Undergraduate Student Speakers

Undergraduate student presentations, Friday 11/16, 5:00-6:30 PM

Session I – Needham Room

5:00-5:12

Delia O'Connor, Plymouth State University

Title: Triangular Taxicab Geometry

Abstract: How do we measure distance between two locations when in New York City where one can only travel along streets and avenues? Taxicab Geometry is modeled after this popular question. Triangular taxicab geometry is an adaptation on the typical taxicab geometry. This alteration will cause alterations with definitions normally taken for granted. We will start from scratch with these definitions and build up.

5:15-5:27

JP Arico, Plymouth State University

Title: The Legend of Question Six

Abstract: The sixth and final question of the 1988 International Math Olympiad has gained a reputation in mathematics for its difficulty. Of the 270 participants in the 1988 IMO, only 12 were able to successfully solve question six, and the problem continues to stump to this day. Question Six states, ``Let a and b be non-negative integers such that ab+1 divides a2+b2. Show that (a2+b2)/(ab+1) is a perfect square." This talk will outline the methods used in solving this question.

5:30-5:42

Isaac Bleecker, Gordon College

Title: A Portrait of Galois

Abstract: Many have heard of Évariste Galois’ mathematical career, and how it ended with his death in a brutal duel at age 20. What led such a young and promising mathematician to that end? The full picture of Galois is one of brash disposition, incredible mathematical intuition, and many unfortunate situations.

5:45-5:57

Caitlyn Miller, Plymouth State University

Title: A Vector Approach to Integration by Parts

Abstract: While integration by parts uses a known formula for evaluating an integral of a product of functions, we suggest an alternative method called the UTP method. Through generalizing patterns that arise from solutions of integrals that require the integration by parts method, we use vectors to represent these patterns. We can evaluate integrals that require several iterations of integration by parts with only one iteration of the UTP method. The UTP method is an organized and fast approach to evaluating an integral that requires the standard parts formula.

Session II - Wellesley Room – Moderator: Prof. Jason Molitierno, Sacred Heart U.

5:00-5:12

Lucinda Cahill, Sacred Heart University

Title: Difference Equations and Lyness' Equation

Abstract: This presentation will delve into Lyness' Equation. Lyness' Equation is a second order difference equation with some interesting properties that will be presented and analyzed throughout this presentation. Equilibrium points and the dynamics of this equation will be discussed and a special case that exhibits ``global periodicity" will be explored.

5:15-5:27

Stephen Clarke, Sacred Heart University

Title: Quantum Computing: A Mathematical Analysis of Shor's Algorithm

Abstract: This presentation will explore how quantum computers work from a base level and look at mathematical functions, specifically finding prime factors of large integers. Quantum computers offer new methods of doing this which led to things like Shor's Algorithm which is tremendously more efficient and faster than classical counterparts. This will lead to understanding real world effects of quantum algorithms on fields such as Cybersecurity, Computer Science, Optimization and Mathematics.

5:30-5:42

Elle Bowe, Trinity College

Title: Methods of Reduction in Finite Spaces

Abstract: This talk introduces the motivation behind open and closed reduction and some intuition based on the equivalence between homotopy groups of finite topological spaces and their associated simplicial complexes.

5:45-5:57

Thu Bui, Trinity College

Title: Spectral Analysis of Hanoi Graphs

Abstract: In this talk we will discuss Hanoi graphs, which can be thought of as finite approximations to the Sierpinski Triangle fractal. Hanoi graphs arise from the classic Towers of Hanoi game, and we will present theoretical and numerical results about the spectrum (eigenvalues) and vibration modes (eigenvectors) of the sequence of Laplacian matrices for the Hanoi graphs.

Session III – Olin Hall Room 101 – Moderator: Rob Poodiack, Norwich University

5:00-5:12

Eric Bauerfeld, Southern Connecticut State University

Title: Constructing Maximally Planar Graphs via Connected Outerplanar Graphs

Abstract: In an honor's thesis, we investigated the four color theorem. We attempted to find an inductive proof of the result, which has been previously proven with the help of a computer (``Every Planar Map is Four Colorable'', Appel and Haken, 1977). Although the computer approach has been improved by Gonthier (``Formal Proof - The Four Color Theorem'', 2008), many would still like to see a ``human'' proof. Realistically, we did not anticipate success, but our investigation yielded some interesting mathematics. Specifically, we describe a characteristic set of induced subgraphs for arbitrary embeddings of maximally planar graphs and explain both how these are constructed and how this set might be used in coloring algorithms.

