Session Index
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Undergraduate Session 1
2:15pm--3:15pm, Frey 241
Speakers: Thomas Stauffer, Nevaeh Sisco, Juniper Haglund and Morgan Hutchinson, Ellery Panaia, Valentyn Sukhyy
Undergraduate Session 2
2:15pm--3:15pm, Frey 243
Speakers: Jordan Boggs , Callie Buffaloe and Aicha Konneh, Rohan Powell and Anna Nguyen
Undergraduate Session 3
2:15pm--3:15pm, Frey 343
Speakers: Olivia Delgiacco, Kathryn Gabriel, Riley Fabing, Benjamin McDowell, Andrew Moody
Undergraduate Session 4
2:15pm--3:15pm, Frey 345
Speakers: Megan Triplett, Kaden Hunter, Alexandra Ziegler, Peyton Haroldson, Madysen Bihun, Clayton Colgan
Undergraduate Session 5
2:15pm--3:15pm, Frey 349
Speakers: Evan Gibbs, Max F Trimmer, Grace Lippert, Dante Mancino
Undergraduate Session 1
Frey 241
2:15pm, Thomas Stauffer (Penn State Brandywine)
Group Theoretic Applications to Music Theory
View Abstract
From the periodicity of sound waves to the fractional counts of time measures, math and music are intrinsically intertwined. In this talk, we will focus on group theoretic applications of music theory. In particular, the set of 12 standard musical notes forms a group, an important object in the study of Abstract Algebra. Particular sets of chord inversions have interesting group and subgroup structures. Studying the interconnection between group theory and music theory provides a deeper understanding of both subjects.
Close Abstract2:30pm, Nevaeh Sisco (Penn State Brandywine)
Exploring color theorems on surfaces through crochet
View Abstract
The famous Four-Color Theorem states that any map requires at most four colors to color in such a way that no two adjacent regions share the same color. However, this theorem changes when the map is drawn on surfaces other than a plane. For example, the Four-Color Theorem actually becomes the Six-Color theorem when the map is drawn on the Mobius band and the Seven-Color theorem when the map is drawn on a torus. Through crocheted models, we will explore patterns of maps that require the maximum number of colors on their respective surfaces.
Close Abstract2:45pm, Juniper Haglund and Morgan Hutchinson (Saint Joseph's University)
Isaac Newton: "The Father of Modern Science"
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Isaac Newton was an English mathematician and physicist who was a key figure of the Scientific Revolution of the 17th century. He contributed many new concepts and theories to the mathematical domain during this time, and was the discoverer of the field of infinitesimal calculus. This presentation will touch on Newton’s early life, but aims to highlight the impacts and consequences of Newton's discoveries on the world of mathematics.
Close Abstract3:00pm, Ellery Panaia, Valentyn Sukhyy (Saint Joseph's University)
Muhammad ibn Musa Al-Khwarizmi Life and Significant Contributions to the History of Mathematics
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Muhammad ibn Musa Al-Khwarizmi (ce. 780-850) was a Persian Mathematician that made significant contributions to the history of algebra, trigonometry, and inspiring algorithms. He is known as the “father of algebra” from major works that he exemplified in his book Kitab al-Mukhtasar fi Hisab al-Jabr W'al-Muqabala. In this presentation we will aim to explain Al-Khwarizmi’s role in the history of mathematics and how he was a major contributor for the Islamic Golden Age.
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Undergraduate Session 2
Frey 243
2:15pm, Jordan Boggs (Saint Joseph's University )
Maryam Mirzakhani
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Among the many remarkable mathematicians in history, Maryam Mirzakhani stands out as one of the most celebrated. Known as a pioneer in mathematics, the “Queen of Mathematics,” and a trailblazer for women in the field, she made history as the first and only woman to receive the prestigious Fields Medal—the highest honor in mathematics. In this presentation, we will explore her life, her groundbreaking contributions to geometry and topology, and the lasting impacts of her work on the world of mathematics.
Close Abstract2:30pm, Callie Buffaloe and Aicha Konneh (Saint Joseph's University )
Bernhard Riemann: The Riemann Hypothesis
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Bernhard Riemann was a German mathematician most notably known for his work with complex analysis and number theory, which subsequently lead him to his famous Riemann Hypothesis. Some of Riemann’s most notable contributions include his Riemann Hypothesis providing Albert Einstein with the mathematical framework for his theory of general relativity. This presentation aims to explain Riemann Hypothesis while highlighting the importance of complex analysis and number theory in relation to the hypothesis.
