups, we will construct explicit families of solvable groups that cannot be embedded as subnor-mal subgroups of any M-group.

2:00-2:15 Generalized splines on cycles (Level 1) LINDSAY DEVER, MEREDITH WILDE (SMITH COLLEGE)A generalized spline is a vertex-labeling on an edge-labeled graph so that the di�erence betweenadjacent vertices is a multiple of the edge label. This generalizes the definition of a spline in ap-plied math. This talk will discuss new results about bases for splines on cycles with edges labeledwith multivariate polynomials.

2:20-2:35 Bases for generalized splines on infinite graphs (Level 1) CLAUDIA YUN, STEPHANIE WEBSTER,JULIA GIBSON (SMITH COLLEGE) Generalized splines are solutions to systems of congruencesmod 𝑛, represented by vertex labels on an edge-labeled graph. They are studied in connectionwith algebraic geometry and cohomology rings. This talk discusses a problem in generalizedsplines significant to a�ne Springer varieties. In particular, we explore finding bases for splineson lattice graphs.

Applied Math IIa JEM 378Chair: Ellen Gasparovic

1:40-1:55 Investigating Extreme Temperatures (Level 1) A. VAN RYZIN (UNION COLLEGE) Climatechange has a big impact on the manifestation of extreme high and low temperatures in the US.What correlations exist between the extreme temperatures and geographic variables? How canwe use statistics, spatial mapping, and data visualization to understand the phenomena? How canwe e�ciently acquire useful data? What trends and patterns do we observe over time? To answerthese questions, we first implement a Python program to extract data from a weather website. Wethen use ArcGIS to compile, analyze, and map the geographic information. Using statistical anal-ysis, we produce correlations between the extreme temperatures and other variables, such as lat-itude, longitude, population, etc. Lastly, we display the data in a way that is accurate, schematic,and provocative.

2:00-2:15 Spinning tops: physics that will make you dizzy (Level 2) NATE HODGE, LOGAN DAVID (SAINTMICHAEL’S COLLEGE) While a spinning top might be considered a kid’s toy, the mathematics be-hind a top’s motion is not child’s play. We will explore the mathematical physics that allows theserigid bodies to seemingly defy gravity. Specifically, we will investigate the movement of tops withsharp and rounded tips, as well as tippe tops, which, when spun at a high enough velocity, becomeinverted and spin with their stems pointing downward. In order to demonstrate the physics thatdepicts the motion of the tops, we will be working with two sets of axes that contain sets of three-dimensional vectors representing the movement in di�erent directions. By delving into the forcesacting on the top, we will describe the e�ect of forces such as the inertial force of rotation, theCoriolis force, and the centrifugal force. We will focus on how these forces influence the angular

velocity and angular momentum of the top’s movement and how each of these interact with thedi�erent dimensions of torque. Three constants of motion — the total energy constant, the Jelletconstant, and the Routh constant — and their relations to the components angular velocity willalso be discussed.

2:20-2:35 Using Weierstrass Elliptic Functions to Look at Motion in an Asymmetric Double Well Potential(Level 2) MELISSA WESTLAND (SAINT MICHAEL’S COLLEGE) This presentation is an overview ofmy work for the senior honors capstone. While previous solutions for the periodic behavior in anasymmetric well rely on stereographic projection, the solution to the problem has in this workbeen found explicitly in terms of the Weierstrass elliptic function. In achieving the solution, wepropose a new way of categorizing the periods of oscillation in the potential.

2:40-2:55 The Uncertainty Principle and Coherent States (Level 2) SERGEY AFINOGENOV (WESTFIELDSTATE UNIVERSITY) The Uncertainty Principle is a fundamental fact about the behavior of quan-tum mechanical systems. I will give a rigorous mathematical statement of the a general form ofthe Uncertainty Principle, and then show how for the original uncertainty relation studied byHeisenberg (relating position and momentum measurements) that a special class of states (knownas coherent states) saturate the inequality. If time permits we’ll explore some interesting propertiesof coherent states.

Applied Math IIb JEM 166Chair: Je� Jauregui

1:40-1:55 Modeling Oyster Guts with a Dual Bead Method (Level 1) SEAN KRAMER (NORWICH UNI-VERSITY) This talk will focus on an interdisciplinary project in which the digestive processes ofbivalve larvae are modeled based on experimental measurements. This talk will be less techni-cal biologically, and will highlight the results from working with a group of marine biologistson the following project. Planktotrophic bivalve larvae form a vital part of the oceanic ecosys-tem. It is therefore extremely valuable to learn about digestive mechanics in order to understandthe extent to which these organisms a�ect ecology in the water column. Using a series of expo-sures to di�erently colored fluorescent polystyrene microbeads, we model larval guts as a contin-uously stirred tank reactor (CSTR), plug flow reactor (PFR) or combinations of the two in se-ries. We also varied several experimental conditions to understand how these a�ected estimates ofgut kinematic parameters. We found the larval guts of M. galloprovincialis aged 2 and 7 days post-fertilization were best described either as a CSTR or CSTR in series with a PFR. Reactor modelsprovided estimates of ingestion rates, which were compared to those obtained by other authorswho measured rates of bead accumulation. Collectively, these studies provide new insight on thedigestive strategy of planktotrophic bivalve larvae.

2:00-2:15 Modeling Communities of Plants in a Patchy Landscape (Level 1) MIEKE VRIJMOET (BENNING-TON COLLEGE) Patches of unusual soil types, such as toxic Serpentine soils, disrupt the uniformdistribution of population of plants in a landscape. While predicting how the density of a singlespecies of plant may respond to these patches is a direct function of the area of patches, deter-mining how assemblages of plants with various physiologies might respond is a more complexquestion for ecologists. Studying these assemblages in the field also o�ers a limited range of spatialconfigurations of landscapes and scales at which to sample, making it di�cult to tease out the in-fluence of the geometric configuration of the patches on these assemblages. Using a model, I testhow possible pools of “species”, distinguished by a reduced set of physiology parameters, mightrespond to various geometric configurations and qualities of patches on a landscape. I then con-sider to what extent the theoretical experiment reflects observed patterns and theories of commu-nity assemblage in Serpentine and other analogous systems.

2:20-2:35 A Study of the E�ects of Greenhouse Gas Emissions on Global Temperature (Level 1) CHERYLHOLMES, ELISE REED, LAUREN WHITE (SMITH COLLEGE) Climate modeling is the study of globalclimate systems using mathematics. As we notice our impact on the environment, it is importantto understand the e�ects of human activity on the long-term health of our planet. A common ande�cient way to model these e�ects is by using computer simulations and di�erential equations.

This talk discusses the relationship between anthropogenic greenhouse gas emissions and climatechange, describing some results about what our climate could look like in the near future.

2:40-2:55 Population Models and the Logistic Equation: the Importance of Being Discrete (Level 1)ZSUZSANNA M. KADAS (SAINT MICHAEL’S COLLEGE) The continuous and discrete logistic equationlook alike. But do they always give the same predictions? We’ll illustrate the “di�erence” made bydiscretization and make the case that discrete time models should be explored alongside continu-ous ones to get the complete picture.

Combinatorics II JEM 364Chair: David Vella

1:40-1:55 The Wonderful World of Generating Functions (Level 1) DAVID VELLA (SKIDMORE COLLEGE)Given a sequence of numbers, the ordinary generating function of the sequence is the formal powerseries in 𝑥 with the coe�cients coming from the sequence: 𝑓(𝑥) =

∑︀𝑛≥0 𝑎𝑛𝑥

𝑛. Another usefulpower series associated with the sequence is the exponential generating function, which is the fol-lowing power series: 𝑔(𝑥) =

∑︀𝑛≥0 𝑎𝑛(𝑥

𝑛/𝑛!). By converting questions about sequences intoquestions about functions, these tools provide a bridge between discrete mathematics (combina-torics, graph theory, number theory, etc.) and continuous mathematics (calculus and analysis). Forexample, either type of generating function can be used to solve recurrence equations easily. Ad-ditionally, I have discovered a way to easily write down what happens when generating functionsare composed, opening up new approaches to discovering and proving combinatorial identities.This spring at Skidmore College, the senior seminar class has been studying generating functions.After I introduce the basic notions in this talk, and explain the method for composing them, therewill follow several talks by the students of the seminar, who will illustrate the use of generatingfunctions in both solving recurrences and in proving combinatorial identities.

