Papers/talks presented by undergraduates
All undergraduate participants in our Section meeting are invited to submit a title and abstract for consideration in our Undergraduate Paper Sessions.
When submitting, please include which of the broad MSC 2020 categories (e.g. “65 Numerical analysis”) is best aligned with your proposal by including it in the “Private Notes” field.
Many polynomial equations cannot be solved by a simple formula like the quadratic formula, yet their roots still follow precise symmetry patterns. In this talk, we study the family *f(x) = x<sup>10</sup> + ax<sup>5</sup> + b* and ask: as the parameters \(a\) and \(b\) vary, what symmetry patterns can its ten complex roots exhibit? These patterns are described by the *Galois group*, which measures how the roots can be rearranged without changing the algebraic relationships they satisfy. Although ten roots could in principle exhibit many different symmetry behaviors, the special structure of this polynomial, particularly the presence of the x<sup>5</sup> term, strongly limits what can occur. By combining theoretical arguments with computer algebra experiments in *Mathematica*, *GAP*, and *Pari/GP*, we show that only four symmetry types arise for irreducible polynomials in this family. Moreover, we determine concrete conditions on \(a\) and \(b\) that distinguish among these four possibilities. This classification highlights how the form of a polynomial shapes the symmetries of its solutions. This is joint research done in collaboration with C. Awtrey and F. Patane.
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Bumpless pipe dreams (BPDs) are combinatorial objects introduced in 2018 to study Schubert polynomials. Using results from the literature, we introduce a graph structure on the set of BPDs for a fixed permutation, where the edges are determined by certain local moves. We implemented these objects in SageMath and used our program to generate all 409,113 graphs for small permutations. We analyzed this data to form a conjecture on which permutations have acyclic graphs, which we proved using induction. In this talk, we give a pattern-avoidance condition that is necessary for a permutation’s BPD graph to be a tree.
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A planar tangle is a planar closed simple smooth curve constructed from quarter circle arcs. These curves represent planar configurations of a fidget toy called a Tangle; which is constructed from connecting freely rotating quarter circular tubes. If you imagine that each pair of connected arcs are allowed to rotate in 3-space around the axis of connection, then there are two natural moves to create new planar tangles from known tangles. These moves induce a adjacency graphs on classes of move-equivalent tangles. This presentation develops properties such as maximum degree and bipartitionability of these graphs and develops an algorithm for generating the adjacency graph of a given class of move-equivalent tangles.
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Manifolds that curve like a sphere are called positively curved, as opposed to those that curve like a saddle, which are called negatively curved. Manifolds with almost positive curvature are positively curved at almost every point in a probabilistic sense, and they are highly sought. The most recent discovery of a positively curved manifold was 2008, and no infinite family of almost positively curved manifolds has been discovered since 2002 until now. We construct infinitely many new examples of manifolds with positive curvature almost everywhere.
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This paper takes a hands-on, exploratory look at whether Mean Shift clustering—a nonparametric method that does not force the data into any particular model—can highlight structural breaks or unusual behavior in FAANG stock returns from 2015 2025. By pairing daily log returns with a rolling volatility measure, we create a simple two-dimensional space that turns each trading day into a point whose location reflects its overall market conditions. The algorithm, developed in python, naturally identifies one large, stable region of data along with several much smaller groups of days that behave differently. To get a better sense of how typical this dominant regime is, a Q–Q plot is used to check how closely its return distribution resembles a normal one. Non dominant clusters are then compared with well-known periods of market stress. The goal of this paper is not to build a predictive tool, but to understand how a flexible, nonparametric clustering method can help reveal market shifts in an intuitive way.
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In this report, we investigate a two-sample procedure for detecting distributional changes in stock returns using a modified energy statistic (EMIC) proposed by Njuki and Ning [2025]. The method compares two adjacent segments of a time series to assess whether they originate from the same underlying distribution. The finite sample properties of the proposed method are conducted to compare its powers and applications. Since the energy-based tests are sensitive to differences in location, scale, skewness, and tail behavior, the proposed approach on modified energy statistics provides a flexible nonparametric framework for identifying potential change-points in financial return data
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Two proper edge-colorings of a graph _G_ are mate-colorings if and only if every vertex of _G_ is incident to the same set of colors under each edge-coloring while each edge receives a different color under each edge-coloring. The color-trade-spectrum (CTS) of a graph _G_ is the set of all _t_ for which there exist two mate-colorings of _G_ using _t_ colors. A generalized theta graph, denoted $$\theta_{n_1,n_2,...,n_k}$$, consists of _k_ paths having only the starting and ending vertices in common with lengths $$n_1,n_2,...,n_k\in\mathbb{N}$$. A generalized wheel graph, denoted $$W^{n}_{k}$$, consists of a central vertex and an _n_-cycle with a path of length _n_ between the central vertex and each vertex in the cycle. We determine the color-trade-spectra of generalized theta graphs and generalized wheel graphs.
