Hudson River Undergraduate Mathematics Conference

HRUMC XXIII Hudson River Undergraduate Mathematics Conference

Saint Michael’s College Colchester Vermont 2016-Apr-02 Schedule overview

8:30–9:50am Registration Dion Center 10:00–10:55am Parallel Sessions One St Edmunds and Jeanmarie Halls 11:05–12:15pm Welcome, Invited Address McCarthy Arts Center 12:25–1:30pm Lunch Dion Center 1:40–2:55pm Parallel Sessions Two St Edmunds and Jeanmarie Halls 3:00–3:25pm Coee and refereshments Dion Center 3:30–4:25pm Parallel Sessions Three St Edmunds and Jeanmarie Halls

Wiﬁ Use 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 help Medical, 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 campus phone 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 dial 2959 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 a unisex rest room on the second floor of JEM, next to room 277. Welcome everyone!

Welcome to the twenty third annual Hudson River Undergraduate Mathematics Conference (HRUMC). Whether you are a student at your first conference or an experienced speaker, we hope that you will find today beneficial, rewarding, and inspiring, and that you will make new friends. Our aim is to build an atmosphere 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. Each year, we invite students and faculty from universities and colleges in New York and New England to send abstracts for the short talks. These describe research projects, independent study projects, or any other 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 your interest. Each of these is marked as Level 1 or Level 2: the Level 1 talks are accessible to everyone while Level 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 that you can easily move from room to room to see talks — you know that each talk that you attend will end on time, and each next one will start when it says it will. If you are a first time presenter then we 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, including any technology questions. The first HRUMC was held at Siena College in 1994, and now it is an annual tradition. For information about previous meetings, pictures from this year’s conference, as well as information about next year’s conference 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 great deal. And, if you can, we hope to see you again, sharing your work, at next year’s conference, hosted by Westfield State University, on Saturday April 8th, 2017.

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

HRUMC Committee Site Arrangements Lauren Childs, Williams College Jim Hefferon Paul Friedman, Union College Lloyd Simons Mohammad Javaheri, Siena College Jesse Johnson, Westﬁeld State University Emelie Kenney, Siena College Joe Kirtland, Marist College Allison Pacelli, Williams College Alejandro Sarria, Williams College David Vella, Skidmore College Edward Welsh, Westﬁeld State University William Zwicker, Union College Institutional Greeting and Invited Address

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

Keynote address: The P vs. NP Problem Scott Aaronson MIT, University of Texas at Austin

Abstract I’ll discuss the status of the famous P ?= NP problem in 2016, oering a personal perspective on what it’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 has been 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 and Ketan Mulmuley, which (in dierent ways) hint at a duality between impossibility proofs and algorithms.

Biography Professor Aaronson holds a Ph.D. in Computer Science from University of California, Berkeley. He is currently Associate Professor of Electrical Engineering and Computer Science at the Massachussets Institute of Technology. Starting in July he will be the David J. Bruton Jr. Centennial Professor of Computer Science at the University of Texas at Austin. He has an international reputation as an expert in the Theory of Computation and Complexity Theory. Scott studies the fundamental limits on what can be eciently computed in the physical world. This en- tails studying quantum computing, the most powerful model of computation that we have. His work has included 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 computing with closed timelike curves; and linear-optical quantum computing. PARALLEL SESSIONS ONE

Abstract Algebra I JEM 380 Chair: Blair Madore

10:00-10:15 The Amazing Connections Between Groups and Symmetric Subgroups (Level 2) Henry Young (Wheaton College) The history of group theory started with the development of permutation groups. However, our mathematical predecessors did not know they were dealing with such an essential algebraic structure. Arthur Caley proved that all groups are isomorphic to a subgroup of a permutation group. Thus, permutations groups contain the structure of all of the famous groups that we know and love. By looking at these isomorphisms and their applications to the 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 commuting through the use of Clifton, Guichard, and Keef’s work. We will explain how many and which elements in a dihedral group commute and use that information to compute the probability of 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) A wealth of structural information about a ﬁnite group can be obtained by studying its irreducible characters. Of particular interest are monomial characters — those induced from a linear character of some subgroup — since Brauer has shown that any irreducible character of a group can be written as an integral linear combination of monomial characters. Our primary focus is the class of M-groups, those groups all of whose irreducible characters are monomial. A classical theorem of Taketa asserts that an M-group is necessarily solvable, and Dade proved that every solvable 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 subnormal subgroups of any M-group.

Analysis JEM 389 Chair: Andrew McIntyre

10:00-10:15 The Equality in Young’s Inequality (Level 1) James Vees (Hamilton College) Young’s Inequality is an important result in functional analysis that relates the areas enclosed by a function and its inverse. This presentation will oer two dierent calculus-based proofs of the conditions that imply equality in Young’s Inequality. The talk will also give an example of an integral that becomes 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 (Fairfield University), Peter Morfe (Cooper Union), Elizabeth C. Yoo (Columbia University) Open dynamical systems are models of physical systems in which mass or energy is allowed to escape from the system. Central questions involve the existence of conditional equilibria (measures that are invariant under the dynamics conditioned on non-escape) which can be realized as limiting distributions under the dynamics of the open system. We study this problem in the context of topological Markov chains, a class of symbolic dynamical systems with a wide variety of applications. Under a combinatorial condition on the Markov chain, we study transfer operators associated with positive recurrent potentials and prove the existence of a spectral gap on a natural function space. This implies the existence (and uniqueness in a certain class) of limiting distributions which represent conditional equilibria for the open system. We also prove a relation between the escape rate from the system and the entropy on the survivor set (the set of points that never enters the hole). 10:40-10:55 Measure theory on the real line (Level 2) Jeff Jauregui (Union College) If you have a bounded subset of the real line, how do you deﬁne its “size”, or “measure”? For an interval [푎, 푏], the measure ought to be just the length of the interval, 푏 − 푎. But what about more complicated sets that have lots of gaps, like the subset of rational numbers in [푎, 푏], or the famous Cantor set? In this talk I will give an introduction to the subject of measure theory, an important branch of analysis.

Applied Mathematics Ia JEM 378 Chair: Amy Wehe

10:00-10:15 Origami Square Twist: Possible to Fold it Rigidly? (Level 1) Autumn Phaneuf (Holyoke Community 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 whether or not a structure can be folded rigidly. “Rigid folding” means that one can replace the paper with a sheet of metal with hinges and still be able to fold the origami model. This presentation will analyze the origami square twist. We know that we can fold a square twist with paper. Is it possible to rigidly fold one? 10:20-10:35 Applications of Inverse Problems to Image Processing (Level 2) Hannah Soper (Norwich University) The goal of this study is to form a basic understanding of inverse problems and their applications, culminating with solving a specific problem relating to image processing: deblurring an image. We do this with the matrix form of convolution used for blurring an image. By calculating the singular value decomposition of this matrix, we can invert the process of blurring an image. When Tikhonov regularization is then imposed on the deblurring process, the problem becomes well-posed and an exact solution can be found. We confirm the viability of this process by 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 (Hamilton College) Continuous symmetries play an important role in modern physics. The study of these symmetries invokes matrix groups; for example, given a system that is invariant about an axis, the set of rotations that leaves the system unchanged is the group 푆푂(2). In this talk we will explore the mathematics of Lie groups, their actions and representations as matrix groups, and how they encode the symmetries of a system.

