Undergraduate Speaker Session

Session Index

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Undergraduate Session 1
2:05pm--3:05pm, Main Building 206
Speakers: Abigail Herrlin, Skyler Hiltebeitel, Noah Andrews, Charlie Humpert, Santure (Haoyuan) Chen, Peter (Ruijie) Liu

Undergraduate Session 2
2:05pm--3:05pm, Main Building 215
Speakers: John Bazaar, Kenneth Huang, Trevor Meintel, Aaron Shabon, Adam LaFountain

Undergraduate Session 3
2:05pm--3:05pm, Main Building 216
Speakers: Tasnia Kader, Joseph Trachtman, Gonghui Lin, Christopher Heitmann

Undergraduate Session 4
2:05pm--3:05pm, Main Building 217
Speakers: Michael Quinnan, Daresalam Ayalew, Brian Fodale, Shuyao Cai, Yuping Luo, Lin Zhu

Undergraduate Session 5
2:05pm--3:05pm, Main Building 218
Speakers: Collin Barber, Dennis Bromley, Ben Crawford, Kelsey DeAcosta, Ashhad Hanafi, Cole Hadley, Trevor Wylezik

Undergraduate Session 1, Main Building 206

2:05pm, Abigail Herrlin, Skyler Hiltebeitel (Messiah University)
Maximally Filled Staircase Tableaux
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Staircase tableaux are combinatorial objects which, similar to Sudoku, consist of boxes containing certain symbols based on a particular set of rules. While these structures have practical applications in areas such as civil engineering and physics, we focus on some theoretical properties related to the maximum number of symbols. In this talk, we will introduce staircase tableaux and present our results, which include the probability of a maximally filled staircase tableaux based on a mapping between tableaux of size $n$ and maximally filled tableaux of size $n+2$.
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2:20pm, Noah Andrews, Charlie Humpert (Messiah University)
Maximally Filled Staircase Tableaux (continued)
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[This continues the previous talk.] Staircase tableaux are combinatorial objects which, similar to Sudoku, consist of boxes containing certain symbols based on a particular set of rules. While these structures have practical applications in areas such as civil engineering and physics, we focus on some theoretical properties related to the maximum number of symbols. In this talk, we will introduce staircase tableaux and present our results, which include the derivation of an algorithm and formula for maximally filling a given diagonal of a staircase tableaux.
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2:35pm, Santure (Haoyuan) Chen (Franklin & Marshall College)
Importance of being prime: Measuring non-unique factorization
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The Fundamental Theorem of Arithmetic states that every integer greater than 1 has a unique factorization. This theorem does not hold in more general settings. By way of a specific example, this talk will introduce non-unique factorization and methods for characterizing the non-uniqueness of factorization.
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2:50pm, Peter (Ruijie) Liu (Franklin & Marshall College)
Importance of being prime: Distance from atom to prime
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By the Fundamental Theorem of Arithmetic, every integer larger than 1 factors uniquely as a product of primes. However, in more general settings, factorization can be nonunique. After using an example to introduce the distinction between irreducible elements (atoms) and primes in a commutative monoid, we will define the $\omega$-invariant, which quantifies the degree to which factorizations can be nonunique, and determine $\omega$ for this example.
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Undergraduate Session 2, Main Building 215