5:15-5:27

Jack-William Barotta, Massachusetts Institute of Technology

Title: The onset of chaos within droplets in steady Stokes flows

Abstract: In this expository work, we will consider the Streamlines inside a neutrally buoyant spherical drop immersed in a general linear flow as shown by Strogatz et. al. 1991. The scope of this project will be to demonstrate the onset of chaotic streamlines that arise from a steady Stokes flow. We show that the chaotic behaviour is entirely governed by both the orientation and magnitude of the vorticity vector.

5:30-5:42

Lisa Cenek, Amherst College

Title: Chorded Pancyclic Properties in Claw-Free Graphs

Abstract: A graph G is doubly chorded pancyclic if G contains a doubly chorded cycle of every possible length m for 4 - m - |V(G)|. We completely characterize the pairs of forbidden subgraphs that guarantee doubly chorded pancyclicity in 2-connected graphs. In this talk, we give a sample of the forbidden subgraph technique used for finding chorded cycles. We also find conditions for generalizations of doubly chorded vertex-pancyclicity and construct graphs that indicate the sharpness of our results.

5:45-5:57

Liubou Klindziuk, Amherst College

Title: Quantum Computing: A Mathematical Analysis of Shor's Algorithm

Abstract: Predicting medical diagnoses early is critical as it can improve treatment outcomes and ultimately save patient lives. Machine learning can help doctors make early predictions by leveraging an abundance of electronic health data. However, medical data is difficult to feed into predictive models because it typically has a large number of missing values. In this work, we propose an early diagnosis prediction model. First, we design an LSTM (Long Short-Term Memory), a type of neural network effective for modeling long time series. Second, to address the issue of missing data values, we implement various data imputation techniques and evaluate their effectiveness when used with the LSTM. Finally, we develop a novel LSTM model named Multi-Label Early Detection (MED), which has the goal of predicting patient diagnoses early in their hospital stay. We compare MED to state-of-the-art baselines using a subset of time series from the MIMIC-III database. We verify that our model obtains comparable AUC scores to that of standard LSTMs while encouraging early predictions.

Session IV – Olin 102 – Moderator: Prof. Maria Fung, Worcester State University

5:00-5:12

Harmony Estabrook, Worcester State University

Title: Math and Pianos

Abstract: Music is beautiful in many ways, not the least of which is the mathematics that weaves its way into the pitches, notation, and rhythms. Narrowing our focus down to the piano, and how it creates sound, helps us to take an in-depth look at what music looks like when represented mathematically. We will look at the behavior of sound waves for specific pitches and consider how different pitches sound together and what that means for the functions that represent the sound.

5:15-5:27

Nathaniel DeVries, Worcester State University

Title: Abstract Algebra and Applications in Music Theory

Abstract: The group of symmetries of a regular polygon may seem like an abstract (no pun intended) topic, but they have remarkably familiar applications. This project will seek to explore the symmetries of the regular dodecagon, and how these symmetries can be heard through their actions on collections of tones from the Western musical scale.

5:30-5:42

Neil Rao, Worcester State University

Title: Theoretical Insights into Predation from the Kolmogorov Standard Model

Abstract: The Kolmogorov Standard Model is a mathematical description of the relationship between the population numbers of predators and prey in a simple system. By further simplifying the system involved so that it is composed of a homogeneous group of unicellular organisms reproducing by mitosis, it is possible to make particular assumptions regarding the behavior of the system as a whole. This provides insight into the general relationship between unspecified predators and prey.

5:45-5:57

Anthony Ter-Saakov, Boston University

Title: Finite Permutation Groups with Few Orbits Under the Action on the Power Set

Abstract: The set-transitive groups acting on an n symbol set (i.e. the groups that have n+1 set-orbits) have been classified in [1]. In this paper we completely classify the groups with n+r set-orbits for 2-r-5 as well as lay out a method that allows one to go further. [1] - R. A. Beaumont and R. P. Peterson, Set-transitive permutation groups, Canad. J. Math. 7 (1955), 35-42.

Session V – Olin 201 – Moderator: Prof. Vince Ferlini, Keene State College

5:00-5:12

MeiRose Neal, Stonehill College

Title: DNA Self-Assembly: Complete Tripartite Graphs

Abstract: We examine self-assembling DNA from a graph theory perspective based on the tile model. These tiles represent junction branched molecules whose arms are double strands of DNA. Given a target graph G, a pot of tiles is designed in order to achieve G in three different scenarios corresponding to distinct levels of laboratory constraints. We find the minimum number of tile types and bond-edge types for each of the three scenarios for complete tripartite graphs and cocktail party graphs.