Applied Mathematics Ib JEM 166 Chair: Joseph Kirtland

10:00-10:15 Portfolio optimization: maximizing return of an investor’s portfolio of assets for a given level of risk (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 theory states that when determining the dierent proportions of wealth an investor should invest in each asset, those assets should not be considered in isolation, but rather relative to every other asset in the portfolio. We will examine how to determine these proportions (called portfolio weights) for any 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 several destinations. To solve these problems a type of linear programming problem used, which is the simplex method. In this talk we will take an example of a transportation problem of transporting produce from farms to retail location. We seek to find the best solution to the constraints of both from what farms can produce and what the retail locations desire. The aim in this problem would be 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 complex phases 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 complex phases of certain links in the network. We first study walks over simple graphs and show that the extent to which transport is enhanced or suppressed depends on the topology of the network, the position of the link in the network, and the values of the chosen phases. In particular, we show that transport can be totally suppressed along the diagonal of an 푁-by-푁 lattice and that transport can be enhanced in graphs containing diverging paths. Finally, we propose a more general procedure for optimizing phases in a complex network. A greedy algorithm is used to reduce the hitting time between the most distant nodes in an Erdős-Rényi network, a preferential attachment network, and a Watts-Strogatz small-world network.

Combinatorics I JEM 364 Chair: 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-patterned fold that, unlike the Miura-ori or the square twist tessellation, contains vertices of degree four and six. This makes it challenging to determine the number of valid mountain-valley (MV) assignments of the snake tessellation, or the number of ways which the tessellation can be folded ﬂat. We present upper bounds on 푆푚,푛, which denotes the number of valid MV assignments for an 푚 × 푛 snake tessellation. We also describe our eorts for ﬁnding a closed formula for 푆(푚, 푛). This work was supported by the NSF ODISSEI grant EFRI-1240441 and advised by Dr. Thomas Hull. 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 the events 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 address questions such as the following: Which partial orders come from schedules? Which come from schedules in which all events have the same length? Which come from schedules in which each event 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 of horizontal lines. We will discuss properties that characterize interval orders and recent research in which we characterize a related class. Our results include an ecient algorithm to construct the intervals in a representation.

Dierential Equations I JEM 362 Chair: 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 the Korteweg-de Vries Equation (KDV): 푢푡 + 푎푢푢푥 + 푢푥푥푥 = 0 and Regularized Long-Wave Equa- tion (RLW): 푢푡 + 푢푥 + 푎푢푢푥 + 푢푥푥푡 = 0. Even though the equations dier from each other, both have been used to describe shallow-water waves because their traveling wave solutions are similar to each other for certain parameters. The objective of our research is to compare numerical properties of the two in terms of speed and stability. We support our results with numerical simulations. 10:20-10:35 Soliton escape velocity after scattering for a nonlinear Klein-Gordon partial dierential equation (Level 2) Salem Moges, Reid Bassette (University of Hartford) We study the scattering of traveling wave solutions (kink and anti-kink) of a nonlinear Klein-Gordon equation: 푢푡푡 + 푏푢푡 = 3 푎푢푥푥 − 푢 + 푢 . Waves traveling in opposite directions collide and the number of collisions depends on 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 velocities and 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-changing nonlinearity coecient. 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. This work is motivated by advancements in the study of Bose Einstein Condensates (BECs). We examine the behavior of these profiles by discretizing the system and varying the control parameters for both the nonlinearity function and the initial profile. We find that for certain parameter regimes we typically get either dispersion or a combination of self-trapping solitons and dispersion, past the sign-change of the nonlinearity coecient.

Geometry I JEM 281 Chair: 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 with the 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 can create. As for the coral reefs they are the most naturally occurring hyperbolic structures that can be found in the world. The crochet coral reef project was originally developed to bring awareness to global warming and its eects on the Great Barrier Reef. Australian twins Margaret and Christine Wertheim brought this project to life in 2005. Not only did this project bring awareness to the Great Barrier Reef, but it brought awareness to math. Through this project we will be able to explore parallel lines, areas, and angles of triangle on hyperbolic planes. We will calculate distances distortions and curvature. There is so much to explore with hyperbolic geometry and crochet 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 the function. We will look at how small (dierentiable) variations of functions give us more information about expressions dependent on these functions. Specifically, Noether’s Theorem states that in a system of functions, each dierential symmetry – or small variation where the system is invariant– constructs a conserved quantity for the system. We will apply this to physical examples under 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’s College) This paper will examine the mathematics behind Archimedean solids with a primary focus on the truncated icosahedron. We will begin with a history of polyedra and the dierence between 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 analyzing its properties. We will look at the object through several lenses, primarily algebraic ones, in which we examine symmetry groups and how they apply to the object. Subsequent sections will look at orbits and stabilizers as well as stellations. The final section will examine our process of rendering a 3D version of a truncated icosahedron and the mathematics utilized to get us there via the computer algebra system Maple.

Linear Algebra JEM 377 Chair: William Zwicker

3:30-3:45 Computing stability of neural activity (or, can neurons compute the 퐿2 norm?) (Level 1) Rebecca Warzer (Bennington College) Recent research has found that the cortical activity corresponding 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 of normed vectors, where the vectors are either MEG sensor data or firing rates of neurons in a network. If neural activity is more stable when stimuli are perceived, then it may be important for a neuron or network of neurons to be able to compute the stability of neural activity such that the stimulus representation can move on to later processing stages. I propose a mathematical model of a neuron that computes the stability and directional variance of a network by implementing the normalization function in a biologically plausible way. 3:50-4:05 A Speculation on “Spectral Analysis of the Supreme Court” (Level 2) Scott Ogle (Wheaton College) 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 the orthogonal decomposition of vectors corresponding to voting splits, we can identify the degree to which a single Supreme Court Justice or minority sub-group aected the outcome of a given court case. In this talk, we will discuss the techniques used in the paper and apply them to data from 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 (SUNY Plattsburgh) Instead of the usual matrix product, inverse, and square root of matrices, we will define the Hadamard (or entry-wise) product, inverse, and square root for matrices. We will show that there are some nice results that come from these alternative definitions.

Number Theory I JEM 168 Chair: 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 ecient method for certifying the primality of an integer 푁 ≡ 1 (mod 3) can be created based on pseudocubes in the ordinary integers Z. Williams and Woodings move this method into the Eisenstein integers Z[푤] and define a new term, Eisen- stein pseudocubes. They create a new algorithm in this context to prove primality of integers 푁 ≡ 1 (mod 3) in a shorter period of time. We will look at the Eisenstein integers and analyze how to use the technique of congruential sieving to compute a table of Eisenstein pseudocubes eciently. 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 푦2 = 푥3 + 푎푥 + 푏 over a field 퐾. These curves form a group over 퐾 under an addition operator that is defined using the curves’ special structure. This talk will focus on elliptic curves over finite fields. Over such curves, we can create secure cryptosystems that work eciently and require smaller keys than cryptosystems based on factoring, such as RSA. We can also create analogs to classic number theoretic algorithms such as Pollard 푝 − 1. 10:40-10:55 Further Studies on the Diophantine Equation (푎2푐푋푘 − 1)(푏2푐푌 푘 − 1) = (푎푏푐푍푘 − 1)2 for 푘 = 6 (Level 2) Catrice Chong, Elizabeth McGrady, GaYee Park (Smith College) In this presentation, we examine the Diophantine equation (푎2푐푋푘 − 1)(푏2푐푌 푘 − 1) = (푎푏푐푍푘 − 1)2 for 푘 = 6. In the previous study by Goedhart and Grundman, it has been proven that there are no positive integer solutions for 푘 ≥ 7. Our research is a continuation of this study, where we attempt to narrow the possible solution sets for 푘 = 6.

Paradoxes JEM 375 Chair: Daniel Velleman

10:00-10:15 The Braess Paradox (Level 1) Nicholas Harding (SUNY Plattsburgh) The topic that is 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 system. 10:20-10:35 The Banach-Tarski Paradox (Level 2) Adam Daniere (Hamilton College ) This presentation 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 of choice leads to this surprising result. 10:40-10:55 A Poker Paradox (Level 1) Gregory Quenell (SUNY Plattsburgh) We rank poker hands according to their frequencies: a rarer hand always beats a more common one. A straight flush (the rarest) beats four of a kind, which beats a full house, and so on. When we introduce one or more wild cards, the frequencies change so that, for example, two pair becomes a rarer hand than 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 is no ranking of the usual poker hands that is consistent with their frequencies.