2:00-2:15 The Wiener Index for Families of Graphs (Level 1) JULIE BRYANT, KHALIL HALL-HOOPER,GUGA VARIASHVILI (SKIDMORE COLLEGE) Let 𝐺 be a connected graph with vertices 𝑣1, 𝑣2, . . . , 𝑣𝑛.Let 𝑑𝑖𝑗 denote the distance from vertex 𝑣𝑖 to vertex 𝑣𝑗 ; that is, the minimal number of edges tra-versed in a path from 𝑣𝑖 to 𝑣𝑗 . The Wiener Index 𝑊 (𝐺) of the graph 𝐺 is defined as the sum ofthe distances between all distinct pairs of vertices on the graph: 𝑊 (𝐺) =

∑︀1≤𝑖<𝑗≤𝑛 𝑑𝑖𝑗 . Finding

the Wiener index of a single graph is simply a matter of finding the distances and adding themup. However, there are many infinite families of graphs indexed by 𝑛 which are important ingraph theory, such as the path graphs 𝑃𝑛, the cycle graphs 𝐶𝑛, the wheel graphs 𝑊𝑛, the completegraphs 𝐾𝑛, and the complete bipartite graphs 𝐾𝑚,𝑛, to name just a few families. For such families,the Wiener index should be a predictable function 𝑤(𝑛) of 𝑛. In this talk we show how to com-pute the Wiener index of some families by deriving a recurrence relation for 𝑤(𝑛) and then usinga generating function approach to solve the recurrence.

2:20-2:35 Counting Restricted Compositions of an Integer n, I (Level 1) CALLUM LANE, DANIEL PINCUS(SKIDMORE COLLEGE) Given a positive integer 𝑛, a partition of 𝑛 is merely a way of writing 𝑛 as asum of positive integers, where the order of the summand doesn’t matter. For example 3+2, 1+4,and 1 + 1 + 3 are all distinct partitions of 5, but we make no distinction between 2 + 3 and 3 + 2.A composition of 𝑛 is merely a partition where the order does matter, so we would distinguish be-tween 2+3 and 3+2. Thus, instead of a set of positive integers whose sum is 𝑛, a composition is anordered 𝑘-tuple of positive integers whose sum is 𝑛, for some 𝑘. So we would write compositionsas (1, 1, 3) or (1, 3, 1) rather than as 1 + 1 + 3 or 1 + 3 + 1.

It is not di�cult to show that the total number of all compositions of 𝑛 is 2𝑛−1. However, in thistalk, we are interested in counting the compositions of 𝑛 which have some restrictions on thesummands (which are known as ‘parts’ of the composition.)

For example, we could ask the question: How many compositions of 𝑛 are there where the last part isan odd number? If we call that number 𝑝(𝑛), we can find a formula for 𝑝(𝑛) as follows. First, wefind a recurrence relation that the numbers 𝑝(𝑛) satisfy. Then we translate the recurrence intoan equation involving the (unknown) generating function of the sequence {𝑝(𝑛)}. Solving this

equation will tell us what the generating function is, and then we can read o� our desired formulafor 𝑝(𝑛) from the generating function.Other types of restrictions on the parts are possible, but this example su�ces to illustrate the tech-nique, which is a routine way of using generating functions in combinatorics — solving recur-rence relations. In the next talk, another group of students from the seminar will discuss othertypes of restrictions on the parts, and will use an entirely di�erent (and not so routine) applicationof generating functions to count the compositions.

2:40-2:55 Counting Restricted Compositions of an Integer 𝑛, II (Level 1) DEREK HALDEN, JAKE RATKEVICH(SKIDMORE COLLEGE) Given a positive integer 𝑛, a partition of n is a way of writing 𝑛 as a sum ofpositive integers, where the order of the summand doesn’t matter. For example 3 + 2, 1 + 4, and1 + 1 + 3 are all distinct partitions of 5, but we make no distinction between 2 + 3 and 3 + 2.A composition of 𝑛 is merely a partition where the order does matter, so we would distinguish be-tween 2+3 and 3+2. Thus, instead of a set of positive integers whose sum is 𝑛, a composition is anordered 𝑘-tuple of positive integers whose sum is 𝑛, for some 𝑘. So we would write compositionsas (1, 1, 3) or (1, 3, 1) rather than as 1 + 1 + 3 or 1 + 3 + 1.It is not di�cult to show that the total number of all compositions of 𝑛 is 2𝑛−1. However, in thistalk, we are interested in counting the compositions of 𝑛 which have some restrictions on thesummands (which are known as ‘parts’ of the composition.)For example, we could ask the question: How many compositions of n are there using only parts at most2? Or how many using only odd numbers as parts? Other types of restrictions on the parts are of in-terest. In a previous talk, other students from the seminar discussed how to answer questions likethese by using generating functions to solve recurrence equations. In this talk, we use generatingfunctions in a completely di�erent way. By expressing these generating functions as compositesof other functions, are able to gain some insight into counting compositions with various types ofrestrictions.

Computer Science JEM 389Chair: William Dundar

1:40-1:55 P = NP and Voting Theory (Level 2) WILLIAM S. ZWICKER (UNION COLLEGE) With morethan two candidates, many di�erent voting rules are possible. For most, calculating who won theelection (given the ballots cast) is computationally simple. Since 1989, however, it’s been knownthat calculating the Kemeny winner (proposed by John Kemeny, former president of DartmouthCollege, and co-inventor of BASIC, the first high-level computer language) is NP-hard. Whatexplains the di�erence between easy and hard here? We locate the exact “intractability boundary”for this context. A key role is played by majority cycles that lie beneath the surface — they arerevealed via the same orthogonal decomposition that serves as the basis for Kirchho�’s Laws inelectric circuit theory.

2:00-2:15 The Dictionary Problem (Level 1) MARGO CHANIN, THADDEUS CLAASSEN (SKIDMORE COL-LEGE) The problem we attempt to solve resides in the field of computational lexicography. Ourquestion is “What are the fewest number of words one would need to know beforehand to then beable to read and understand the rest of the dictionary?” We demonstrate how we solved the prob-lem and where we assumed facts about the dictionary due to its imperfections. Finally, we showour list of words and their potential use for those interested in lexicography.

2:20-2:35 Semantic and Contextual Insight from Word Vectorization (Level 1) PETER ORZELL (CHAMPLAINCOLLEGE) Extracting meaning and context from written text has long proved to be a di�cult taskin computing. Manipulation of word data mapped to a vector space known as a word embedding isa recent strategy that has been used to gain an enormous amount of insight into the meaning be-hind text. This scheme has largely been brought to community attention by word2vec and its au-thors, T. Mikolov, K. Chen, G. Corrado, and J. Dean, in their 2013 ICLR paper, E�cient Estima-tion of Word Representations in Vector Space. In their described method, the cosine distance betweenmapped word nodes can be used as a determiner of conceptual similarity. One completely emer-gent benefit of the system is that a “conceptual algebra” can even be performed by preserving a

relationship between two concept nodes as a di�erence vector. This di�erence vector can then beapplied as a transformation to another concept node to yield a conceptually analogous result, e.g.,vector(king) − vector(man) + vector(woman) ≈ vector(queen). In this presentation, I will briefly ex-amine the mathematics behind creating a word2vec word embedding, methods for navigating theembedding mathematically, and a few of the potential insights that can be obtained from workingwith such an embedding.

2:40-2:55 Lambda Calculus: A Computational System of Mathematics (Level 1) SCOTT BARRETT (CHAM-PLAIN COLLEGE) Lambda calculus is a system formalized by Alonzo Church as a way of expressingcomputation to solve any mathematical problem. It was formalized around the same time as itsbetter-known cousin, the Turing machine, and was invented to solve the same problem in math-ematics. Church’s paper was released a year before Turing’s. Lambda calculus is an extremely el-egant system that is logically equivalent to Turing machines, but is less intuitive to grasp. Bothformal systems have inspired many influential programming languages, but the languages thatspawned from each school of thought are as radically di�erent as the systems themselves. This talkwill explain the basics of lambda calculus and explore how it relates to modern-day computer sci-ence.

Di�erential Equations II JEM 362Chair: Jenna Reis

2:00-2:15 Fourier Series (Level 1) PAULETTE KLEIN (RUSSELL SAGE) Fourier Series are used to solveLaPlace’s equation, Poisson’s equation, the di�usion of heat flow equation, wave equation,Helmholtz equation, and Schrödinger equation. The results will be how to solve all di�erent typesof equations using Fourier Series. Also looking at di�erences and similarities between Fourier Se-ries, Taylor Series, and LaPlace Transformers.