Close Abstract2:45pm, Rohan Powell and Anna Nguyen (Saint Joseph's University)
The History of Sudoku
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This talk will be about the mathematical puzzle Sudoku. We will begin by defining what a Sudoku is. After the brief explanation, we will talk a little bit about its inventor, Howard Garns (1905-1989), and what led him to create Sudoku, and how it became popular in Japan. Lastly, we will also talk about the benefits of doing Sudoku. There is also a chance for the audience to try solving one for themselves! We will call on nine volunteers. Each will pick a number from 1-9 to fill in the puzzle, such that each row, column, and subgrid contains numbers 1-9 with no repeats.
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Undergraduate Session 3
Frey 343
2:15pm, Olivia Delgiacco (Lebanon Valley College)
Exploring Matchgate Circuits
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This presentation explores a pictorial representation of matchgate circuits, which are quantum circuits that have the same power as classical computers. Using the representation, we aim to simplify the input quantum states into the most basic generalized form achievable through matchgate circuits, otherwise known as matchgate equivalence. In our studies, we’ve shown a reduced state equivalence for up to five qubits.
Close Abstract2:30pm, Kathryn Gabriel (Kutztown University of Pennsylvania)
Interference Effects in Quantum Nonlinear Scattering
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Scattering by external potentials is a central dynamical feature of quantum systems. In coherent media, quantum interference causes scattering to become very sensitive to the barrier position. In a noninteracting (linear) system, this scattering pattern is sinusoidal and well described by analytical stationary solutions to the Schrödinger equation. However, in media with interparticle interactions, like those found in Bose-Einstein condensates (BECs), the dynamics becomes nonlinear. In prior work, the nonlinear scattering could in general only be described numerically, due to the breakdown of the superposition principle. We have developed a method to use analytical solutions of stationary states. We describe the BEC with a nonlinear Schrödinger equation, in a hydrodynamic picture where the stationary density distribution obeys a cubic equation. The roots take on a special significance, with two of them determining the amplitude of oscillations for solutions and the third defining the impacts of nonlinearity. They also are related to observable physical quantities, such as the current density and chemical potential. We relate these analytical solutions to the numerical results for scattering dynamics.
Close Abstract2:45pm, Riley Fabing (Kutztown University)
Collatz Conjecture
View Abstract
The Collatz Conjecture is a mathematical problem created by Lothar Collatz in 1937. The
expression is (3x+1)/2. This conjecture proposes that any integer will eventually become one
after repeatedly applying the following operation: if even, divide by two, if odd, multiply by three
and add one. The integers react differently depending on whether they are even, odd, negative,
or positive. The Conjecture follows a law that is often seen in large data sets. We will review
the problem's origins and how it has developed since then. Lastly, we will see what the goal of
the conjecture is, and why we as mathematicians continue to research it.
Close Abstract3:00pm, Benjamin McDowell, Andrew Moody (Kutztown University)
A First Look at Niven Numbers in Negative Bases
View Abstract
Niven numbers are numbers that are divisible by the sum of their digits. We take this concept and apply it in the rarely explored world of the negative bases. Our talk will explore brief novelties arising from negative bases, and then derive proofs and theorems for Niven numbers in negative bases. We will first walk through consequences of the counterintuitive way that negative bases affect digit sums, and use that fact to examine the longest possible sequence of Niven numbers.
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Undergraduate Session 4
Frey 345
2:15pm, Megan Triplett (Dickinson College)
On the Existence of Partition of the Hypercube Graph Into 3 Initial Segments of Given Lengths
View Abstract
Let $Q_n$ be a hypercube graph of dimension n. The initial segment $I_k ⊆ Q_n$ is the subset consisting
of the first $k$ vertices of $Q_n$ in the binary order. A pair of integers $(a, b) >0$ is said to be fit if, whenever
$2n ≥ a + b$, there exists $g1, g2 ∈ $Aut$(Q_n)$ such that $g1(I_a) ∪ g2(I_b) = I_a+I_b$, and $(a, b)$ is unfit otherwise. For $a + b + c = 2n$, there is a partition of $Q_n$ into $3$ initial segments of length $a, b,$ and $c$ if and only if $(a, b)$ is a fit pair. Thus, the notion of fit and unfit pairs is closely related to the graph-partition problem for hypercube graphs. This talk introduces a new criterion in determining whether $(a, b)$ is fit using an easy-to-compute
point-counting function and applies this criterion to generate the set of all unfit pairs. It further shows that the number of unfit pairs $(a, b)$, where corresponds to the number of surjection of an $n$-element set to a $4$-element set.
Close Abstract2:30pm, Kaden Hunter, Alexandra Ziegler, Peyton Haroldson (Millersville University of Pennsylvania)
Modeling Charged Particle Motion in a Uniform Magnetic Field Using ODEs
View Abstract
The motion of a charged particle in a uniform magnetic field is governed by the Lorentz force, which yields a system of ordinary differential equations (ODEs). This project explores the numerical and analytical methods used to solve these ODEs to model the trajectory of a proton in a uniform magnetic field. We analyze the proton's motion under specific initial conditions, including velocity, position, and magnetic field strength, assuming the absence of an electric field. Through this approach, we aim to better understand the behavior of charged particles in magnetic fields and validate theoretical predictions.