Statistics Ia STE 104 Chair: Phil Yates

10:00-10:15 A Sports Analytics Consulting Course: Part I (Level 1) Michael Schuckers (St. Lawrence University) In this talk, I describe a course taught at St. Lawrence University during the Fall 2015 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’s hockey over the course of the semester. This talk will discuss the motivation for the course, the structure of the course, the outcomes of the course and lessons learned from running the course from 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. Lawrence University during the Fall 2015 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’s hockey over the course of the semester. Throughout the semester, the student collected and analyzed data for each sport. A summary of the work done for each team will be presented. Further, we present some of the student final projects that were part of this course as well 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 could predict 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 upon those results, which were obtained via a weighted, linear regression model. This Monte Carlo approach produced a model whose Spearman correlation coecients were roughly equal to 0.85 for the top 15, top 25 and top 35 teams, respectively, with regards to said final poll. This article will describe 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 102 Chair: 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 films nominated for Best Picture from 2006-2014 using R and Minitab. We discuss the dierences found between lead screen times according to year and director gender, the relationship between how long a female lead is on screen and how likely it is that a film will win the Oscar, a logistic regression model that we built to predict Oscar Winners, and also the lack of racial diversity in these 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 to find ways for people to extract meaning from the data eectively. We discuss the process of data scraping, building a database, and giving a user tools to investigate the data. We build these tools using R and provide a user interface with Shiny apps. We illustrate these ideas using data from the Citi Bike service in New York City that covers 330 stations and millions of rentals in 2015.

Topology JEM 373 Chair: Adam Lowrance

10:00-10:15 Non-Orientable Surfaces (Level 1) Marcella Daley (St. Michael’s College) In this talk, we will define non-orientable surfaces and discuss their essential properties. Three examples of non-orientable surfaces are the Möbius strip, Klein bottle, and Boy’s surface. Möbius strips are 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 Möbius strip. Klein bottles are non-orientable, closed manifold, one-sided surfaces. Attaching two Möbius strips together creates a Klein bottle, although unlike a Möbius strip, they cannot be embedded in R3. Boy’s surfaces are real projective planes immersed in three dimensions. A Boy’s surface is created 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 Arise Upon Them (Level 2) Evan Gall (Bennington College) With two poles of a galvanic battery in hand we ask, how do dierent topological spaces aect the types of steady steamings (electric field 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 are the most general class of functions that describe electrostatics on surfaces, how the construction of these functions changes depending on the surface they are applied to, and how these surfaces aect 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 existence of an invariant for generic, smooth, closed curves in the plane. Since then, the result has seen ex- 2 tensions to curves with cusps, as well as to curves in other topological spaces, including 푆2, RP , and R3. This presentation will include the statement and a proof of the original result for planar curves. Let 훾 be a generic, smooth, closed curve in the plane. Let 퐶 be the number of crossings in 훾. Let 퐼 be the number of inflections in 훾. We sort the doubly-tangent lines to 훾 into two types; the number of type-one doubly-tangent lines denoted by 푇 and the number of type-two doubly tangent lines denoted by 푆. A doubly-tangent line 퐿 is called type one if both its nearby neighborhoods in 훾 occur on the same side of 퐿. A doubly-tangent line 퐿 is called type two if its nearby neighborhoods in 훾 occur on opposite sides of 퐿. Our invariant is given by the surprisingly sim- 1 ple formula 푇 − 푆 = 퐶 + 2 퐼. PARALLEL SESSIONS TWO

Abstract Algebra II JEM 373 Chair: John McHugh

1:40-1:55 Subnormal Subgroups of M-Groups (Level 2) John McHugh (University of Vermont) A wealth of structural information about a finite group can be obtained by studying its irreducible characters. Of particular interest are monomial characters — those induced from a linear character of some subgroup — since Brauer has shown that any irreducible character of a group can be written as an integral linear combination of monomial characters. Our primary focus is the class of M-groups, those groups all of whose irreducible characters are monomial. A classical theorem of Taketa asserts that an M-group is necessarily solvable, and Dade proved that every solvable 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 subnormal 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 dierence between adjacent vertices is a multiple of the edge label. This generalizes the definition of a spline in applied math. This talk will discuss new results about bases for splines on cycles with edges labeled with 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 congruences mod 푛, represented by vertex labels on an edge-labeled graph. They are studied in connection with algebraic geometry and cohomology rings. This talk discusses a problem in generalized splines significant to ane Springer varieties. In particular, we explore finding bases for splines on lattice graphs.

Applied Math IIa JEM 378 Chair: Ellen Gasparovic

1:40-1:55 Investigating Extreme Temperatures (Level 1) A. Van Ryzin (Union College) Climate change 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 can we use statistics, spatial mapping, and data visualization to understand the phenomena? How can we eciently acquire useful data? What trends and patterns do we observe over time? To answer these questions, we first implement a Python program to extract data from a weather website. We then use ArcGIS to compile, analyze, and map the geographic information. Using statistical analysis, 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 (Saint Michael’s College) While a spinning top might be considered a kid’s toy, the mathematics behind a top’s motion is not child’s play. We will explore the mathematical physics that allows these rigid bodies to seemingly defy gravity. Specifically, we will investigate the movement of tops with sharp and rounded tips, as well as tippe tops, which, when spun at a high enough velocity, become inverted and spin with their stems pointing downward. In order to demonstrate the physics that depicts 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 dierent directions. By delving into the forces acting on the top, we will describe the eect of forces such as the inertial force of rotation, the Coriolis 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 the dierent dimensions of torque. Three constants of motion — the total energy constant, the Jellet constant, and the Routh constant — and their relations to the components angular velocity will also 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 of my work for the senior honors capstone. While previous solutions for the periodic behavior in an asymmetric well rely on stereographic projection, the solution to the problem has in this work been found explicitly in terms of the Weierstrass elliptic function. In achieving the solution, we propose 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 (Westfield State University) The Uncertainty Principle is a fundamental fact about the behavior of quantum mechanical systems. I will give a rigorous mathematical statement of the a general form of the Uncertainty Principle, and then show how for the original uncertainty relation studied by Heisenberg (relating position and momentum measurements) that a special class of states (known as coherent states) saturate the inequality. If time permits we’ll explore some interesting properties of coherent states.

Applied Math IIb JEM 166 Chair: 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 of bivalve 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 biologists on 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 understand the extent to which these organisms aect ecology in the water column. Using a series of expo- sures to dierently 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 series. We also varied several experimental conditions to understand how these aected estimates of gut 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 models provided estimates of ingestion rates, which were compared to those obtained by other authors who measured rates of bead accumulation. Collectively, these studies provide new insight on the digestive 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 uniform distribution of population of plants in a landscape. While predicting how the density of a single species of plant may respond to these patches is a direct function of the area of patches, determining how assemblages of plants with various physiologies might respond is a more complex question for ecologists. Studying these assemblages in the field also oers a limited range of spatial configurations of landscapes and scales at which to sample, making it dicult to tease out the influence of the geometric configuration of the patches on these assemblages. Using a model, I test how possible pools of “species”, distinguished by a reduced set of physiology parameters, might respond to various geometric configurations and qualities of patches on a landscape. I then consider to what extent the theoretical experiment reflects observed patterns and theories of community assemblage in Serpentine and other analogous systems. 2:20-2:35 A Study of the Eects of Greenhouse Gas Emissions on Global Temperature (Level 1) Cheryl Holmes, Elise Reed, Lauren White (Smith College) Climate modeling is the study of global climate systems using mathematics. As we notice our impact on the environment, it is important to understand the eects of human activity on the long-term health of our planet. A common and ecient way to model these eects is by using computer simulations and dierential equations. This talk discusses the relationship between anthropogenic greenhouse gas emissions and climate change, 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 equation look alike. But do they always give the same predictions? We’ll illustrate the “dierence” made by discretization and make the case that discrete time models should be explored alongside continuous ones to get the complete picture.