2:05pm, John Bazaar (Penn State)
The Gathering Number of a Graph and Vertex Degrees
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Let G be a non-complete, finite simple graph with vertex set V (G). We define the gathering number of G, denoted by g(G), to be the minimum value of |X| − ω(G − X) for all vertex cut-sets X ⊂ V (G), where ω(G − X) is the number of components of ⟨G − X⟩. Note that g(G) = −s(G), where s(G) is the scattering number of G. We discuss the motivation behind the definition of the gathering number, and we provide conditions on the degree sequence of G which imply g(G) ≥ k for a fixed integer k, with −|V (G)| ≤ k ≤ |V (G)|. We also indicate specifically how these degree conditions are best possible.
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2:20pm, Kenneth Huang (Penn State Brandywine )
Component Order Edge Connectivity and Vertex Degrees
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Given a finite, simple graph G, the k-component order edge connectivity of G is the minimum number of edges whose removal results in a subgraph for which every component has order at most k-1. In general, determining the k-component order edge connectivity of a graph is NP-hard. In light of this fact, we look to determine conditions on the vertex degrees of G that can be used to imply a lower bound on the k-component order edge connectivity of G. We will discuss the process for generating such conditions for a lower bound of 1 or 2, and we explore how the complexity increases when the desired lower bound is 3 or more. Along the way, we will present connections with extremal graphs and integer partition theory. These results are part of an undergraduate research project with Dr. Yatauro at Penn State Brandywine.
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2:35pm, Trevor Meintel, Aaron Shabon (Penn State Brandywine)
Cubic Knot Mosaics
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Knot projections can be constructed out of set of specific mosaic tiles, which were first introduced by Lomonaco and Kauffman in 2008. Typically, the tiles are arranged in a square grid and invariants such as the mosaic number (minimial size grid needed to build the knot with tiles) can be defined. We were interested in constructing knot mosaics on the surface of a three-dimensional object - in particular, a cube. In this presentation, we will introduce the notion of cubic knot mosaics, and define the cubic mosaic number of a knot. We will present some families of knots for which we have computed the cubic mosaic numbers, as well as explore some general bounds on the relationship between crossing number and cubic mosaic number. Finally, we will share some work in progress as well as some open questions that can extend this project further.
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2:50pm, Adam LaFountain (Penn State Brandywine)
Visualizing the 4th Dimension
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This talk is an exploration of visualizing 4-dimensional shapes using different methods. We first consider 3-dimensional objects and how to depict them through 2 dimensional shadows, or via a “movie” of cross sections. We then look into the 3-dimensional shadows that 4-dimensional shape will cast, and portray a 4-dimensional shape from a “movie” of 3-dimensional cross sections.
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Undergraduate Session 3, Main Building 216

2:05pm, Tasnia Kader (Temple University)
Orthogonal and Special Orthogonal Groups in Lean Theorem Prover
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Lean is a proof assistant created by Leonardo De Moura in 2013 as a Microsoft Research project. Its mathematical library, mathlib, is a collective endeavor to establish a comprehensive repository of formalized mathematics within the Lean proof assistant. In an effort to further expand the extensive mathlib, we worked on defining the orthogonal and special orthogonal groups in Lean. Then, with these definitions as a foundation, we were able to prove lemmas and theorems that were previously absent from the mathlib.
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2:20pm, Joseph Trachtman (Temple University)
Strategic Range Study
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This study seeks to address the reasons why Range and Borda disagree on who wins an election a significant portion of the time when looking at the theoretical generated election data. We will explain the two voting methods and show some examples for both, and show how they may disagree. We hypothesize that a portion of this discrepancy in the voting methods is caused by ballots who leave at least one option blank, as borda and range counts this differently. We attempt to understand the significance of using various values for blank range votes, and see if this change in value causes less of a discrepancy in the voting methods.
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2:35pm, Gonghui Lin (Temple University)
Gambling with Uncertainty
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In “Gambling Under Unknown Probabilities as Proxy for Real World Decisions Under Uncertainty”, David Aldous and F. Thomas Bruss brought up the allowance issue. The allowance issue explores strategies that take account of knowing the existence of error in a prediction. Our research focuses on adjusting preexisting strategies by considering the allowance issue. Simulations of coin flips will be involved to test strategies and the effectiveness of the allowance issue.
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2:50pm, Christopher Heitmann (Temple University)
Music in Group Theory: Finding Group Structure in Harmony
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It is clear that music and mathematics are linked in a multitude of ways, many of which are still unexplored. While music is first and foremost an art form, it is deeply rooted in mathematical structures and patterns. I will examine how transformations between musical triads can be formalized as group generators and how the resulting Cayley graphs show a profound link between these chords. Research on this topic has shown that the dihedral group of order 24 can be generated from the operations REL, PAR, and LT (defined by David Lewin) that transform major and minor triads. I aim to extend the collection of chords to include diminished triads and to define musical transformations that reveal underlying group structure. I will then construct Cayley graphs that highlight other musical relationships and symmetries not previously considered.
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Undergraduate Session 4, Main Building 217