Close Abstract2:45pm, Madysen Bihun (Shippensburg University )
Decay of Botox Within the Human Body: A Mathematical Study
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In this mathematical study, we propose a mathematical model that simulates the dynamics of how Botulinum Toxin interacts with in the cholinergic nerve terminal. This process involves three steps, which includes initial binding, translocational, and lytic steps. We present some numerical results.
Close Abstract3:00pm , Clayton Colgan (Bucks County Community College)
Using Inferential Statistics to Study Growing Conditions
View Abstract
This study is to determine advisable practices on planting trees based on current growing conditions. Three candidate species were chosen which all had thier own, unique growing conditions (Northern Red Oak, Black Cherry and American Sycamore).
I. The soil was sampled across 30 different locations on the property.
II. Soil pH was tested using a pH meter at each location. The data was then collected and a confidence interval was calculated to ensure that if growing conditions are favorable without soil amendment, it is safe to assume with a certain level of confidence that anywhere on the property is also favorable.
III. Soil texture was tested using the traditional jar method (to separate the layers of sand, silt and clay). The layers were then recorded as percentages and measured on the USDA Textural Triangle for soil.
IV. Data from the soil pH study and soil texture study were then combined to determine if there is any linear correlation between the two data sets.
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Undergraduate Session 5
Frey 349
2:15pm, Evan Gibbs (York College of Pennsylvania)
Elucidating the Heterogeneity of Wing Morphology in D. melanogaster using Wasserstein Geometry
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Quantitatively understanding variation in the morphology of an organism is a difficult task regardless of species. In our research, we aim to tackle this problem with the wings of wild-type D. melanogaster. Through the novel application of Wasserstein Geometry, we interpret the boundaries and veins of a given wing as the support of a probability measure on $\mathbb{R}^2$ and calculate the 2-Wasserstein distance ($W_2$) between wings. This embeds the space of all wings as a submanifold of the Wasserstein manifold, allowing us to effectively leverage its structure to learn about the modes of variation of D. melanogaster wings. We discuss the process by which we are able to represent the movement between wings via Riemannian logarithms of this submanifold, thereby providing a more intuitive representation of the morphological differences compared to previous works. Further, as we can locally represent the manifold linearly, we make use of a unique linear technique, multiscale Singular Value Decomposition (mSVD), to gauge the intrinsic dimensionality and degrees of freedom of our space at a given wing. Through both techniques, we aim to effectively characterize the ways in which D. melanogaster wings can vary and present a new methodology for realizing morphological variation.
Close Abstract2:30pm, Max F Trimmer (Muhlenberg College)
Word Length of Reflections in Dihedral Groups Under 3-Reflection Presentations
View Abstract
In this talk, I will present my current research with a faculty member on word lengths of elements in dihedral groups. More specifically, I will present an upper bound for the word length of a reflection in a dihedral group with respect to a three-reflection generating set obtained using techniques from additive combinatorics. This solves a previous conjecture about a quantity $\lambda_1$ which measures the stability of a group under perturbations in the words corresponding to certain elements with respect to certain presentations.
Close Abstract2:45pm, Grace Lippert (Messiah University)
Alpha Beta Patterns on the First Diagonal in Staircase Tableaux
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In this talk, we will introduce combinatorial objects called staircase tableaux. Staircase tableaux are connected to an interesting particle model in statistical mechanics called the asymmetric simple exclusion process. The steady state distribution of this model can be expressed in terms of staircase tableaux. In this talk, we generalize some previous results about the distribution of alphas followed by betas on the first diagonal of staircase tableaux which relates to potential movement within the model.
Close Abstract3:00pm, Dante Mancino (Bucknell University )
Linearizable Instances of the Quadratic Minimum Spanning Tree Problem on 3-connected Graphs
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The quadratic minimum spanning tree problem (QMSTP) is graph-based combinatorial optimization problem that, in general, is notoriously difficult to solve. We study special easily-solvable instances of the QMSTP that are known as linearizable, meaning that the quadratic objective can be equivalently rewritten as linear in a manner that preserves the objective function value of each feasible solution. Previous work has shown that a sufficient condition for linearizability is that the (symmetric) matrix of cost coefficients is a weak sum matrix. This work also showed that this sufficient condition becomes necessary for linearizability when the underlying graph is a complete graph. In our work, we broaden this result by proving that this sufficient linearizability condition is necessary if and only if the QMSTP instance is defined on a 3-connected graph.
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