Combinatorics II JEM 364 Chair: 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 power ∑︀ 푛 series in 푥 with the coecients coming from the sequence: 푓(푥) = 푛≥0 푎푛푥 . Another useful power series associated with the sequence is the exponential generating function, which is the fol- ∑︀ 푛 lowing power series: 푔(푥) = 푛≥0 푎푛(푥 /푛!). By converting questions about sequences into questions about functions, these tools provide a bridge between discrete mathematics (combinatorics, graph theory, number theory, etc.) and continuous mathematics (calculus and analysis). For example, 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 functions are 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, there will follow several talks by the students of the seminar, who will illustrate the use of generating functions 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 퐺 be a connected graph with vertices 푣1, 푣2, . . . , 푣푛. Let 푑푖푗 denote the distance from vertex 푣푖 to vertex 푣푗; that is, the minimal number of edges tra- versed in a path from 푣푖 to 푣푗. The Wiener Index 푊 (퐺) of the graph 퐺 is defined as the sum of ∑︀ the distances between all distinct pairs of vertices on the graph: 푊 (퐺) = 1≤푖<푗≤푛 푑푖푗. Finding the Wiener index of a single graph is simply a matter of finding the distances and adding them up. However, there are many infinite families of graphs indexed by 푛 which are important in graph theory, such as the path graphs 푃푛, the cycle graphs 퐶푛, the wheel graphs 푊푛, the complete graphs 퐾푛, and the complete bipartite graphs 퐾푚,푛, to name just a few families. For such families, the Wiener index should be a predictable function 푤(푛) of 푛. In this talk we show how to compute the Wiener index of some families by deriving a recurrence relation for 푤(푛) and then using a 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 푛, a partition of 푛 is merely a way of writing 푛 as a sum 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 푛 is merely a partition where the order does matter, so we would distinguish between 2+3 and 3+2. Thus, instead of a set of positive integers whose sum is 푛, a composition is an ordered 푘-tuple of positive integers whose sum is 푛, for some 푘. So we would write compositions as (1, 1, 3) or (1, 3, 1) rather than as 1 + 1 + 3 or 1 + 3 + 1. It is not dicult to show that the total number of all compositions of 푛 is 2푛−1. However, in this talk, we are interested in counting the compositions of 푛 which have some restrictions on the summands (which are known as ‘parts’ of the composition.) For example, we could ask the question: How many compositions of 푛 are there where the last part is an odd number? If we call that number 푝(푛), we can find a formula for 푝(푛) as follows. First, we find a recurrence relation that the numbers 푝(푛) satisfy. Then we translate the recurrence into an equation involving the (unknown) generating function of the sequence {푝(푛)}. Solving this equation will tell us what the generating function is, and then we can read o our desired formula for 푝(푛) from the generating function. Other types of restrictions on the parts are possible, but this example suces to illustrate the technique, which is a routine way of using generating functions in combinatorics — solving recurrence relations. In the next talk, another group of students from the seminar will discuss other types of restrictions on the parts, and will use an entirely dierent (and not so routine) application of generating functions to count the compositions. 2:40-2:55 Counting Restricted Compositions of an Integer 푛, II (Level 1) Derek Halden, Jake Ratkevich (Skidmore College) Given a positive integer 푛, a partition of n is a way of writing 푛 as a sum 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 푛 is merely a partition where the order does matter, so we would distinguish between 2+3 and 3+2. Thus, instead of a set of positive integers whose sum is 푛, a composition is an ordered 푘-tuple of positive integers whose sum is 푛, for some 푘. So we would write compositions as (1, 1, 3) or (1, 3, 1) rather than as 1 + 1 + 3 or 1 + 3 + 1. It is not dicult to show that the total number of all compositions of 푛 is 2푛−1. However, in this talk, we are interested in counting the compositions of 푛 which have some restrictions on the summands (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 most 2? Or how many using only odd numbers as parts? Other types of restrictions on the parts are of interest. In a previous talk, other students from the seminar discussed how to answer questions like these by using generating functions to solve recurrence equations. In this talk, we use generating functions in a completely dierent way. By expressing these generating functions as composites of other functions, are able to gain some insight into counting compositions with various types of restrictions.

Computer Science JEM 389 Chair: William Dundar

1:40-1:55 P = NP and Voting Theory (Level 2) William S. Zwicker (Union College) With more than two candidates, many dierent voting rules are possible. For most, calculating who won the election (given the ballots cast) is computationally simple. Since 1989, however, it’s been known that calculating the Kemeny winner (proposed by John Kemeny, former president of Dartmouth College, and co-inventor of BASIC, the first high-level computer language) is NP-hard. What explains the dierence 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 are revealed via the same orthogonal decomposition that serves as the basis for Kirchho’s Laws in electric 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. Our question is “What are the fewest number of words one would need to know beforehand to then be able to read and understand the rest of the dictionary?” We demonstrate how we solved the problem and where we assumed facts about the dictionary due to its imperfections. Finally, we show our 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 (Champlain College) Extracting meaning and context from written text has long proved to be a dicult task in computing. Manipulation of word data mapped to a vector space known as a word embedding is a recent strategy that has been used to gain an enormous amount of insight into the meaning behind text. This scheme has largely been brought to community attention by word2vec and its authors, T. Mikolov, K. Chen, G. Corrado, and J. Dean, in their 2013 ICLR paper, Ecient Estima- tion of Word Representations in Vector Space. In their described method, the cosine distance between mapped 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 dierence vector. This dierence vector can then be applied 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 examine the mathematics behind creating a word2vec word embedding, methods for navigating the embedding mathematically, and a few of the potential insights that can be obtained from working with 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 expressing computation to solve any mathematical problem. It was formalized around the same time as its better-known cousin, the Turing machine, and was invented to solve the same problem in mathematics. 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. Both formal systems have inspired many influential programming languages, but the languages that spawned from each school of thought are as radically dierent as the systems themselves. This talk will explain the basics of lambda calculus and explore how it relates to modern-day computer science.

Dierential Equations II JEM 362 Chair: Jenna Reis

2:00-2:15 Fourier Series (Level 1) Paulette Klein (Russell Sage) Fourier Series are used to solve LaPlace’s equation, Poisson’s equation, the diusion of heat ﬂow equation, wave equation, Helmholtz equation, and Schrödinger equation. The results will be how to solve all dierent types of equations using Fourier Series. Also looking at dierences 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 New York New Paltz) For every linear partial dierential or ordinary equation, one can consider the corresponding linear partial dierential or ordinary operator. Darboux Transformations of Type I are transformations of linear partial dierential operators, that can be used, for example, to transform the operator corresponding to a dicult to solve equation into the operator corresponding to 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 dierential invariants of the corresponding operator. Dierential invariants encode the essential information about the operator, 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 will look at shapes and patterns that they give rise to. 2:20-2:35 Supersymmetric Darboux Transformations and Wronskian (Level 2) Simon Li (SUNY New Paltz) Darboux transformations are a way of solving dierential 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 theorem and its extension in the supercase and using the properties of the berezinian, we construct the 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 solutions.