2:05pm, Michael Quinnan (University of Scranton)
Network Analysis on the Spread of COVID-19 in Northeastern Pennsylvania
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COVID-19 has become a modern global phenomenon, as the disease has killed millions of individuals around the world. Globalization has only increased the risk of new pandemics arising in the future. Thus, it is crucial to study the COVID-19 pandemic, so that proper preparations can be taken to mitigate the risk of new pandemics. This presentation will attempt to aid in the understanding of COVID-19 on a more local scale, specifically for the Northeastern region of Pennsylvania. We will talk about various techniques we apply to simulate and analyze the COVID-19 dynamics in the NEPA area and how we utilize the Markov process to access different vaccination plans along with other quantitative analyses.
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2:20pm, Daresalam Ayalew (Garland High School, Garland Texas)
The Impact of Covid-19 on students learning in Mathematics in Garland High School, Garland Texas : A Case study on students understanding and clarity of Mathematics .
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The covid 19 pandemic has impacted many of learning opportunity especially those in large groups or involving in person interaction with peers. Due to the pandemic most of the curriculum is adapted to an online format so the changed format is likely to impact learning pedagogy effecting students. In my study I investigated the online learning challenges that students experience with a focus on student’s home learning , self-regulation , virtual learning environment and students over all learning experience in Mathematics in Garland High school and discuss the new results with possible recommendation.
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2:35pm, Brian Fodale (Millersville University)
Sensitivity Assessment of a Two-Step Method in Skin Image Identification
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Image processing is a field of an increased interest in the scientific/medical community, where the use of Red Green Blue (RGB) decomposition to extract image information has proven to be valuable. It has been shown that skin identification can be achieved by a two-step process: (1) partitioning the input data into clusters, and (2) estimating a binary predictor for each cluster for a final classification. In this project we focus on the problem of skin identification through cluster analysis, predictive modeling, and classification. Generalized Linear Models and Feed-Forward Neural Networks are evaluated for their performance in detecting skin images. We investigate each model’s performance and the techniques are applied to melanoma images. (Research conducted during the 2022 University of Iowa Summer Institute in Biostatistics)
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2:50pm, Shuyao Cai, Yuping Luo, Lin Zhu (Arcadia University)
Safer Home Program
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To protect Storslysia from risks resulting from extreme weather events caused by climate change, our team has designed a social insurance program with the goal of safeguarding the well-being, survival of individuals, and preservation of property and the environment. By assessing catastrophic risks based on historical hazard data, we made projections for future economic costs. Strategies for incentivizing domestic migrations are then proposed for hedging the risks arising from disasters. This work was submitted in participation for the 2023 SOA Research Institute Student Research Case Study Challenge.
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Undergraduate Session 5, Main Building 218

2:05pm, Collin Barber (Penn State Brandywine)
Rainbow Calculus
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In this presentation, we will delve into the fascinating calculus of rainbows and explore why they form at a 42-degree angle. We will explore the intricate interplay of light, water droplets, and geometry that creates this beautiful natural phenomenon. We will also discover how to identify the different classifications of rainbows.
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2:20pm, Dennis Bromley (Penn State Brandywine)
Golden Hexagons and Honeybees
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In this presentation, we will present various crocheted models that exhibit connections between honeybees, golden hexagons, and the Fibonacci sequence. We begin with a crocheted model mirroring the family tree of honeybees through the tenth generation, which follows the Fibonacci sequence. This model uses brown beads to represent male honeybees and yellow beads to represent female honeybees. Our second model uses regular hexagons with side lengths following the Fibonacci sequence to create an ever-increasing approximation of a golden hexagon. In this model, we have assigned colors to the leading digits 1-9 of the Fibonacci sequence as follows: 1 2 3 4 5 6 7 8 9. This color-coding allows us to observe a phenomenon known as Benford’s law. Our final model uses regular hexagons with side lengths of 5 units and 8 units. With these hexagons we are able to create an approximate golden hexagon. The string art in this model highlights the interesting geometry of a golden hexagon.
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2:35pm, Ben Crawford (Penn State Brandywine)
History and applications of Fractals
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The Mandelbrot set is a famous set of numbers named after mathematician Benoit Mandelbrot. The Mandelbrot set is defined as the set of all complex numbers that can be iterated through a particular recursive equation and converge to a constant value. This set of numbers can be represented by a picture on the complex plane that uses different colors to represent how fast different numbers converge. In this talk, we will discuss iterative functions, fractals, complex numbers, and how each of these work to form the well-known visualization of the Mandelbrot set.
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2:50pm, Kelsey DeAcosta, Ashhad Hanafi, Cole Hadley, Trevor Wylezik (Penn State/Kutztown)
Consecutive Integer Matrices
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We study square matrices of consecutive integers, i.e. $n\times n$ matrices where the $i^{th}$ row, $j^{th}$ column entry is $a_{ij} = k + (i-1)n + (j-1), (k \in \mathbb{Z})$. For example, \[A = \begin{bmatrix} k &&& k+1 &&& k+2 \\ k+3 &&& k+4 &&& k+5 \\ k+6 &&& k+7 &&& k+8 \\ \end{bmatrix}.\] We have discovered the general form of the characteristic polynomial for all sizes $n$ and starting values $k$, and other properties such as the trace, determinant, and eigenvalues. Other variations and generalizations of consecutive integer matrices are discussed.
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