1:40-1:55 Darboux Transformations in Type 1 (Level 1) GABRIELLE BUCK (STATE UNIVERSITY OF NEWYORK NEW PALTZ) For every linear partial di�erential or ordinary equation, one can consider thecorresponding linear partial di�erential or ordinary operator. Darboux Transformations of Type Iare transformations of linear partial di�erential operators, that can be used, for example, to trans-form the operator corresponding to a di�cult to solve equation into the operator correspondingto an easier equation.We concentrate on equations of third order and of two independent variables. The action of Dar-boux Transformations can be seen with the consideration of gauge di�erential invariants of thecorresponding operator. Di�erential invariants encode the essential information about the opera-tor, and by studying their transformations, we see how that essential information changes.Using these tools we investigate possible orbits of Darboux Transformations of Type I. We willlook at shapes and patterns that they give rise to.

2:20-2:35 Supersymmetric Darboux Transformations and Wronskian (Level 2) SIMON LI (SUNY NEWPALTZ) Darboux transformations are a way of solving di�erential equations such as the Strum-Louville and KdVs. In the past decades the study of supersymmetry has introduced “super” ver-sions of these familiar problems. Using what we know of the classical Darboux and Crum the-orem and its extension in the supercase and using the properties of the berezinian, we constructthe super-Wronskian, an analogue of the Wronskian determinant, to find a way to express Dar-boux tranformations of non-degenerate supersymmetric integrable systems in terms of their solu-tions.

Geometry II JEM 281Chair: Ockle Johnson

1:40-1:55 Generating Spirographs and the Mathematical Rose (Level 1) KATELYN BANIA (SAINT MICHAEL’SCOLLEGE) The purpose of this research is to delve into the mathematical background of an aes-thetically appealing Spirograph as well as focus on the similar concept of the Maurer Rose. Wewill examine what a Spirograph is, what characteristics it has, and what field of mathematics these

types of designs (and other related objects) belong to. Most importantly, we will discuss the pro-cess of how to generate Spirograph designs without the use of interlocking templates (such asgears and wheels) and instead with their underlying mathematical equations.

2:00-2:15 Equidistant Sets of Curves (Level 1) MARK HUIBREGTSE (SKIDMORE COLLEGE) The equidistantset, or midset, of two sets in the plane is the set of points that are equidistant from each of the twosets. We will consider a couple of methods for computing the midset (at least approximately) oftwo curves that are given as graphs of functions. One possible application of such methods wouldbe to locate the boundary between two jurisdictions that are separated by a river, where theboundary line (running through the river) is to be the equidistant set defined by the two shore-lines.

2:20-2:35 Triples of Integers and Associated Triangles (Level 1) VINCENT FERLINI (KEENE STATE COLLEGE)If 𝑚 and 𝑛 are integers with 0 < 𝑛 < 𝑚 then the triple (𝑚2 − 𝑛2, 2𝑚𝑛,𝑚2 + 𝑛2) are integersides of a triangle with a right angle. These are also known as Pythagorean triples. In this talk,we shall present three other triples of integers based on 𝑚 and 𝑛 that characterize other types oftriangles.

2:40-2:55 Stereographic Projections of Loxodromes on a Sphere (Level 1) KRISTEN MCCARTHY, (SAINTMICHAEL’S COLLEGE) The piece I have created is a stereographic projection of multiple sphericalspirals which are specific forms of loxodromes. In order to understand this we must understandthe basics of a logarithmic spiral. A logarithmic spiral to a plane is like a loxodrome to a sphere.A loxodrome, also known as a rhumb line, cuts all meridians at the same angle. Common to ge-ography of the earth, a part of a loxodrome is not the shortest distance between two points on asphere like the globe, but rather the direction you would take if you followed your compass at anexact direction throughout your entire trip. A spherical spiral is a special case of loxodrome andcan be given by a set of three parametric equations which I have altered to make the 3D printedshape. As for the projection there is a specific formula for projection radiating from a certain pointto a plane tangent to the sphere. As seen in my piece spherical spirals project logarithmic spirals.Altering the point of projection can give very interesting projections onto a plane.

Graph Theory JEM 380Chair: Adam Lowrance

1:40-1:55 Expose’ of Running Clubs — Combinatorial Investigation (Level 1) JOHNATHON HOLBROOKS(SUNY POTSDAM) Consider the graph consisting of 𝑛-squares joined together in a straight line.Following the techniques presented by Nissen and Taylor, we will show how many distinct trailssuch a graph contains. We will give both a recursive and a direct formula for the solution.

2:00-2:15 Generating Solutions to the N-Queens Problem Using 2-Circulants (Level 1) SEBASTTIANHOWARD, RYAN BARSTOW (SUNY POTSDAM) The 𝑁-Queens problem is to place 𝐵 mutuallynonattacking queens on an 𝑁 ×𝑁 chessboard. We show a constuction, due to Erbas and Tanik, togenerate one solution for any 𝑁 ≥ 4 using 2-circulants.

2:20-2:35 The Prison Epidemic: Using Graph Theory to Model Contagion (Level 1) SAMANTHA TREM-BLAY, CONOR DISHER (SAINT MICHAEL’S COLLEGE) The rate of incarceration in the UnitedStates has reached a point that it can be modeled as an epidemic. Various sources have used theSusceptible-Infected-Susceptible statistical model to show the spread of incarceration. Using graphtheory and network theory, the data can be clarified, along with the relationship between incar-ceration of an individual and future incarceration of the family members of that person. This canbe explained by the financial and social strain put on the family of an inmate. We examine thebest ways to model incarceration as an infectious disease. We will demonstrate simple networkexamples, as well as using Python code and current data to model incarceration in the UnitedStates.

2:40-2:55 Minimal Length Maximal Green Sequences for Type A Quivers (Level 1) E. CORMIER, P.DILLERY, J. RESH, J. WHELAN (VASSAR COLLEGE) The study of maximal green sequences (MGS)is motivated by string theory, in particular Donaldson-Thomas invariants and the BPS spectrum.This concept can also be examined through the framework of 𝜏-tilting modules in representation

theory. It is known that triangulations of disks with no punctures yield type A quivers. B. Kellerintroduced green mutations and the corresponding MGS’s. These sequences can be studied boththrough the combinatorial transformations of directed graphs as well as through triangulations ofdisks.Our research focuses on maximal green sequences of minimal length for quivers mutation equiv-alent to type A quivers. It is known that each acyclic quiver has at least one minimal length MGSof length 𝑛, where 𝑛 is the number of vertices in the quiver. First, we define an algorithm thatproduces such a sequence of mutations for any given acyclic type A quiver. For cyclic type Aquivers, we define an algorithm that produces an MGS of length 𝑛 + 𝑡 where 𝑛 is the number ofvertices and 𝑡 is the number of 3-cycles in a quiver. We then proceed to show that 𝑛 + 𝑡 is alwaysthe minimal length of MGS’s corresponding to any type A quiver.

Graph Theory and Topology JEM 377Chair: Andrew McIntyre

1:40-1:55 Unmarked Length Spectrum Rigidity of Metric Graphs (Level 1) MELISSA MISCHELL, NARIN LU-ANGRATH (WESLEYAN UNIVERSITY) Our project looks at how much geometric information abouta graph is contained in the unmarked length spectrum. The length spectrum of a metric graphis the set containing all the lengths of the loops in the graph, and how many di�erent loops haveeach length. We show that if two complete graphs with rationally independent metrics have thesame length spectrum, then they are isometric. We also prove that there are only finitely manymetrics on a graph with the same length spectrum.

2:00-2:15 The 17 types of plane tiling patterns, and the Euler characteristic (Level 1) ANDREW MCINTYRE(BENNINGTON COLLEGE) The classification of the 17 symmetry types of plane tiling patterns isoften a high point of an undergraduate course in abstract algebra. The Euler characteristic of sur-faces is often a central point of an undergraduate course in topology. In fact, the two are con-nected: the classification can be proved by means of topology, using the Euler characteristic. Onewrinkle is that a tiling symmetry type is associated with a 2-dimensional orbifold, a generaliza-tion of a surface which allows cone points and fold lines. I will present the main idea of this proof.None of this is original; the proof is due to Bill Thurston, and has been popularized by John Con-way. (This is a topic from an undergraduate course, The Art of Mathematics, which was developedby Katie Montovan and which I am co-teaching with her.)