Geometry II JEM 281 Chair: Ockle Johnson

1:40-1:55 Generating Spirographs and the Mathematical Rose (Level 1) Katelyn Bania (Saint Michael’s College) 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. We will 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 process of how to generate Spirograph designs without the use of interlocking templates (such as gears and wheels) and instead with their underlying mathematical equations. 2:00-2:15 Equidistant Sets of Curves (Level 1) Mark Huibregtse (Skidmore College) The equidistant set, or midset, of two sets in the plane is the set of points that are equidistant from each of the two sets. We will consider a couple of methods for computing the midset (at least approximately) of two curves that are given as graphs of functions. One possible application of such methods would be to locate the boundary between two jurisdictions that are separated by a river, where the boundary 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 푚 and 푛 are integers with 0 < 푛 < 푚 then the triple (푚2 − 푛2, 2푚푛, 푚2 + 푛2) are integer sides 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 푚 and 푛 that characterize other types of triangles. 2:40-2:55 Stereographic Projections of Loxodromes on a Sphere (Level 1) Kristen McCarthy, (Saint Michael’s College) The piece I have created is a stereographic projection of multiple spherical spirals which are specific forms of loxodromes. In order to understand this we must understand the 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 a sphere like the globe, but rather the direction you would take if you followed your compass at an exact direction throughout your entire trip. A spherical spiral is a special case of loxodrome and can be given by a set of three parametric equations which I have altered to make the 3D printed shape. As for the projection there is a specific formula for projection radiating from a certain point to 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 380 Chair: Adam Lowrance

1:40-1:55 Expose’ of Running Clubs — Combinatorial Investigation (Level 1) Johnathon Holbrooks (SUNY Potsdam) Consider the graph consisting of 푛-squares joined together in a straight line. Following the techniques presented by Nissen and Taylor, we will show how many distinct trails such 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) Sebasttian Howard, Ryan Barstow (SUNY Potsdam) The 푁-Queens problem is to place 퐵 mutually nonattacking queens on an 푁 × 푁 chessboard. We show a constuction, due to Erbas and Tanik, to generate one solution for any 푁 ≥ 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 United States has reached a point that it can be modeled as an epidemic. Various sources have used the Susceptible-Infected-Susceptible statistical model to show the spread of incarceration. Using graph theory and network theory, the data can be clarified, along with the relationship between incarceration of an individual and future incarceration of the family members of that person. This can be explained by the financial and social strain put on the family of an inmate. We examine the best ways to model incarceration as an infectious disease. We will demonstrate simple network examples, as well as using Python code and current data to model incarceration in the United States. 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 휏-tilting modules in representation theory. It is known that triangulations of disks with no punctures yield type A quivers. B. Keller introduced green mutations and the corresponding MGS’s. These sequences can be studied both through the combinatorial transformations of directed graphs as well as through triangulations of disks. Our research focuses on maximal green sequences of minimal length for quivers mutation equivalent to type A quivers. It is known that each acyclic quiver has at least one minimal length MGS of length 푛, where 푛 is the number of vertices in the quiver. First, we define an algorithm that produces such a sequence of mutations for any given acyclic type A quiver. For cyclic type A quivers, we define an algorithm that produces an MGS of length 푛 + 푡 where 푛 is the number of vertices and 푡 is the number of 3-cycles in a quiver. We then proceed to show that 푛 + 푡 is always the minimal length of MGS’s corresponding to any type A quiver.

Graph Theory and Topology JEM 377 Chair: 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 about a graph is contained in the unmarked length spectrum. The length spectrum of a metric graph is the set containing all the lengths of the loops in the graph, and how many dierent loops have each length. We show that if two complete graphs with rationally independent metrics have the same length spectrum, then they are isometric. We also prove that there are only finitely many metrics 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 is often a high point of an undergraduate course in abstract algebra. The Euler characteristic of surfaces is often a central point of an undergraduate course in topology. In fact, the two are connected: the classification can be proved by means of topology, using the Euler characteristic. One wrinkle 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 developed by 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 (Dartmouth College), Patricia Cahn (Smith College) A continuous deformation of a loop on a surface is called a homotopy. Given two homotopy classes of loops 훼1, 훼2 on an oriented surface, it is natural to ask how to compute the minimum number of intersection points 푚(훼1, 훼2) of loops in these two classes.

We show that for 훼1 ̸= 훼2 the number of terms in the Andersen-Mattes-Reshetikhin Poisson bracket of 훼1 and 훼2 is equal to 푚(훼1, 훼2). Chas found examples showing that a similar statement does not, in general, hold for the Goldman Lie bracket of 훼1 and 훼2.

Mathematics Education I JEM 168 Chair: 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 content standards as well as the Standards for Mathematical Practice. Students who demonstrate the characteristics of the eight practice standards in the Common Core illustrate behaviors and skills that mathematicians show. Exercises that illustrate these practice standards at the high school level will 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 transformed mathematics. I will also be focusing on how the mathematic standards have aected students (K- 12) with learning disabilities/diculties (MD) since there are rigorous demands associated with these new standards. I hope to discover statistics that can back whether the Common Core State Standards have improved student comprehension and test scores or if it has been detrimental to student’s achievement. By completing this research I hope it gives others and myself a deeper understanding 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 want to construct hard data on how students with learning disabilities are articulating this new way of education. 2:40-2:55 What is problem solving in a high school math class supposed to look like? (Level 1) Michael Czupryna (Westfield State University) Problem solving is a crucial skill to possess when graduating high school. The teaching strategies in most high school math classes almost assure that 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 examples on this issue will be presented.

Number Theory II JEM 375 Chair: John Trono

1:40-1:55 Arithmetic Derivative (Level 1) Heather Paight (Keene State College) The arithmetic derivative of a natural number 푛 , denoted 푛′ , is defined as follows: If 푛 is prime then 푛′ = 1 and for natural numbers 푎 and 푏, then (푎푏)′ = 푎푏′ + 푏푎′. This rule for (푎푏)′ has the same form as the Leibniz Rule in Calculus. This talk will explore some basic properties of the arithmetic derivative with an emphasis on those that have analogs in Calculus. The definition will then be extended to apply to integers and rational numbers. Connections between the arithmetic derivative and two famous unsolved problems in number theory will be included. 2:00-2:15 Hyperbolic Numbers (Level 1) Kegan Landfair (Keene State College) The algebraic equation 푥2 = 1 has two solutions 푥 = −1 and 푥 = 1. We assume the existence of a new number 푢, called the unipotent, which has the property that 푢 ̸= −1 or 1 and that 푢2 = 1. The hyperbolic numbers then are of the form 푎 + 푏푢 where 푎 and 푏 are real numbers. These are similar to complex numbers which are of the form 푎 + 푏푖 where 푖2 = −1. This talk will present the basic properties of hyperbolic numbers and emphasize the similarity with complex numbers. In addition, the Lorentz equations that relate the times and positions of an event as measured by two observers in relative motion 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) Everyone knows that the rational numbers are countable (i.e., can be made into a list) but not everyone knows how to explicitly do that. Such a listing is called an enumeration. We look at three closely related 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,... , and 2/1, 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 (︀ 푛 )︀푛 (Wheaton College) Stirling’s formula 푛! ≈ 2휋푛 푒 is one of my all-time favorite “Are you serious?” results in mathematics. The factorial is simply a product of natural numbers, but yet we get a very good approximation for 푛! involving the square root of 휋 and powers of 1/푒, which are definitely not natural numbers. In this talk we’ll use nothing more complicated than integration by parts and a few diagrams to develop an approximation for 푛! that is very close to Stirling’s formula.