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Patients with aortic valve stenosis often have calcified valve leaflets that impede blood flow. Transcatheter aortic valve replacement (TAVR) offers patients a minimally invasive option to replace their aortic valve by guiding a catheter through their blood vessels to deploy a bioprosthetic valve. We developed a 0-D model of the left heart to investigate TAVR performance in a patient-specific context. To achieve this, we constructed a lumped-parameter representation of cardiovascular dynamics, incorporating flows, pressures, resistances, and compliances of the heart chambers and valves. These physiological elements were represented through a system of differential equations, which we solved numerically using Backward Euler. We simulated flow and pressure dynamics upstream and downstream of the aortic valve to better capture post-TAVR behavior. By tuning the model to post-TAVR clinical data found in the literature, we demonstrated its ability to capture patient-specific hemodynamics. This tuning allows for more accurate simulation of post-TAVR cardiac dynamics, providing cardiologists with a tool to optimize patient outcomes.
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This presentation investigates a recursively defined number triangle that generalizes classical structures such as Pascal’s and Fibonacci’s triangles. The first diagonal is constant, while subsequent diagonals are generated using patterns related to Fibonacci and triangular numbers. A central result of this study is the discovery that all 5×5 matrices along the left side of the triangle have a determinant of 2. This matrix captures the recursive structure of the triangle and provides a systematic way to describe how entries depend on one another. Closed-form expressions for the third and fourth diagonals are derived to support and justify the matrix representation. This work demonstrates how matrix methods offer a new framework for understanding generalized number triangles and their connections to classical sequences.
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Given a simple finite graph $G=(V(G), E(G))$, a vertex subset $D\subseteq V(G)$ is a distance $k$-dominating set if every vertex $v\in V(G)-D$ lies within distance $k$ of some vertex in $D$. The distance $k$-domination number $\gamma_k(G)$ is the cardinality of a minimum distance $k$-dominating set. The distance k-bondage number $b_k(G)$ is the size of a minimum edge set B suchthat $\gamma_k(G-B)>\gamma_k(G)$. In this paper, we establish upper bounds on $b_k(G)$ in respect to degree sums and exact values of a k-distance bondage number for common graph classes. Our results generalize previous bounds by incorporating distance-based domination constraints. We also introduce a refined bound in respect to the number of internally disjoint paths. These findings contribute to the broader study of domination and bondage parameters in restricted graph classes and provide insights into combin atorial optimization.
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Motivated by frequency-assignment problems in cellular networks, we study complete conflict-free colorings of graphs. A coloring of a graph $G$ is conflict-free if every vertex has a uniquely colored neighbor in its open neighborhood. The minimum number of colors required for such a coloring is the conflict-free chromatic number, denoted $\chi_{CF}(G)$. We determine the exact values of $\chi_{CF}(G)$ for several graph families such as paths, cycles, trees, complete graphs, and windmill graphs and provide conjectures on circulant and grid graphs.
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Ehrenborg and Happ introduced the concept of parking sequence, an extension of parking function for cars of different sizes. A graph presentation for parking sequences will be introduced, and this yields another proof for the theorem proved by Ehrenborg and Happ that enumerates the number of parking sequences.
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Given a group action on a set of combinatorial objects, a statistic on these objects is called homomesic if its average value is the same across all orbits. The notion was coined in 2015 by Propp and Roby, who were motivated by earlier examples of this phenomenon from chip-firing on graphs (2008) and rowmotion on antichains (2009). In this talk, we will present a family of posets whose number of inversions are homomesic with respect to the promotion action.