2:20-2:35 Counting intersection points of loops on a surface (Level 2) VLADIMIR CHERNOV (DARTMOUTHCOLLEGE), PATRICIA CAHN (SMITH COLLEGE) A continuous deformation of a loop on a surface iscalled a homotopy. Given two homotopy classes of loops 𝛼1, 𝛼2 on an oriented surface, it is natu-ral to ask how to compute the minimum number of intersection points 𝑚(𝛼1, 𝛼2) of loops in thesetwo classes.We show that for 𝛼1 ̸= 𝛼2 the number of terms in the Andersen-Mattes-Reshetikhin Poissonbracket of 𝛼1 and 𝛼2 is equal to 𝑚(𝛼1, 𝛼2). Chas found examples showing that a similar statementdoes not, in general, hold for the Goldman Lie bracket of 𝛼1 and 𝛼2.

Mathematics Education I JEM 168Chair: Li-Mei Lim

1:40-1:55 Understanding the Common Core Practice Standards (Level 1) ELIZABETH RAYMOND (WEST-FIELD STATE UNIVERSITY) The Common Core State Standards for Mathematics include both con-tent standards as well as the Standards for Mathematical Practice. Students who demonstrate thecharacteristics of the eight practice standards in the Common Core illustrate behaviors and skillsthat mathematicians show. Exercises that illustrate these practice standards at the high school levelwill be shared.

2:00-2:15 The Common Core State Standards in Mathematics for Students with Learning Disabilities(Level 1) KATHARINE BOUCHARD (RUSSELL SAGE COLLEGE) My program, Childhood Educa-tion/Mathematics, has influenced the direction of my research paper. Currently, I have been

researching about the Common Core State Standards (CCSS) and how they have transformedmathematics. I will also be focusing on how the mathematic standards have a�ected students (K-12) with learning disabilities/di�culties (MD) since there are rigorous demands associated withthese new standards. I hope to discover statistics that can back whether the Common Core StateStandards have improved student comprehension and test scores or if it has been detrimental tostudent’s achievement. By completing this research I hope it gives others and myself a deeper un-derstanding of these fairly new mathematic standards since there is much controversy with them.My research is not to say whether common core is the right or wrong choice, however I wantto construct hard data on how students with learning disabilities are articulating this new way ofeducation.

2:40-2:55 What is problem solving in a high school math class supposed to look like? (Level 1) MICHAELCZUPRYNA (WESTFIELD STATE UNIVERSITY) Problem solving is a crucial skill to possess whengraduating high school. The teaching strategies in most high school math classes almost assurethat students will not retain the material being taught. What strategies and questions can educa-tors ask their students to increase mind growth and problem solving skills? Exploration and exam-ples on this issue will be presented.

Number Theory II JEM 375Chair: John Trono

1:40-1:55 Arithmetic Derivative (Level 1) HEATHER PAIGHT (KEENE STATE COLLEGE) The arithmeticderivative of a natural number 𝑛 , denoted 𝑛′ , is defined as follows: If 𝑛 is prime then 𝑛′ = 1 andfor natural numbers 𝑎 and 𝑏, then (𝑎𝑏)′ = 𝑎𝑏′ + 𝑏𝑎′. This rule for (𝑎𝑏)′ has the same form as theLeibniz Rule in Calculus. This talk will explore some basic properties of the arithmetic derivativewith an emphasis on those that have analogs in Calculus. The definition will then be extended toapply to integers and rational numbers. Connections between the arithmetic derivative and twofamous unsolved problems in number theory will be included.

2:00-2:15 Hyperbolic Numbers (Level 1) KEGAN LANDFAIR (KEENE STATE COLLEGE) The algebraicequation 𝑥2 = 1 has two solutions 𝑥 = −1 and 𝑥 = 1. We assume the existence of a new number𝑢, called the unipotent, which has the property that 𝑢 ̸= −1 or 1 and that 𝑢2 = 1. The hyperbolicnumbers then are of the form 𝑎+ 𝑏𝑢 where 𝑎 and 𝑏 are real numbers. These are similar to complexnumbers which are of the form 𝑎+ 𝑏𝑖 where 𝑖2 = −1. This talk will present the basic properties ofhyperbolic numbers and emphasize the similarity with complex numbers. In addition, the Lorentzequations that relate the times and positions of an event as measured by two observers in relativemotion in Einstein’s Theory of Relativity will be derived using hyperbolic numbers.

2:20-2:35 Enumerating the Rationals (Level 1) SAM NORTHSHIELD (SUNY-PLATTSBURGH) Everyoneknows that the rational numbers are countable (i.e., can be made into a list) but not everyoneknows how to explicitly do that. Such a listing is called an enumeration. We look at three closelyrelated enumerations of the (positive) rationals: 1/1, 2/1, 1/2, 3/1, 2/3, 3/2, 1/3, 4/1, 3/4, 5/3, 2/5, . . . , and2/1, 1/1, 4/1, 3/2, 2/3, 3/1, 4/3, 1/2, 6/1, 5/3, 4/5, 7/2, . . . , and 3/1, 2/1, 3/2, 1/1, 6/1, 5/2, 9/5, 4/3, 3/4, 5/1, . . . ,and how they are generated.

2:40-2:55 Stirling’s Formula: Factorials, square roots and transcendentals, oh my! (Level 1) TOMMY RATLIFF(WHEATON COLLEGE) Stirling’s formula 𝑛! ≈

√2𝜋𝑛

(︀𝑛𝑒

)︀𝑛 is one of my all-time favorite “Are youserious?” results in mathematics. The factorial is simply a product of natural numbers, but yet weget a very good approximation for 𝑛! involving the square root of 𝜋 and powers of 1/𝑒, which aredefinitely not natural numbers. In this talk we’ll use nothing more complicated than integrationby parts and a few diagrams to develop an approximation for 𝑛! that is very close to Stirling’s for-mula.

Statistics IIa STE 104Chair: Philip Yates

1:40-1:55 Methods and Applications of Quantile Regression Models (Level 1) ELIZABETH ESCOBAR (ST.LAWRENCE UNIVERSITY) Quantile regression is a type of regression function that models rela-tionships among predictor variables using various quantiles of the response variable. In contrastto standard linear regression which models changes in the average response, quantile regressionexamines changes in the quantiles of the response variable based on changes in the predictors, giv-ing us more solid estimates based on these variations. Ecologists often use quantile regression tomeasure causal relationships, as it is a very e�ective method of predicting how di�erent quantilesof organism responses, such as population size, are a�ected by various environmental factors. It isalso an e�ective tool for detecting abnormal growth patterns in growth charts. Quantile regres-sion allows us to distinguish the complex manner in which predictors a�ect the response variable.This presentation will introduce the concept of quantile regression and discuss how some of thecommon standard regression ideas, such as model selection and resampling based inference, can beincorporated into quantile regression.

2:00-2:15 Exploring Robust Alternatives to Least Squares Regression (Level 1) CURTIS HURLBUT (ST.LAWRENCE UNIVERSITY) Robust regression is a form of regression analysis designed to not beoverly a�ected by violations of assumptions. Robust regression models are not as sensitive to out-liers as ordinary least squares estimates are. In the case of the presence of outliers, least squares es-timation is ine�cient and can be biased, in which case robust regression is a viable alternative.This presentation will introduce the method of robust regression and examine how model selec-tion and resampling based inference is done in the robust case.

2:40-2:55 Principal Component Analysis and Outliers E�ect (Level 2) STEVEN MARTINEZ (WESTERN NEWENGLAND UNIVERSITY) Our interest is to study Principal Component Analysis and some of its ap-plications in the same time we would like to study the e�ect of outliers on the components of thePCA. PCA is a multivariate statistical procedure that uses an orthogonal transformation, of a ma-trix data, to transform a set of variables into a set of linearly uncorrelated variables called principalcomponents. The first component is the one with highest variability of the data. The second com-ponent has the second largest variability explained of the data and so on. Hence the number ofcomponents is less or equal to the number of original variables. We take as many components aswe need to satisfy the “parsimonious principle,” but explain as much variability as possible. We areinterested in defining the PCA, explain its properties and show some few working examples withthe e�ect of outliers included.