Statistics IIa STE 104 Chair: 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 relationships among predictor variables using various quantiles of the response variable. In contrast to standard linear regression which models changes in the average response, quantile regression examines changes in the quantiles of the response variable based on changes in the predictors, giving us more solid estimates based on these variations. Ecologists often use quantile regression to measure causal relationships, as it is a very eective method of predicting how dierent quantiles of organism responses, such as population size, are aected by various environmental factors. It is also an eective tool for detecting abnormal growth patterns in growth charts. Quantile regression allows us to distinguish the complex manner in which predictors aect the response variable. This presentation will introduce the concept of quantile regression and discuss how some of the common standard regression ideas, such as model selection and resampling based inference, can be incorporated 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 be overly aected by violations of assumptions. Robust regression models are not as sensitive to outliers as ordinary least squares estimates are. In the case of the presence of outliers, least squares estimation is inecient 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 selection and resampling based inference is done in the robust case. 2:40-2:55 Principal Component Analysis and Outliers Eect (Level 2) Steven Martinez (Western New England University) Our interest is to study Principal Component Analysis and some of its applications in the same time we would like to study the eect of outliers on the components of the PCA. PCA is a multivariate statistical procedure that uses an orthogonal transformation, of a matrix data, to transform a set of variables into a set of linearly uncorrelated variables called principal components. The ﬁrst component is the one with highest variability of the data. The second component has the second largest variability explained of the data and so on. Hence the number of components is less or equal to the number of original variables. We take as many components as we need to satisfy the “parsimonious principle,” but explain as much variability as possible. We are interested in deﬁning the PCA, explain its properties and show some few working examples with the eect of outliers included. Statistics IIb STE 102 Chair: 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 what enables students to thrive at 4-year postsecondary institutions is crucial in supporting their academic and future success. In particular, it is essential that incoming students be in the best possible position 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 into these courses according to their score on the mathematics section of the SAT (unless otherwise challenged). The research conducted in “Investigating Student Success at Southern: The Role of Mathematics” an Honor’s Thesis in the Mathematics department, aims to investigate mathematics education at SCSU. Specifically, the study investigates the ecacy of mathematics course placement based on the math section score of the SAT, the ecacy of math course placement based on pre-college information such as high school GPA, math courses taken in high school, etc., and how performance in their first math course at Southern influences students’ cumulative GPA, per- sistence, and graduation. Substantial empirical research in postsecondary mathematics education supports 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 success in mathematics and overall success in college. The results of the study will inform admission pro- cedures, influence mathematics course placement policies, and inform educational perspectives involving mathematical competency and overall academic success at Southern Connecticut State University. 2:00-2:15 A Propensity Score Analysis of High School Type on Post-Graduate Outcomes (Level 1) Sarah Markiewicz, Michael Lopez (Skidmore College) Multiple linear regression is a popular tool to 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 populations analyzed have dierent covariates’ distributions. For example, a model estimating the eect of public versus private secondary education could be confounded by underlying dierences in the characteristics of students attending each type of school. One tool that can account for these types of baseline dierences is the propensity score, defined as the probability of receiving a certain treatment. We implement a propensity score analysis to estimate the eect of secondary school choice 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% for each digit, 1 through 9. For some sets of data this is not the case. Rather, the number 1 is more frequently 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 history of Benford’s Law, from its inception in 1881 to its rejuvenation in 1938. Following the ideas of Havil, using a result gained from the Weyl Equidistribution Theorem and probability theory, we will 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. Lawrence University) Linear discriminant analysis (LDA) is a classification technique commonly used for dimensionality reduction. LDA uses existing information to compute latent explanatory variables that maximizes separation between multiple classes. Like other classification techniques (such as linear regression), LDA can be used for predictions and to determine important variables in the model. 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 102 Chair: Je Jauregui

3:30-3:45 Gyroid Surface (Level 1) Rachel Field (Saint Michael’s College) In this article we will discuss the formation of the Schoen gyroid surface as well as some of its applications in the natural world 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 other natural occurrences of the gyroid. We will also discuss the history of the gyroid as a minimal surface, its discovery and what makes it so special. 3:50-4:05 Numerical Boundary Control of Wave Equation (Level 2) John Curtin (State University of New York at New Paltz) The goal of this project is to find the boundary controls to bring the wave function from one prescribed state to the other resulting in minimum energy. This will be done using numerical methods on the wave equation, namely those involving the finite dierence method. This will be done using MATLAB. 4:10-4:25 Mathematics of engineering: building a supersonic rocket (Level 2) Noah Turner (Western Connecticut State University) A project aimed at designing and analyzing flight and performance 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 mathematical issues involved in the project such as simulation of motion and optimization of physical parameters for better performance characteristics. The presentation will include visualizations of the computational results, as well as a brief discussion of the autonomous on-board control. It will conclude with a video of a launch of a similar supersonic rocket previously built by the presenter.

Applied Math IIIb JEM 378 Chair: 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 the Turning-Grille Method. This can be done on a square grid or a linear grid. We will examine the dierences between a square and linear grid by going through an example in detail. We will also discuss the mathematics behind each grid. 3:50-4:05 Programmatic Art Through Ray Marching Primitives (Level 1) David Johnston (Champlain College) Ray marching is a graphics programming technique that is used primarily in games and graphics demos. While ray marching can be used for general rendering, it is more commonly used for cooking up speciﬁc eects such as displacement mapping, soft shadows, and sub-surface scattering. In contrast to classic ray casting, ray marching is simpler to implement, parallelizes well on modern graphics processing units (GPUs), and can be used on shapes that do not have analytic intersection functions. In this presentation we will focus on ray marching primitives using the OpenGL Shading Language (GLSL), with a little dip into shading those primitives if time allows.

Combinatorics III JEM 362 Chair: William Zwicker

3:30-3:45 Catalan Numbers and Fine Numbers (Level 1) Ajay Barde (Skidmore College) The Catalan numbers 퐶푛 are among the most celebrated integer sequences in mathematics. They are known to enumerate scores of dierent combinatorial objects, from legal strings of pairs of parentheses, to Dyck paths in the plane, to edge triangulations of an 푛-gon, to non-crossing set partitions. The Fine numbers 퐹푛 are another integer sequence which are related to the Catalan numbers.

In this talk we present some recently discovered identities which express 퐶푛 in terms of other Catalan numbers, as well as an identity which expresses 퐹푛 in terms of the Catalan numbers. The approach 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 approach is 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 푠(푛, 푘) can be defined as the coecients in the expansion of the “falling factorial” function: 푥푛 = 푥(푥 − 1)(푥 − ∑︀푛 푘 2) ··· (푥 − 푛 + 1) = 푘=1 푠(푛, 푘)푥 . Stirling numbers of the second kind, denoted 푆(푛, 푘) count the number of ways to break up or partition a set with 푛 elements into 푘 nonempty, disjoint subsets. There are many combinatorial identities known for both types of Stirling numbers, including identities that relate the two kinds together. In this talk we present a recently discovered identity which expresses 푆(푛, 푘) in terms of the 푠(푛, 푘)’s. The approach 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. This approach has previously led to proofs of identities (including some new discoveries) of several famous integer sequences, such as the Bernoulli numbers, Euler numbers, Bell numbers, and Catalan numbers. We can now add Stirling numbers to the list. 4:10-4:25 Grandmama de Bruijn Sequence (Level 1) Pat Dragon, Oscar Hernandez, Aaron Williams (Bard College at Simon’s Rock) A de Bruijn sequence is a binary string of length 2푛 which, when viewed cyclically, contains every binary string of length 푛 exactly once as a sub- string. For example, 000100110101111 is a de Bruijn sequence for 푛 = 4. The most popular de Bruijn sequence, often called the Ford sequence, for each value of 푛 is the first in lexicographic order 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 rotation. This talk will focus on a new construction of de Bruijn sequences for each 푛 used to generate 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 diers from the FKM algorithm 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 푛 bits in the sequence, that builds the Grandmama de Bruijn Sequence. We will prove that either of these constructions results in a valid de Bruijn sequence and that the resulting sequence is distinct from the Ford sequence. If time permits, we will outline the proof for arbitrarily large alphabets (instead of the binary alphabet {0, 1}).