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Viral social media campaigns can generate millions of dollars in donations, yet predicting their spread remains challenging. This study applies epidemiological models to analyze the 2014 ALS Ice Bucket Challenge (IBC) using published data, which models the campaign using Susceptible-Infected-Recovered (SIR) differential equations, branching process theory, and network epidemic frameworks. Using the reported values for the basic reproduction number R₀=1.43 and serial interval τ=2.1 days, I calculated the transmission rate and recovery rate. I also used branching process and SIR eigenvalue analysis to verify R₀. Using graph theory metrics (degree centrality, clustering coefficient, and network heterogeneity), I analyzed how a scale-free topology can amplify spread. Extending this framework, this study considers why the similar Rice Bucket Challenge (RBC) did not achieve comparable global impact. I develop an exponential barrier model p(b)=p₀e^(-b) to quantify participation costs. This model predicts a reproduction number R₀ ≈ 0.70 < 1, which is consistent with observed limited diffusion. The comparison highlights five factors associated with viral success: 1) R₀ > 1; 2) short serial intervals; 3) favorable network topology; 4) low participation barriers; and 5) strategic seeding among highly connected individuals. These findings suggest that epidemiological models can provide a useful framework for forecasting the virality of social media campaigns.
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The Atlantic Meridional Overturning Circulation (AMOC) plays a crucial role in regulating Earth’s climate by transporting heat and nutrients throughout the ocean. Warm, saline surface waters flow northward, where cooling in subpolar regions increases their density, causing them to sink and return southward as part of the deep ocean circulation. Over the past century, rising greenhouse gas concentrations have begun to alter this circulation, leading to gradual changes in its structure and variability. Within the subpolar North Atlantic, regions such as the Labrador Sea, the West Greenland Current and Disko Bay play a key role, as deep convection, where surface waters cool, sink, and mix into the deep ocean, strongly influences AMOC variability. Improving our understanding of the processes that control convection in these regions will enhance our ability to assess ongoing changes in the AMOC, with the broader goal of informing climate policy. This work develops a new, computationally efficient framework that combines Finite-Time Lyapunov Exponents, Lagrangian Coherent Structures, and graph-theoretic measures to find regions of coherent flow to diagnose how deep convection redistributes water within the subpolar North Atlantic.
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This project comes from utilizing my major of math and statistics to study a passion of mine: music. With this project, I aim to highlight possible patterns, shifts, and broader trends within popular music in America over the past several decades. Using a historical dataset of all Billboard Hot 100 number one hits, I explore how artist gender, musical genre, audio characteristics, and more relate to chart success and longevity. To connect my statistical training with my interest in music, I built regression and machine learning models incorporating audio features, musical key, genre indicators, and temporal information. I also examined broad associations in the data, such as the uneven distribution of gender across genres. Throughout the project, I use cross validation, temporal splits, and sensitivity analyses to ensure that the results are robust and reproducible. More broadly, this work reflects my commitment to applying mathematical and statistical tools to questions in music research—showing how data can deepen our understanding of the cultural forces behind the music that define different eras.
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For a random parallelogram ABCD with AB parallel to DC, if BD intersects CE at F, where E is a random point on AB, what is the area of ADFE if the areas of CBF and EBF are provided? This geometric problem seems simple, but finding the solution is not easy. In this talk, we will explore some properties regarding area of triangles, area of trapezoids, and similar triangles. We then will use these properties, including one only requires middle school mathematics knowledge, to find the solution.
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This presentation will discuss the periodic makeup of laminations and Julia sets. First, I used the original polynomial to create our Julia sets. Next, displaying additional functions over the image, helping to explain the periodic makeup of the Julia sets created. Finally, I’ll explain how this strategy helped me to understand the behavior of our generated Julia sets to identify the Julia set to lamination correspondence more easily.
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Theorems by Thue, Pólya, and Mahler show that if a polynomial with integer coefficients $f(x)$ has at least two distinct complex roots, the greatest prime factor of $f(n)$ tends to infinity as $n$ approaches infinity. Consequentially, for every finite set of prime $P$, the intersection of $f(\mathbb{Z})$ and the $S$-unit of $P$ is finite. An extension on $f(\mathbb{Q})$ will be explored, by mimicking the technique used by Pólya, and applying a corollary of Falting's theorem for quartic binary forms.
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Rated Charleston as best city in north America to travel and live by Travel magazine, the home prices of Charleston area became to soar. It is demanding from home buyers to address the factors affecting home prices in Charleston area. We selected several candidate factors contributing to the home prices: regions, number of bedrooms, number of bathrooms, square feet, median income, and waterfront view or not. The data are collected from a sample of 210 homes on Zillow. We fit data to a general linear model. Among the candidate factors, the analysis results show that all except the number of bedrooms/bathrooms significantly affect the home prices. We further apply the Bonferroni method for simultaneous testing of the region effects vs. the baseline effect of home prices from downtown Charleston. Even though the estimates of home prices of the non-downtown regions are lower than that of downtown, such difference is not significant between the pair Isle of Palms and downtown Charleston.