Statistics IIb STE 102Chair: Amy Wehe

1:40-1:55 The Role of Mathematics in Understanding Student Success at a Four-Year Public University(Level 1) BRIAN DARROW, JR. (SOUTHERN CONNECTICUT STATE UNIVERSITY) Understanding whatenables students to thrive at 4-year postsecondary institutions is crucial in supporting their aca-demic and future success. In particular, it is essential that incoming students be in the best possibleposition for success upon entry. When students attend Southern Connecticut State University(SCSU) as freshman, they are enrolled in a required mathematics course. Students are placed intothese courses according to their score on the mathematics section of the SAT (unless otherwisechallenged). The research conducted in “Investigating Student Success at Southern: The Role ofMathematics” an Honor’s Thesis in the Mathematics department, aims to investigate mathematicseducation at SCSU. Specifically, the study investigates the e�cacy of mathematics course place-ment based on the math section score of the SAT, the e�cacy of math course placement based onpre-college information such as high school GPA, math courses taken in high school, etc., andhow performance in their first math course at Southern influences students’ cumulative GPA, per-sistence, and graduation. Substantial empirical research in postsecondary mathematics educationsupports the methods of the project. Results gleaned from statistical analyses conducted on real,SCSU longitudinal student cohort data provide a summary of the relationship between successin mathematics and overall success in college. The results of the study will inform admission pro-cedures, influence mathematics course placement policies, and inform educational perspectivesinvolving mathematical competency and overall academic success at Southern Connecticut StateUniversity.

2:00-2:15 A Propensity Score Analysis of High School Type on Post-Graduate Outcomes (Level 1) SARAHMARKIEWICZ, MICHAEL LOPEZ (SKIDMORE COLLEGE) Multiple linear regression is a popular toolto estimate the associations between a set of covariates and a continuous response variable. How-ever, these models can perform poorly when improperly specified or when the underlying popu-lations analyzed have di�erent covariates’ distributions. For example, a model estimating the e�ectof public versus private secondary education could be confounded by underlying di�erences in thecharacteristics of students attending each type of school. One tool that can account for these typesof baseline di�erences is the propensity score, defined as the probability of receiving a certaintreatment. We implement a propensity score analysis to estimate the e�ect of secondary schoolchoice on post-graduate outcomes.

2:20-2:35 Benford’s Law (Level 2) ROBERT CAMERON, AMANDA LYONS (SUNY POTSDAM) Intuitively,one would assume that the first digits of a set of data would be distributed evenly — ∼ 11.1% foreach digit, 1 through 9. For some sets of data this is not the case. Rather, the number 1 is morefrequently the first digit than the number 2, 2 is more frequently the first digit than 3, and so on.This mysterious statistical phenomenon is known as Benford’s Law. We will delve into the historyof Benford’s Law, from its inception in 1881 to its rejuvenation in 1938. Following the ideas ofHavil, using a result gained from the Weyl Equidistribution Theorem and probability theory, wewill be able to give some rationale as to why Benford’s Law works.

2:40-2:55 Using Linear Discriminant Analysis to Predict Beer (Level 1) BROOKE MCGRAW (ST. LAWRENCEUNIVERSITY) Linear discriminant analysis (LDA) is a classification technique commonly used fordimensionality reduction. LDA uses existing information to compute latent explanatory variablesthat maximizes separation between multiple classes. Like other classification techniques (such aslinear regression), LDA can be used for predictions and to determine important variables in themodel. This talk will introduce LDA and apply it to the classic Fischer’s Iris dataset as well as clas-sifying beer styles based on home-brew recipes.

PARALLEL SESSIONS THREE

Applied Math IIIa STE 102Chair: Je� Jauregui

3:30-3:45 Gyroid Surface (Level 1) RACHEL FIELD (SAINT MICHAEL’S COLLEGE) In this article we willdiscuss the formation of the Schoen gyroid surface as well as some of its applications in the naturalworld and the artificial world. Applications that will be discussed are gyroid lattices as the causa-tion of color in butterfly wings, the possibility of gyroids being used as bone implants, and othernatural occurrences of the gyroid. We will also discuss the history of the gyroid as a minimal sur-face, its discovery and what makes it so special.

3:50-4:05 Numerical Boundary Control of Wave Equation (Level 2) JOHN CURTIN (STATE UNIVERSITY OFNEW YORK AT NEW PALTZ) The goal of this project is to find the boundary controls to bring thewave function from one prescribed state to the other resulting in minimum energy. This will bedone using numerical methods on the wave equation, namely those involving the finite di�erencemethod. This will be done using MATLAB.

4:10-4:25 Mathematics of engineering: building a supersonic rocket (Level 2) NOAH TURNER (WESTERNCONNECTICUT STATE UNIVERSITY) A project aimed at designing and analyzing flight and per-formance of a supersonic rocket is a multidisciplinary endeavor requiring the use of mathematics,physics, programming, and electrical engineering. The focus of this presentation is on mathe-matical issues involved in the project such as simulation of motion and optimization of physicalparameters for better performance characteristics. The presentation will include visualizations ofthe computational results, as well as a brief discussion of the autonomous on-board control. It willconclude with a video of a launch of a similar supersonic rocket previously built by the presen-ter.

Applied Math IIIb JEM 378Chair: John McHugh

3:30-3:45 Turning Grille Method: Odd Grids and Linear Grid Codes (Level 1) KIMBERLY WOOD (WEST-ERN NEW ENGLAND UNIVERSITY) In this talk, we will discuss a transposition cipher called theTurning-Grille Method. This can be done on a square grid or a linear grid. We will examine thedi�erences between a square and linear grid by going through an example in detail. We will alsodiscuss the mathematics behind each grid.

3:50-4:05 Programmatic Art Through Ray Marching Primitives (Level 1) DAVID JOHNSTON (CHAMPLAINCOLLEGE) Ray marching is a graphics programming technique that is used primarily in gamesand graphics demos. While ray marching can be used for general rendering, it is more commonlyused for cooking up specific e�ects such as displacement mapping, soft shadows, and sub-surfacescattering. In contrast to classic ray casting, ray marching is simpler to implement, parallelizes wellon modern graphics processing units (GPUs), and can be used on shapes that do not have ana-lytic intersection functions. In this presentation we will focus on ray marching primitives usingthe OpenGL Shading Language (GLSL), with a little dip into shading those primitives if time al-lows.

Combinatorics III JEM 362Chair: William Zwicker

3:30-3:45 Catalan Numbers and Fine Numbers (Level 1) AJAY BARDE (SKIDMORE COLLEGE) The Catalannumbers 𝐶𝑛 are among the most celebrated integer sequences in mathematics. They are knownto enumerate scores of di�erent combinatorial objects, from legal strings of pairs of parentheses,

to Dyck paths in the plane, to edge triangulations of an 𝑛-gon, to non-crossing set partitions. TheFine numbers 𝐹𝑛 are another integer sequence which are related to the Catalan numbers.In this talk we present some recently discovered identities which express 𝐶𝑛 in terms of otherCatalan numbers, as well as an identity which expresses 𝐹𝑛 in terms of the Catalan numbers. Theapproach we take is to express the generating function of these sequences as composite functions,then analyze the composition using a theorem published in 2008 by D. Vella. A ‘counting’ proof(or ‘bijective’ proof ) of these identities was later discovered, but the generating function approachis how they were originally discovered.

3:50-4:05 Stirling Numbers from Stirling Numbers (Level 1) SAM ARMSTRONG (SKIDMORE COLLEGE)There are two kinds of Stirling numbers. Stirling numbers of the first kind, denoted 𝑠(𝑛, 𝑘) can bedefined as the coe�cients in the expansion of the “falling factorial” function: 𝑥𝑛 = 𝑥(𝑥 − 1)(𝑥 −2) · · · (𝑥− 𝑛+ 1) =

∑︀𝑛𝑘=1 𝑠(𝑛, 𝑘)𝑥

𝑘.Stirling numbers of the second kind, denoted 𝑆(𝑛, 𝑘) count the number of ways to break up or parti-tion a set with 𝑛 elements into 𝑘 nonempty, disjoint subsets. There are many combinatorial iden-tities known for both types of Stirling numbers, including identities that relate the two kinds to-gether.In this talk we present a recently discovered identity which expresses 𝑆(𝑛, 𝑘) in terms of the𝑠(𝑛, 𝑘)’s. The approach we take is to express the generating function of these sequences as com-posite functions, then analyze the composition using a theorem published in 2008 by D. Vella.This approach has previously led to proofs of identities (including some new discoveries) of sev-eral famous integer sequences, such as the Bernoulli numbers, Euler numbers, Bell numbers, andCatalan numbers. We can now add Stirling numbers to the list.