Dierential Equations III JEM 166 Chair: 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 appropriate for 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 spread through water or in water with the presence of Heteroptera water bugs (transmitted when biting humans). 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 model for the bug population as seen in some malaria models. The system requires several parameters that are adjusted for realistic results. The parameters are then tuned to admit epidemic and non-epidemic solutions, showing that the model admits both types 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 the disease spread can be stopped. This may be more cost-eective than antibiotics. 3:50-4:05 Internal Dierential Equations (Level 1) Benjamin Oltsik (Hamilton College) What started as a mere passing thought at prom night developed into a characterization of solutions to an important class of dierential equations. We are calling these internal dierential equations because they have the form 푦′ = 푦(푥 − 1)(푦 OF 푥 − 1). In this talk, we will describe a unique approach to solving and generalizing solutions to internal dierential equations. We will also discuss why internal dierential 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 for understanding solutions to dierential equations, but time consuming and not visual in nature. In this presentation I will talk about using the Runge-Kutta equations and the software GeoGebra to construct a tool that numerically estimates solutions to any ordinary dierential equation. The tool takes as input a dierential equation with an initial condition and produces a numerical estimation 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 solutions of a dierential equation.

Fractals and Chaos JEM 377 Chair: Daniel M. Look

3:30-3:45 Fractal Landscape Generation (Level 1) Brianna Healy (Saint Michael’s College) We investigate the fractal qualities of natural landscapes, examine the length British coastline, and discuss computer generation of fractal landscapes. True landscapes are generated using height maps while realistic landscapes are generated randomly using various algorithms. We focus primarily on the diamond-square algorithm and the smoothing eects created by multiplying two or more fractal landscapes together. Fractal landscapes can be applied to computer-generated imagery techniques 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 dynamical systems arise naturally when modeling physical phenomena. In this presentation we investigate the behaviors of the Rikitake Model of Geomagnetic Reversal. This model provides insight into how 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 of the fixed points change under various parameter regimes. In the Rikitake Model, disparate time scales lead to dierent 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 exist in phase space. Solutions to the model are illustrated using numerical methods in MATLAB.

History of Mathematics JEM 281 Chair: 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 for granted. What were number systems like in their infancy? Who is credited with the first written examples of number systems, operations, and fractions? Take a journey back in time to discover the lasting eects of ancient mathematics and the reasons why we still use the same ideas today. 3:50-4:05 휋 = 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 to be about 3; however, the exact value, known as pi, was unknown. Pi was first calculated by the 10 1 ancient Greek Archimedes to be between 3 71 and 3 7 . Since then, mathematicians have been trying to find the best approximation for the irrational number. Today, pi has been calculated to trillions of digits, but that could not have been done without the work of mathematicians such as Archimedes, Euler, Wallis, and others. This presentation will examine the ways pi has been calculated — 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 seen Girolamo Cardano’s method for solving the general cubic equation 푥3 + 푏푥2 + 푐푥 + 푑 = 0. Car- dano published his solution in his Ars Magna or The Rules of Algebra in 1545 CE. We will look at the history of his solution, the method that he used, and its modern formulation. We shall see that his solution was a process (or an algorithm), not an algebraic formula, and that instead of solving the general cubic, he solved thirteen separate cases of the cubic.

Knot Theory JEM 389 Chair: 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 crossings alternate under, over, under, over, etc. as one travels along each component of the link. A link diagram is almost-alternating if one crossing can be changed so that the diagram is alternating. 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, we discuss 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) We will examine the relationship between the number of components of a link and the Conway notation associated with that link. After showing that a rational link possesses either one or two components, we will then provide a method for determining the number of components in the link based 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 introduction to knot theory will include group theory concepts for knots, knot isomorphisms, and knot classiﬁcation. Alexander polynomials will be discussed after sucient background is provided.

Mathematics Education II JEM 375 Chair: 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 Governors University) In this talk we will share various elements used in flipped, blended, and online classrooms that can enrich your face-to-face course. Your presenters have many years of experience in face-to-face, flipped, blended, and online teaching, particularly in calculus. While the focus will be 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 (Champlain College) At Champlain College we have a popular Game Programming major packed with industry-specific course work. Their curriculum does not have the credit allowance for the standard 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 within the context of Game Programming applications and projects. This carefully constructed 3-credit course can have students performing moderately sophisticated mathematics by the end of the first year of course work. What started out as a math class for game-programming majors is now a popular math course for all of our undergraduate students who want or need exposure to upper level mathematics but do not have the credit allowance for the standard sequence of math courses to 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 triples and the Pythagorean theorem. Also, we’ll examine a few of its derivations and some of its extensions, including an interesting result due to ancient Greek mathematician Pappus of Alexandria.

Number Theory III JEM 364 Chair: Blair Madore

3:30-3:45 A Conjecture on Sums of Consecutive Polygonal Numbers (Level 1) Samantha Wyler (SUNY New Paltz) A polygonal number 푃 (푔, 푛) is a number represented as dots arranged in the shape of a 푔-sided regular polygon. In this talk, we will show that every other triangular number (푔 = 3, 푛 even) starts of the next group of consecutive polygonal numbers such that if you have one more number in the lower group (푛 + 1 terms) then in the higher group (푛 terms), then the sum of the lower numbers will equal the sum of the higher numbers + (4 − 푔)푃 (3, 푛). 3:50-4:05 Proofs and Applications of Fibonacci Numbers (Level 2) Alec Covey (SUNY Potsdam) This expository talk will show the Fibonacci sequence is a basis for many profound, mathematical concepts. We will show connections between Pascal’s Triangle and the Fibonacci, demonstrate that the Fibonacci sequence can be expressed by an application of the Binomial Theorem, and will prove the Euler-Binet formula for Fibonacci numbers. The problem of tiling a (2 × 푛)-board with (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 this talk, we will explore some key concepts related to sound waves: the harmonic series, fundamental frequency, and the phenomenon known as the missing fundamental. In a combination of music theory and number theory, we will discuss how these concepts relate to the Greatest Common Divisor (i.e. GCD) and Least Common Multiple (i.e. LCM).

Probability and Statistics JEM 168 Chair: Ada Morse

3:30-3:45 How To Brew a Better Cup of Coee (Level 1) Scott Bianco (Siena College) Ever won- der how water goes through your coee grounds to make that perfect cup of joe? Well there is a whole theory behind it called Percolation Theory. We will look at the two of the main types of percolation thought a set of regular lattice points. The first type called a bond percolation considers the lattice graph edges as the relevant entities. Site percolation considers the lattice graph vertices as the relevant entities. The relevant entities are either occupied with probability 푝 or va- cant with probability 1 − 푝 which either allow of block fluid flow respectively. This allows us to study more than how to brew coee. We can use it model heat flux, electric current, spread of infections 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 commonly collected and studied in fields such as geology, ecology, and the social sciences. Researchers often 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. In this 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 ) Janelle Fredericks (St. Lawrence University) There are many ways to understand how dierent datasets can be clustered together. First, we will take an in depth look at Gaussian Mixture Model Clustering via the EM algorithm to see how points are determined to be in each individual cluster. Second, we will use simulated data with a predetermined number of cluster to examine how dierent 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 푘-means algorithm, and hierarchical clustering techniques to determine which methods are optimal in dierent clustering situations.

Statistics III STE 104 Chair: 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 beliefs about location of a parameter. We use the R shiny package to create an interactive web app that allows users to specify a prior distribution, input data and observe the resulting posterior distribution. We discuss how this app can be used to demonstrate Bayesian inference for parameters such as 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 on sample 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 formula based 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 many samples from dierent populations and then compare the coverage rates and widths of the intervals with each method. We vary the population proportion and sample sizes to explore which methods might work best in dierent 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 data samples. It enjoyed recently successes in detecting human faces, automatic diagnostics, and speech recognition, to name just a few fields. Machine learning algorithms are built based on powerful mathematical tools and techniques from statistics, linear algebra, and optimization theory. The focus of this presentation is on analysis of mathematical underpinnings of a popular classification algorithm of Machine Learning and the eect of parameters involved in the optimization stage on the algorithm performance. Applications to real data sets will be also provided.