5:15-5:27

Cameron Spiess, Keene State College

Title: The Hodgkin-Huxley Model and Mathematical Neuroscience

Abstract: The human brain communicates with many specialized cells called neurons. Neurons send electrical signals to each other through connections called synapses. We can model how a neuron sends these electrical signals with a system of differential equations. The specific system of equations that we will use to model the behavior of a neuron is called the Hodgkin-Huxley (HH) Model. The HH Model shows how a neuron behaves as it sends an electrical current throughout its membrane and releases the current through its synapse, which is called an action potential. We will discuss the pieces of the HH Model and how each piece helps show the behavior of a neuron going through an action potential. We will then show some examples of how the HH Model has helped advance mathematical neuroscience to get to our understanding of mathematical neuroscience that we have today.

5:30-5:42

Nicole De Almeida, Keene State College

Title: Integers, Permuted Digits and Averages

Abstract: This presentation starts with the following example: Begin with the positive integer 629 and write down the permutation of its digits, i.e. all ways to write a 3-digit number using the digits 6,2, and 9. In this case they will be 629, 692, 269, 296, 926 and 962. The average of these six permutations is 3774/6 = 629 which is one of the permutations. Along with this example I will present additional cases of numbers that also have this property. This presentation is the result of an investigation of why some integers have this property and others do not.

Session VI – Olin 202 -

5:00-5:12

Courtney DesRoches, Keene State College

Title: Fractions in Simplest Form

Abstract: A University of California, Berkeley Math Circle Problem gives the fraction (2n+2)/(2n+3) where n is an odd integer and asks whether it is in simplified form. This note will look at the general sequence of fractions (an+b)/(cn+d) having integer constants a,b,c,d and variable integer n where we assume that at least one of a and c is nonzero. Our objective is to give conditions on the constants that will yield fractions in the sequence that are all in simplified form, some in simplified form, and none in simplified form.

5:15-5:27

Andrew Nelson, Keene State College

Title: Tiling Polyhedra

Abstract: Have you wondered what properties exist for the shapes that infinitely tile a 2-D plane? In this presentation, I will discuss how you can break down a convex polyhedron to create an equivalent flat representation and how we know it is a tile-maker. Through visual examples, I will show the process of stamping a tetrahedron. I will use the facts gained to help compile a proof showing that a regular convex tetrahedron can tile a 2-D plane infinitely.

5:30-5:42

Lydia Ahlstrom, Keene State College

Title: Knot Games

Abstract: Take a string, tie knots in it, and connect the ends. You have created a mathematical knot in three-space! The necessary background information regarding knots will preface the introduction of the game To Knot or Not to Knot. This is one of many games that can be played on the shadow of a knot, where all crossings are left undetermined. Two players take turns choosing which strand crosses over the other. One player wishes to create the unknot, while the other attempts to create a knot.

Session VII – Olin 202 - Moderator: Professor Karen Stanish, Keene State College

5:00 – 5:12

Matthew Pittendreigh, Keene State College

Title: An Introduction to Machine Learning for Mathematicians

Abstract: Machine learning is a method for automatically building analytical models. When trained on an appropriate dataset, machine learning models can recognize patterns to accomplish tasks, and suppress those features that do not contribute to performance. This process is completed by applying familiar concepts from Calculus and Linear Algebra. In fact, the recent resurgence in machine leaning is largely due to our ability to complete complex mathematical operations more quickly or in parallel.

Machine learning models have permeated the technological landscape, contributing to problem domains such as: image recognition, advertisement targeting, language processing, and much more. We aim to answer such questions as: What is machine learning? What types of machine learning algorithms are prevalent? How is mathematics used to train and optimize models? What is parallel processing? We will also showcase examples of machine learning applications developed at Keene State College, programmed in both MATLAB and Python.

5:15-5:27

Danielle Wiley, Keene State College

Title: HS Sequences

Abstract: Consider a sequence that begins with 2 followed by 3. The product is 6 and the sequence now becomes 2, 3, 6. The next product 3×6=18 creates two values 1 and 8 and the sequence is extended to 2, 3, 6, 1, 8. The next product 6×1=6 results in 2, 3, 6, 1, 8, 6. The sequence will continue infinitely in this manner because each multiplication produces a single or double-digit integer, so either one or two values will always be added on the “tail” of the sequence. This sequence first appeared in a problem devised by Hugo Steinhaus. A sequence generated in this way can begin with any two nonnegative single digits and it will be referred to as an HS sequence. This talk will analyze the long-term behavior of these sequences.

5:30-5:42

Erin Granger, Western New England University

Title: Finding Inverses in Galois Fields

Abstract: The process of finding multiplicative inverses in Galois fields plays an important role in cryptography. These inverses are used in the computations of the Advanced Encryption Standard (AES). AES is the encryption standard used by companies, including Apple, and the U.S. government to encrypt sensitive information. The computations of AES are done in Galois fields and multiplicative inverses are used for one of the transformations of AES. Wang's Method can be used to calculate these multiplicative inverses.