4:10-4:25 Grandmama de Bruijn Sequence (Level 1) PAT DRAGON, OSCAR HERNANDEZ, AARONWILLIAMS (BARD COLLEGE AT SIMON’S ROCK) A de Bruijn sequence is a binary string of length2𝑛 which, when viewed cyclically, contains every binary string of length 𝑛 exactly once as a sub-string. For example, 000100110101111 is a de Bruijn sequence for 𝑛 = 4. The most popular deBruijn sequence, often called the Ford sequence, for each value of 𝑛 is the first in lexicographicorder and can be constructed by means of a greedy algorithm [A Problem in Arrangements;Martin 1934]. There is also a necklace concatenation algorithm, known as the FKM algorithm,that relies on strategically ordering representatives of equivalence classes of strings under rota-tion. This talk will focus on a new construction of de Bruijn sequences for each 𝑛 used to gen-erate what is called the Grandmama De Bruijn Sequence [The Grandmama de Bruijn Sequence;Dragon, H, Williams 2016]. It is interesting to note that this construction di�ers from the FKMalgorithm only in choosing which order to apply on the representatives of equivalence classes.There is also a successor rule, an algorithm that looks at the previous 𝑛 bits in the sequence, thatbuilds the Grandmama de Bruijn Sequence. We will prove that either of these constructions re-sults in a valid de Bruijn sequence and that the resulting sequence is distinct from the Ford se-quence. If time permits, we will outline the proof for arbitrarily large alphabets (instead of thebinary alphabet {0, 1}).

Di�erential Equations III JEM 166Chair: Zsuzsanna Kádas

3:30-3:45 A Multi-Component Model for the Buruli Ulcer (Level 1) SCOTT LE FEVRE (NORWICH UNI-VERSITY) Buruli Ulcers are a flesh eating disease, found in several regions throughout the world.The World Health Organization considers the ulcers a neglected disease. We build a multi-component model to describe the spread of disease in a population with parameters appropriatefor Benin, one of the endemic regions. Unfortunately the method of transmission is not under-stood. Given the literature on cases of exposure and infection, the disease is thought to spreadthrough water or in water with the presence of Heteroptera water bugs (transmitted when bitinghumans). We begin from within the SIR framework and build systems to reflect postulated out-break scenarios. Given the perceived importance of water bugs in transmission, we build a modelfor the bug population as seen in some malaria models.

The system requires several parameters that are adjusted for realistic results. The parameters arethen tuned to admit epidemic and non-epidemic solutions, showing that the model admits bothtypes of observed behavior. The most sensitive parameter is found to be the biting rate of Het-eroptera Water Bugs. If the biting rate of the bugs can be decreased, the model predicts that thedisease spread can be stopped. This may be more cost-e�ective than antibiotics.

3:50-4:05 Internal Di�erential Equations (Level 1) BENJAMIN OLTSIK (HAMILTON COLLEGE) Whatstarted as a mere passing thought at prom night developed into a characterization of solutions toan important class of di�erential equations. We are calling these internal di�erential equationsbecause they have the form 𝑦′ = 𝑦(𝑥 − 1)(𝑦 OF 𝑥 − 1). In this talk, we will describe a unique ap-proach to solving and generalizing solutions to internal di�erential equations. We will also discusswhy internal di�erential equations are useful for modeling real world phenomena.

4:10-4:25 Development of a Runge-Kutta Numerical Estimator in GeoGebra (Level 2) SHAYNA BENNETT,GREGORY PETRICS (JOHNSON STATE COLLEGE) Numerical estimation is an important process forunderstanding solutions to di�erential equations, but time consuming and not visual in nature. Inthis presentation I will talk about using the Runge-Kutta equations and the software GeoGebrato construct a tool that numerically estimates solutions to any ordinary di�erential equation. Thetool takes as input a di�erential equation with an initial condition and produces a numerical esti-mation of the solution. For equations of order 3 or lower, it also displays the estimate graphically.The tool is fast and easy to use, ideal for anyone interested in exploring the nature of the solutionsof a di�erential equation.

Fractals and Chaos JEM 377Chair: Daniel M. Look

3:30-3:45 Fractal Landscape Generation (Level 1) BRIANNA HEALY (SAINT MICHAEL’S COLLEGE) We in-vestigate the fractal qualities of natural landscapes, examine the length British coastline, and dis-cuss computer generation of fractal landscapes. True landscapes are generated using height mapswhile realistic landscapes are generated randomly using various algorithms. We focus primarily onthe diamond-square algorithm and the smoothing e�ects created by multiplying two or morefractal landscapes together. Fractal landscapes can be applied to computer-generated imagerytechniques for film and television.

3:50-4:05 Analysis and Applications of Nonlinear Systems: The Rikitake Model of Geomagnetic Pole Rever-sals (Level 2) LANCE OSTBY, JOCELYN LATULIPPE (NORWICH UNIVERSITY) Nonlinear dynamicalsystems arise naturally when modeling physical phenomena. In this presentation we investigatethe behaviors of the Rikitake Model of Geomagnetic Reversal. This model provides insight intohow the Earth’s magnetic field is changing. By varying model parameters and initial conditions,the dynamics and properties of the system will change. For example, the behavior and stability ofthe fixed points change under various parameter regimes. In the Rikitake Model, disparate timescales lead to di�erent stability conditions and chaotic behavior. In this presentation we use Du-lac’s criterion to establish limit cycles for the Rikitake Model and show that periodic solutions existin phase space. Solutions to the model are illustrated using numerical methods in MATLAB.

History of Mathematics JEM 281Chair: Paul Friedman

3:30-3:45 Necessity: the Mother of Mathematics (Level 1) ADAM HEMINGWAY (WESTFIELD STATE UNI-VERSITY) Modern mathematics is rooted in a number system that is so familiar, many take it forgranted. What were number systems like in their infancy? Who is credited with the first writtenexamples of number systems, operations, and fractions? Take a journey back in time to discoverthe lasting e�ects of ancient mathematics and the reasons why we still use the same ideas today.

3:50-4:05 𝜋 = 3.141592653589793 . . .: How Do We Know That? (Level 1) KATHERINE MARINOFF(KEENE STATE COLLEGE) Since ancient times, mathematicians have been aware that the perime-ter of a circle divided by its diameter yields a constant quantity. The value was first measured tobe about 3; however, the exact value, known as pi, was unknown. Pi was first calculated by the

ancient Greek Archimedes to be between 3 1071 and 3 1

7 . Since then, mathematicians have beentrying to find the best approximation for the irrational number. Today, pi has been calculated totrillions of digits, but that could not have been done without the work of mathematicians such asArchimedes, Euler, Wallis, and others. This presentation will examine the ways pi has been calcu-lated — geometrically, through infinite series, and as a probability — throughout history.

4:10-4:25 Cardano’s Solution of the Cubic Equation (Level 1) ROSS GINGRICH (SOUTHERN CONNECTI-CUT STATE UNIVERSITY) While the quadratic formula is well known, many students have not seenGirolamo Cardano’s method for solving the general cubic equation 𝑥3 + 𝑏𝑥2 + 𝑐𝑥 + 𝑑 = 0. Car-dano published his solution in his Ars Magna or The Rules of Algebra in 1545 CE. We will look atthe history of his solution, the method that he used, and its modern formulation. We shall see thathis solution was a process (or an algorithm), not an algebraic formula, and that instead of solvingthe general cubic, he solved thirteen separate cases of the cubic.

Knot Theory JEM 389Chair: Richard Bedient

3:30-3:45 Almost-alternating links and the Jones polynomial (Level 1) ADAM LOWRANCE (VASSAR COL-LEGE), OLIVER DASBACH (LOUISIANA STATE UNIVERSITY) A link diagram is alternating if the cross-ings alternate under, over, under, over, etc. as one travels along each component of the link. Alink diagram is almost-alternating if one crossing can be changed so that the diagram is alter-nating. Alternating links are those links that have alternating diagrams, and similarly, almost-alternating links are non-alternating links that have almost-alternating diagrams. In this talk, wediscuss an obstruction for a link to be almost-alternating arising from the Jones polynomial.

3:50-4:05 Conway Links and Continued Fractions (Level 1) JOHN BENNETT (HAMILTON COLLEGE) Wewill examine the relationship between the number of components of a link and the Conway nota-tion associated with that link. After showing that a rational link possesses either one or two com-ponents, we will then provide a method for determining the number of components in the linkbased on the continued fraction given by the Conway notation.

4:10-4:25 Topics in Knot Theory (Level 1) MELISSA WESTLAND (SAINT MICHAEL’S COLLEGE) A brief in-troduction to knot theory will include group theory concepts for knots, knot isomorphisms, andknot classification. Alexander polynomials will be discussed after su�cient background is pro-vided.