Tropical Mathematics JEM 380 Chair: 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. There are, however, other ways one can define these operations. In tropical mathematics, one defines the tropical addition, ⊕, of two real numbers 푎, 푏 ∈ R as the minimum of the two numbers, or 푎 ⊕ 푏 = min {푎, 푏}, and the tropical multiplication, ⊙, as the sum of the two numbers such that 푎 ⊙ 푏 = 푎 + 푏. This talk will present many of the interesting properties that can be deduced from these 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 computational biologists. In tropical mathematics (also called a max-plus algebra), the operation of addition becomes the maximum of the two values and the operation of multiplication changes to standard addition. Tropical mathematics can be used to address problems in industrial processes, capacity assessment, and network analysis. In this talk we will take graphical and tropical linear algebra approaches to describe processes (from trac to industry to vending machines) and introduce methods 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 project is using Fibonacci sequence to investigate patterns on pineapples. The initial approach of this project is the investigation of 2-D planar models. The arrangement of sunflowers’ seeds can be represented by Maple. It gives a general idea about how to use Fibonacci sequence in this application. 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 mainly focused 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 in it. The model will be printed by using a 3-D printer and laser cutter for the sunflower seed patterns. Index of Authors Afinogenov, Sergey Applied Math IIa Fredericks, Janelle Probability and Statistics Armstrong, Sam Combinatorics III Ashline, George Mathematics Education II Gall, Evan Topology Gibson, Julia Abstract Algebra II Bania, Katelyn Geometry II Gingrich, Ross History of Mathematics Barde, Ajay Combinatorics III Barrett, Scott Computer Science Haehl, Alexandria J. Statistics Ia Barstow, Ryan Graph Theory Halden, Derek Combinatorics II Bartlett, Josiah Statistics Ia Hall-Hooper, Khalil Combinatorics II Bassette, Reid Dierential Equations I Hamelin, Ryan Tropical Mathematics Bell, Sydney A. Statistics Ia Harding, Nicholas Paradoxes Bennett, John Knot Theory Healy, Brianna Fractals and Chaos Bennett, Shayna Dierential Equations III Hemingway, Adam History of Mathematics Bianco, Scott Probability and Statistics Hernandez, Oscar Combinatorics III Blue, Jennifer Mathematics Education II Hodge, Nate Applied Math IIa Bouchard, Katharine Mathematics Education I Holbrooks, Johnathon Graph Theory Boyadzhiyska, Simona Combinatorics I Holmes, Cheryl Applied Math IIb Bryant, Julie Combinatorics II Howard, Sebasttian Graph Theory Buck, Gabrielle Dierential Equations II Huibregtse, Mark Geometry II Bulger, Shauna Statistics Ia Hurlbut, Curtis Statistics IIa Hurlbut, Curtis J. Statistics Ia Cahn, Patricia Graph Theory and Topology Cameron, Robert Statistics IIb Ianzano, Chris Analysis Chanin, Margo Computer Science Isham, Kelly Number Theory I Chernov, Vladimir Graph Theory and Topology Cheverie, Jacob Tropical Mathematics Jauregui, Je Analysis Chong, Catrice Number Theory I Jia, Miaoqing Number Theory I Claassen, Thaddeus Computer Science Johnson, Tristan Geometry I Cormier, E. Graph Theory Johnston, David Applied Math IIIb Covey, Alec Number Theory III Curtin, John Applied Math IIIa Kadas, Zsuzsanna M. Applied Math IIb Czupryna, Michael Mathematics Education I Kenney, Kristen Combinatorics I Klein, Paulette Dierential Equations II Daley, Marcella Topology Kramer, Sean Applied Math IIb Daniere, Adam Paradoxes Darrow, Jr. , Brian Statistics IIb Landfair, Kegan Number Theory II Dasbach, Oliver Knot Theory Lane, Callum Combinatorics II David, Logan Applied Math IIa LaPlante, Sara Probability and Statistics Davis, Andrew Statistics III Latulippe, Jocelyn Fractals and Chaos De Silva, Namini Applied Mathematics Ib Le Fevre, Scott Dierential Equations III Demers, Mark F. Analysis Lengieza, Michael L. Statistics Ia Dever, Lindsay Abstract Algebra II Lerma, Rebecca Applied Mathematics Ib Dillery, P. Graph Theory Li, Boris Tropical Mathematics Disher, Conor Graph Theory Li, Simon Dierential Equations II Dragon, Pat Combinatorics III Liu, Nanjiang Statistics III Dragon, Patrick Topology Lopez, Michael Statistics IIb Lowrance, Adam Knot Theory Edmondson, Mackenzie Geometry I Luangrath, Narin Graph Theory and Topology Escobar, Elizabeth Statistics IIa Lumbra, Celsey Geometry I Lyons, Amanda Statistics IIb Ferlini, Vincent Geometry II Field, Rachel Applied Math IIIa Mao, Jessica Probability and Statistics Marino, Katherine History of Mathematics Soper, Hannah Applied Mathematics Ia Markiewicz, Sarah Statistics IIb Stevens, Scott Mathematics Education II Martinez, Steven Statistics IIa Mayer, Philip Analysis Tank III, John H. Statistics Ia McCarthy, Kristen Geometry II Theobald, Michael A. Statistics Ia McGrady, Elizabeth Number Theory I Tremblay, Samantha Graph Theory McGraw, Brooke Statistics IIb Trenk, Ann Combinatorics I McHugh, John Abstract Algebra II Trono, John A. Statistics Ia McHugh, John Abstract Algebra I Turner, Noah Applied Math IIIa McIntyre, Andrew Graph Theory and Topology McLaren, Samuel Dierential Equations I Van Ryzin, A. Applied Math IIa Mischell, Melissa Graph Theory and Topology Vandermause, Jonathan Applied Mathematics Ib Moges, Salem Dierential Equations I Variashvili, Guga Combinatorics II Molino, Gianmarco Dierential Equations I Vees, James Analysis Monhart, Matthew Statistics Ia Vella, David Combinatorics II Morfe, Peter Analysis Vo, MyVan Probability and Statistics Vrijmoet, Mieke Applied Math IIb Nasser, Calista Number Theory III Warzer, Rebecca Linear Algebra Northshield, Sam Number Theory II Webster, Stephanie Abstract Algebra II O’Keee, Bailey Statistics Ib West, Kelsey A. Statistics Ia O’Keee, Bailey J. Statistics Ia Westland, Melissa Applied Math IIa Ogle, Scott Linear Algebra Westland, Melissa Knot Theory Oltsik, Benjamin Dierential Equations III Whelan, J. Graph Theory Orzell, Peter Computer Science White, Lauren Applied Math IIb Ostby, Lance Fractals and Chaos Wilde, Meredith Abstract Algebra II Williams, Aaron Combinatorics III Paight, Heather Number Theory II Wood, Kimberly Applied Math IIIb Pan, Xuehang Statistics Ib Wyler, Samantha Number Theory III Park, GaYee Number Theory I Peterson, Amanda Abstract Algebra I Yoo, Elizabeth C. Analysis Petrics, Gregory Dierential Equations III Young, Henry Abstract Algebra I Phaneuf, Autumn Applied Mathematics Ia Yun, Claudia Abstract Algebra II Pincus, Daniel Combinatorics II Zachos, Damaris Dierential Equations I Qi, Scarlett Statistics III Zwicker, William S. Computer Science Quenell, Gregory Paradoxes

Ramrath, Bethany Geometry I Ransom, Colton F. Statistics Ia Ratkevich, Jake Combinatorics II Ratli, Kenneth Applied Mathematics Ia Ratli, Tommy Number Theory II Raymond, Elizabeth Mathematics Education I Reams, Robert Linear Algebra Reed, Elise Applied Math IIb Resh, J. Graph Theory Rockmore, Daniel Applied Mathematics Ib Romagna, Christopher J. Statistics Ia

Schuckers, Michael Statistics Ia Shuchat, Alan Combinatorics I Shull, Randy Combinatorics I Silver, Jennifer Mathematics Education II Simoes, Julia H. Statistics Ia 1 P

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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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