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The traditional pairwise comparison voting method is not widely used, as evidenced by the fact that no national-level elections use it. One of the main reasons for this is that, unlike in elections with few candidates, national elections tend to have a large number of candidate, which dramatically increase the time required to evaluate the votes and determine the winners. In this research, we propose a new method for conducting elections, the stagewise comparison method, to hold elections with a large number of candidates and return results faster than using the traditional pairwise comparison method. This method requires dividing candidate pools into successively smaller pools and making pairwise comparisons among the candidates. We conclude by demonstrating the effectiveness of this approach in terms of time and resources required to conduct such an election.
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This presentation shows the process of finding a corresponding Julia Set from a particular lamination and delves deeper into the understanding of a specific Julia Set, the “Sea Turtle.” The presentation will cover the manipulation of the Sea Turtle, aka the “$d$-Armed Turtle” and the proof of showing affine conjugacy as well as prediction of the amount of solutions to the Turtle. 37E10 Dynamical systems involving maps of the circle
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Uniform distribution under irrational rotation is a classical result in dynamical systems. If a point on the unit circle is iteratively rotated by an irrational multiple of 2π, its orbit is equidistributed with respect to arc length measure, a consequence of the equidistribution theorem and the circle’s rotational symmetry. This research stems from a difference equation in the complex plane that creates an ellipse when -2 < m < 2. Applying an irrational rotation to the ellipse, it is found that although the resulting orbit remains dense, approaching every point on the ellipse arbitrarily closely, it fails to be uniformly distributed with respect to arc length.
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We consider two nonlinear (damped and undamped) partial differential equations which model, among other systems from fluid dynamics, special solutions of the n-dimensional incompressible Euler equations. We show that the incorporation of damping effects may or may not suppress the finite-time blowup that occurs in solutions of the associated undamped equation. In particular, we discuss how the amount of damping required to suppress blowup is determined by a relaxation time which depends on the blowup time of solutions of the undamped system.
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"Thick data" captures the qualitative richness of human experience including emotions, behaviors, and perceptions, and while it is central to the design process, synthesizing it risks losing nuance as researchers cluster experiences into discrete themes. This data is often identified as unquantifiable, but through modern natural language processing (NLP) techniques, the bridge between qualitative and quantitative is shortened. This research introduces a methodology for identifying sentiment between speakers in an interview study. The process includes (1) partitioning a sentence sequence into subsequences representing each speaker (2) embedding sentences into vectors representing sentiment polarity using a BERT-based embedding model architecture, and (3) softmax normalization to produce probability distributions used for visualization. This process is then expressed through new visualizations and explores various visual encoding characteristics to create effective visuals, defined by three principles: consistency, accuracy, and meaningfulness. The goal is to provide a new synthesis methodology for real-world research operations.
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A *total prime labeling* of a graph is an extension of a prime labeling in which we distinctly label the vertices and edges with the integers $1, 2, \ldots, \lvert V \rvert + \lvert E \rvert$. In a total prime labeling, the labels on adjacent vertices are relatively prime, and for each vertex of degree at least 2, the greatest common divisor of the labels on its incident edges is 1. In addition to introducing total prime labelings for various classes of graphs, this talk will highlight certain classes of graphs that do not allow a total prime labeling.
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Ever since Newton and Kepler, science has made extensive use of mathematical models for systems, at times in fields where they are probably simpler than useful. Ecology is among the fields where formulae are useful, but can never be truly precise. One of the more common metrics calculated about an ecosystem is the rarefaction curve. This curve approximates how many species within a specified group will be found, with sample size as the independent variable. They may be used to compare species richness between sites with differing levels of sampling, approximate the total diversity of a site, estimate the total global diversity of a clade, or perform other similar calculations. They are generally assumed to follow either an inverse decay curve—\(\hat{s}=a(1-e^{-bx})\)—or a logarithmic curve—\(\hat{s}=a\ln({bx})\). Much of my research work has been focused on documenting the fauna of the early Pleistocene Carolinian Waccamaw Formation. As part of this, I have also done systematics/taxonomy and paleoecology. Since much of the material for the faunal documentation consists of bulk samples, this allows for calculation of a rarefaction curve across a very wide range of sample sizes. Plotting these data points reveals that they match neither of the expected types of curves, but have a much longer slow increase than expected. The closest match found so far is the integral of a cumulative lognormal, however, suggestions of other possible curves are welcome.
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