Mathematics Education II JEM 375Chair: George Ashline

3:30-3:45 Spice up your Face-to-Face Class with Features from Flipped, Blended, and Online Classrooms(Level 1) JENNIFER BLUE (SUNY EMPIRE STATE COLLEGE), JENNIFER SILVER (WESTERN GOVERNORSUNIVERSITY) In this talk we will share various elements used in flipped, blended, and online class-rooms that can enrich your face-to-face course. Your presenters have many years of experience inface-to-face, flipped, blended, and online teaching, particularly in calculus. While the focus willbe on calculus, the material presented applies to any math subject and course.

3:50-4:05 A Math Course for Game Programming Majors (Level 1) SCOTT STEVENS (CHAMPLAINCOLLEGE) At Champlain College we have a popular Game Programming major packed withindustry-specific course work. Their curriculum does not have the credit allowance for the stan-dard sequence of five to six math courses found in a typical computer science degree. Our course,Matrices, Vectors, and 3D Math, teaches standard topics of Calculus III and Linear Algebra withinthe context of Game Programming applications and projects. This carefully constructed 3-creditcourse can have students performing moderately sophisticated mathematics by the end of the firstyear of course work. What started out as a math class for game-programming majors is now apopular math course for all of our undergraduate students who want or need exposure to upperlevel mathematics but do not have the credit allowance for the standard sequence of math coursesto get there. I will present the content/structure of the course and some student projects.

4:10-4:25 The Pythagorean Theorem: Some History, Derivations, and Extensions (Level 1) GEORGE ASH-LINE (ST. MICHAEL’S COLLEGE) We’ll consider some historical background for Pythagorean triplesand the Pythagorean theorem. Also, we’ll examine a few of its derivations and some of its exten-sions, including an interesting result due to ancient Greek mathematician Pappus of Alexandria.

Number Theory III JEM 364Chair: Blair Madore

3:30-3:45 A Conjecture on Sums of Consecutive Polygonal Numbers (Level 1) SAMANTHA WYLER (SUNYNEW PALTZ) A polygonal number 𝑃 (𝑔, 𝑛) is a number represented as dots arranged in the shapeof a 𝑔-sided regular polygon. In this talk, we will show that every other triangular number (𝑔 = 3,𝑛 even) starts of the next group of consecutive polygonal numbers such that if you have one morenumber in the lower group (𝑛 + 1 terms) then in the higher group (𝑛 terms), then the sum of thelower numbers will equal the sum of the higher numbers + (4− 𝑔)𝑃 (3, 𝑛).

3:50-4:05 Proofs and Applications of Fibonacci Numbers (Level 2) ALEC COVEY (SUNY POTSDAM) Thisexpository talk will show the Fibonacci sequence is a basis for many profound, mathematical con-cepts. We will show connections between Pascal’s Triangle and the Fibonacci, demonstrate thatthe Fibonacci sequence can be expressed by an application of the Binomial Theorem, and willprove the Euler-Binet formula for Fibonacci numbers. The problem of tiling a (2 × 𝑛)-boardwith (2× 1)-dominos will be solved as a direct application of the Fibonacci sequence.

4:10-4:25 Pitch Perfect: A Brief Discussion on Sound Waves & Number Theory (Level 1) CALISTA NASSER(SUNY EMPIRE STATE COLLEGE) What is a pitch and how does it relate to mathematics? In thistalk, we will explore some key concepts related to sound waves: the harmonic series, fundamentalfrequency, and the phenomenon known as the missing fundamental. In a combination of musictheory and number theory, we will discuss how these concepts relate to the Greatest CommonDivisor (i.e. GCD) and Least Common Multiple (i.e. LCM).

Probability and Statistics JEM 168Chair: Ada Morse

3:30-3:45 How To Brew a Better Cup of Co�ee (Level 1) SCOTT BIANCO (SIENA COLLEGE) Ever won-der how water goes through your co�ee grounds to make that perfect cup of joe? Well there is awhole theory behind it called Percolation Theory. We will look at the two of the main types ofpercolation thought a set of regular lattice points. The first type called a bond percolation con-siders the lattice graph edges as the relevant entities. Site percolation considers the lattice graphvertices as the relevant entities. The relevant entities are either occupied with probability 𝑝 or va-cant with probability 1 − 𝑝 which either allow of block fluid flow respectively. This allows us tostudy more than how to brew co�ee. We can use it model heat flux, electric current, spread ofinfections and other flows though porous mediums.

3:50-4:05 Methods to Address Area-to-Area Change of Support and Modifiable Areal Unit Problems(Level 1) SARA LAPLANTE, JESSICA MAO, MYVAN VO (SMITH COLLEGE) Spatial data are commonlycollected and studied in fields such as geology, ecology, and the social sciences. Researchers of-ten cull data from a variety of sources, which may not contain values from compatible regions.This process raises questions as to how to aggregate the data from incompatible spatial regions(source units) in order to make statistical inference about data in the regions of interest (target units).The preferred method of determining the aggregated values for target units remains unclear. Inthis paper, we explore known methods for making area-to-area spatial data compatible.

4:10-4:25 An Introduction to Gaussian Mixture Modeling for Model-Based Clustering (Level ) JANELLEFREDERICKS (ST. LAWRENCE UNIVERSITY) There are many ways to understand how di�erentdatasets can be clustered together. First, we will take an in depth look at Gaussian Mixture ModelClustering via the EM algorithm to see how points are determined to be in each individual clus-ter. Second, we will use simulated data with a predetermined number of cluster to examine howdi�erent classic clustering algorithms perform compared to Gaussian Mixture Modeling. Specif-ically, I will be comparing Gaussian Mixture Modeling (as implemented through the R package

“mclust”), the 𝑘-means algorithm, and hierarchical clustering techniques to determine whichmethods are optimal in di�erent clustering situations.

Statistics III STE 104Chair: Joseph Kirtland

3:30-3:45 Shiny Bayes: Developing an App to Illustrate Bayesian Inference (Level 1) SCARLETT QI (ST.LAWRENCE UNIVERSITY) Bayesian inference is a statistical method for using data to update beliefsabout location of a parameter. We use the R shiny package to create an interactive web app thatallows users to specify a prior distribution, input data and observe the resulting posterior distribu-tion. We discuss how this app can be used to demonstrate Bayesian inference for parameters suchas a binomial proportion, Poisson mean, or normal mean.

3:50-4:05 Comparing Methods for Constructing Confidence Intervals Using Simulations in R (Level 1) NAN-JIANG LIU (ST. LAWRENCE UNIVERSITY) A confidence interval (CI) is a pair of numbers, based onsample data, designed to capture the value of some population parameter. There are several dif-ferent methods for constructing CI. For example, we can find a CI for proportion using a formulabased on normal distribution, a bootstrap distribution of simulation in proportions, a “plus 4” ad-justments to proportion, or Bayesian credible interval. W e use R simulations to generate manysamples from di�erent populations and then compare the coverage rates and widths of the in-tervals with each method. We vary the population proportion and sample sizes to explore whichmethods might work best in di�erent situations.

4:10-4:25 Machine Learning Logistic Classifier: mathematical analysis of the algorithm, computer implemen-tation, and applications (Level 1) ANDREW DAVIS (WESTERN CONNECTICUT STATE UNIVERSITY)Machine learning is a scientific toolbox that can be used to reveal patterns in data or classify datasamples. It enjoyed recently successes in detecting human faces, automatic diagnostics, and speechrecognition, to name just a few fields. Machine learning algorithms are built based on powerfulmathematical tools and techniques from statistics, linear algebra, and optimization theory. Thefocus of this presentation is on analysis of mathematical underpinnings of a popular classificationalgorithm of Machine Learning and the e�ect of parameters involved in the optimization stage onthe algorithm performance. Applications to real data sets will be also provided.

Tropical Mathematics JEM 380Chair: Thomas Ratli�

3:30-3:45 Tropical Mathematics (Level 1) JACOB CHEVERIE (KEENE STATE COLLEGE) At an early age,children learn about numbers and two important operations: addition and multiplication. Thereare, however, other ways one can define these operations. In tropical mathematics, one definesthe tropical addition, ⊕, of two real numbers 𝑎, 𝑏 ∈ R as the minimum of the two numbers, or𝑎 ⊕ 𝑏 = min {𝑎, 𝑏}, and the tropical multiplication, ⊙, as the sum of the two numbers such that𝑎 ⊙ 𝑏 = 𝑎 + 𝑏. This talk will present many of the interesting properties that can be deduced fromthese definitions along with potential applications.

3:50-4:05 Applications of Tropical Mathematics (Level 1) RYAN HAMELIN (FITCHBURG STATE UNIVERSITY)Tropical Mathematics is a field studied by computer scientists, mathematicians, and computationalbiologists. In tropical mathematics (also called a max-plus algebra), the operation of addition be-comes the maximum of the two values and the operation of multiplication changes to standardaddition. Tropical mathematics can be used to address problems in industrial processes, capacityassessment, and network analysis. In this talk we will take graphical and tropical linear algebra ap-proaches to describe processes (from tra�c to industry to vending machines) and introduce meth-ods to optimize the use of time and resources.

4:10-4:25 Fibonacci Pineapples (Level 1) BORIS LI (ST. MICHAEL’S COLLEGE) The purpose of the projectis using Fibonacci sequence to investigate patterns on pineapples. The initial approach of thisproject is the investigation of 2-D planar models. The arrangement of sunflowers’ seeds can berepresented by Maple. It gives a general idea about how to use Fibonacci sequence in this appli-cation. The 3-D approach is the main part of this project. When we peel the seeds of pineapples,

we always goes spirally. This is also related to the Fibonacci sequence. The investigation is mainlyfocused on the cylindrical body of the pineapple, and how the seeds are arranged on the surface.By using Maple, the 3-D model will be generated and illustrate the pure math that is involved init. The model will be printed by using a 3-D printer and laser cutter for the sunflower seed pat-terns.

Index of AuthorsAfinogenov, Sergey Applied Math IIaArmstrong, Sam Combinatorics IIIAshline, George Mathematics Education II

Bania, Katelyn Geometry IIBarde, Ajay Combinatorics IIIBarrett, Scott Computer ScienceBarstow, Ryan Graph TheoryBartlett, Josiah Statistics IaBassette, Reid Di�erential Equations IBell, Sydney A. Statistics IaBennett, John Knot TheoryBennett, Shayna Di�erential Equations IIIBianco, Scott Probability and StatisticsBlue, Jennifer Mathematics Education IIBouchard, Katharine Mathematics Education IBoyadzhiyska, Simona Combinatorics IBryant, Julie Combinatorics IIBuck, Gabrielle Di�erential Equations IIBulger, Shauna Statistics Ia

Cahn, Patricia Graph Theory and TopologyCameron, Robert Statistics IIbChanin, Margo Computer ScienceChernov, Vladimir Graph Theory and TopologyCheverie, Jacob Tropical MathematicsChong, Catrice Number Theory IClaassen, Thaddeus Computer ScienceCormier, E. Graph TheoryCovey, Alec Number Theory IIICurtin, John Applied Math IIIaCzupryna, Michael Mathematics Education I

Daley, Marcella TopologyDaniere, Adam ParadoxesDarrow, Jr. , Brian Statistics IIbDasbach, Oliver Knot TheoryDavid, Logan Applied Math IIaDavis, Andrew Statistics IIIDe Silva, Namini Applied Mathematics IbDemers, Mark F. AnalysisDever, Lindsay Abstract Algebra IIDillery, P. Graph TheoryDisher, Conor Graph TheoryDragon, Pat Combinatorics IIIDragon, Patrick Topology

Edmondson, Mackenzie Geometry IEscobar, Elizabeth Statistics IIa

Ferlini, Vincent Geometry IIField, Rachel Applied Math IIIa

Fredericks, Janelle Probability and Statistics

Gall, Evan TopologyGibson, Julia Abstract Algebra IIGingrich, Ross History of Mathematics

Haehl, Alexandria J. Statistics IaHalden, Derek Combinatorics IIHall-Hooper, Khalil Combinatorics IIHamelin, Ryan Tropical MathematicsHarding, Nicholas ParadoxesHealy, Brianna Fractals and ChaosHemingway, Adam History of MathematicsHernandez, Oscar Combinatorics IIIHodge, Nate Applied Math IIaHolbrooks, Johnathon Graph TheoryHolmes, Cheryl Applied Math IIbHoward, Sebasttian Graph TheoryHuibregtse, Mark Geometry IIHurlbut, Curtis Statistics IIaHurlbut, Curtis J. Statistics Ia

Ianzano, Chris AnalysisIsham, Kelly Number Theory I

Jauregui, Je� AnalysisJia, Miaoqing Number Theory IJohnson, Tristan Geometry IJohnston, David Applied Math IIIb

Kadas, Zsuzsanna M. Applied Math IIbKenney, Kristen Combinatorics IKlein, Paulette Di�erential Equations IIKramer, Sean Applied Math IIb

Landfair, Kegan Number Theory IILane, Callum Combinatorics IILaPlante, Sara Probability and StatisticsLatulippe, Jocelyn Fractals and ChaosLe Fevre, Scott Di�erential Equations IIILengieza, Michael L. Statistics IaLerma, Rebecca Applied Mathematics IbLi, Boris Tropical MathematicsLi, Simon Di�erential Equations IILiu, Nanjiang Statistics IIILopez, Michael Statistics IIbLowrance, Adam Knot TheoryLuangrath, Narin Graph Theory and TopologyLumbra, Celsey Geometry ILyons, Amanda Statistics IIb

Mao, Jessica Probability and Statistics

Marino�, Katherine History of MathematicsMarkiewicz, Sarah Statistics IIbMartinez, Steven Statistics IIaMayer, Philip AnalysisMcCarthy, Kristen Geometry IIMcGrady, Elizabeth Number Theory IMcGraw, Brooke Statistics IIbMcHugh, John Abstract Algebra IIMcHugh, John Abstract Algebra IMcIntyre, Andrew Graph Theory and TopologyMcLaren, Samuel Di�erential Equations IMischell, Melissa Graph Theory and TopologyMoges, Salem Di�erential Equations IMolino, Gianmarco Di�erential Equations IMonhart, Matthew Statistics IaMorfe, Peter Analysis

Nasser, Calista Number Theory IIINorthshield, Sam Number Theory II

O’Kee�e, Bailey Statistics IbO’Kee�e, Bailey J. Statistics IaOgle, Scott Linear AlgebraOltsik, Benjamin Di�erential Equations IIIOrzell, Peter Computer ScienceOstby, Lance Fractals and Chaos

Paight, Heather Number Theory IIPan, Xuehang Statistics IbPark, GaYee Number Theory IPeterson, Amanda Abstract Algebra IPetrics, Gregory Di�erential Equations IIIPhaneuf, Autumn Applied Mathematics IaPincus, Daniel Combinatorics II

Qi, Scarlett Statistics IIIQuenell, Gregory Paradoxes

Ramrath, Bethany Geometry IRansom, Colton F. Statistics IaRatkevich, Jake Combinatorics IIRatli�, Kenneth Applied Mathematics IaRatli�, Tommy Number Theory IIRaymond, Elizabeth Mathematics Education IReams, Robert Linear AlgebraReed, Elise Applied Math IIbResh, J. Graph TheoryRockmore, Daniel Applied Mathematics IbRomagna, Christopher J. Statistics Ia

Schuckers, Michael Statistics IaShuchat, Alan Combinatorics IShull, Randy Combinatorics ISilver, Jennifer Mathematics Education IISimoes, Julia H. Statistics Ia

Soper, Hannah Applied Mathematics IaStevens, Scott Mathematics Education II

Tank III, John H. Statistics IaTheobald, Michael A. Statistics IaTremblay, Samantha Graph TheoryTrenk, Ann Combinatorics ITrono, John A. Statistics IaTurner, Noah Applied Math IIIa

Van Ryzin, A. Applied Math IIaVandermause, Jonathan Applied Mathematics IbVariashvili, Guga Combinatorics IIVees, James AnalysisVella, David Combinatorics IIVo, MyVan Probability and StatisticsVrijmoet, Mieke Applied Math IIb

Warzer, Rebecca Linear AlgebraWebster, Stephanie Abstract Algebra IIWest, Kelsey A. Statistics IaWestland, Melissa Applied Math IIaWestland, Melissa Knot TheoryWhelan, J. Graph TheoryWhite, Lauren Applied Math IIbWilde, Meredith Abstract Algebra IIWilliams, Aaron Combinatorics IIIWood, Kimberly Applied Math IIIbWyler, Samantha Number Theory III

Yoo, Elizabeth C. AnalysisYoung, Henry Abstract Algebra IYun, Claudia Abstract Algebra II

Zachos, Damaris Di�erential Equations IZwicker, William S. Computer Science

1 I-89 Exit 15 is a quarter mile this way

2 Enter here, park on your left

3 McCarthy Arts building; invited address

4 Saint Edmunds Hall (STE); parallel sessions

5 Jeanmarie Hall (JEM); parallel sessions

6 Dion Center; register, lunch, and break

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