Mathematics

Session I

Alethea Callahan, "Measurements of Population Patterns in Satellite Images: Overcoming Noise in Spatial Data Using Algebraic Topology"

This project uses algebraic topology to create a method of extracting information about a population from corrupted satellite images.

The methodology for this project is to start with a population model and introduce salt and pepper noise (salt and pepper noise mimics missing/corrupted data) using MatLab, to simulate the noise observed in population data. Then, morphological operations are used to smooth out the corruption in the data. Persistence diagrams are created, using python, which display topological features– high density population patches as well as holes in the population. Persistence diagrams of the simulated model before noise is introduced and after the noise is smoothed are compared to see which topological features persist. Next, this method of morphologically smoothing noisy data and extracting information about topological features is applied to satellite images. A major result is that when the matrix is smoothed, finer details are lost, but the major topological features are still apparent on the persistence diagram. Hence, we can draw conclusions about significant topological features from the smoothed persistence diagram.

Student Major(s): Mathematics

Advisor: Dr. Sarah Day

John Introne, "Fault Detection and Accommodation of Time Series Pressure Measurements in Hypersonic Vehicles"

The development of air-breathing hypersonic vehicles has significant implications in military defense and reconnaissance, allowing for more maneuverable and quicker-striking missiles and planes. When these vehicles reach supersonic speeds, shockwaves form inside the engines, causing drastic increases in pressure and temperature and decreases in air speed. These changes aid combustion, and the degree of change correlates with the position of the shock-train leading edge (STLE), the point at which multiple shockwaves intersect and pressure increases. Pressure measurements throughout the engine provide a fundamental way of tracking the STLE, but the hostile environment may distort sensor information, severely limiting such tracking systems. We propose multiple machine learning approaches, including a convolution-based autoencoder and a Gaussian process regression model, to identify and correct faulty pressure measurements over time that result from a variety of failure modes. These methods accurately find problematic faults and improve STLE prediction under uncertainty, allowing for safer and more efficient flight.

Student Major(s): Mathematics, Computer Science

Advisor: Dr. Gregory Hunt

Elizabeth Stebner, "Modeling the Population Dynamics of Oysters Crassostrea virginica"

Recent declines in oyster populations have caused significant economic and ecological losses in the Chesapeake Bay area. Changing environmental and external conditions, such as increased disease prevalence and fishing, alter the requirements for successful restoration of oyster populations. So, the purpose of this research is to inform restoration efforts in the Chesapeake Bay by mathematically modeling the population dynamics of the modern eastern oyster over long periods of time. The project uses an integral projection model to track the age and size distribution of the population from year to year as the oysters grow, survive, and reproduce. By running the simulation for long periods of time, it can provide important insights on the equilibrium size and age distributions of a population, as well as determine whether a population will achieve long-term survival or go extinct based on initial conditions.

Student Major(s): Mathematics and Physics

Advisor: Dr. Leah Shaw

Xuzhong Wang, "Exploring Linear Bound For Graph Coloring Reconfiguration"

My summer research project focuses on investigating linear upper bounds in graph coloring reconfiguration. Specifically, we aim to prove that for any two proper graph colorings, it is possible to transform one coloring into another while recoloring each vertex only a constant number of times. Our methodology primarily relies on a combination of greedy algorithms and the discharging method. By applying greedy recoloring steps, we can identify families of reducible structures within any minimal counterexample, which in turn allows us to conclude that certain classes of graphs admit linear-time reconfiguration sequences. This result carries significant implications, not only in the field of combinatorics and graph theory, but also in providing rigorous theoretical foundations for practical algorithms in computer science. Establishing a guaranteed upper bound on state transformation steps can contribute to more predictable and efficient runtime behavior in software systems, enabling advances in algorithm design and optimization.

Student Major(s): Mathematics, Computer Science

Advisor: Dr. Gexin Yu

Georgia Wensell, "Modeling Blue Crab Dynamics: Fishing, Predation, and Hematodinium"

Outbreaks of Bitter Crab Disease in blue crabs have caused economic losses in the fishery industry in the Chesapeake Bay. This project builds upon the mathematical model of the blue crab population from previous years using differential equations to predict disease dynamics. The goal of this research is to begin expanding the model to an open system and adjusting fishing-related mortality to reflect fishing practices. The open system allows for the young blue crabs that are entering the system to be at a constant level for more consistent simulations. This new model takes crab migration into account as a factor in disease transmission and simulations indicate that fishing behavior with retaining versus discarding infected adult crabs does not have a large impact on disease levels. Thus, efforts should be focused on other environmental impacts on disease prevalence in the Chesapeake Bay.

Student Major(s) and Minor: Mathematics Major, Data Science Minor

Advisor: Dr. Junping Shi

Session II

Anaiel Jibrelle Dayrit, "Seed Dispersal Strategies: Evaluating Spatial Population Patterns and Predicting Extinction"

In an age of climate change, certain spatial patterns can spell disaster for a population, but what patterns can spell prosperity and longevity? The research question for this project is, “How do dispersal strategies influence the lifespan of a species in the long run?” This will focus on the ways flora can repopulate by measuring simulated spatial population patterns. The project will build on existing mathematical models for population growth and dispersal, including simulations of the model in MATLAB. It will generate images and moving graphics that will depict population density patterns resulting from varying dispersal strategies. Results from this project demonstrate that dispersal strategies stemming from the exponential-power distribution survive extended periods of time within specific ranges of input values. A long term goal of this work is to develop methods for detecting the influence of dispersal strategies in satellite images and other measured population data.

Student Major: Undeclared

Advisor: Dr. Sarah Day

Xinyue Fu, "Statistical Validation and Refinement of Synthetic Neuroimaging Data"

This project aims to address this limitation by developing a rigorous statistical framework that integrates functional data analysis and triangulated spline smoothing to assess fidelity. Additionally, a transformation technique will be created to correct identified discrepancies, ensuring that synthetic datasets align with the statistical properties of original neuroimaging data. By improving the reliability of synthetic imaging, this research has significant implications for advancing neurodegenerative disease studies while mitigating ethical and financial constraints associated with real patient data.

Student Major(s): Mathematics and Economics

Advisor: Dr. Guannan Wang

Merryn Grindstaff, "Modeling Oyster Reefs with Partial Differential Equations"

Oysters are a keystone species that are vital to the environment. Currently, oyster reefs are at approximately one percent of historic levels in the Chesapeake Bay. This project aims to aid in efforts to restore oyster reefs in the Chesapeake Bay through information gathered from a system of partial differential equations, a one dimensional spatial model, composed of juvenile oyster volume, adult oyster height, dead shell, or reef, height, and sediment height. Numerical simulations run with these equations provide information about the complicated relationship between different sizes and shapes of oyster reefs and their behavior over time, including insight into the minimum size at which restored reefs must be built under different conditions in order to not be suffocated by sediment.

Student Major(s): Applied Mathematics & Data Science

Advisor: Dr. Leah Shaw

Ellen Jin, "Scalable Functional Data Analysis in Distributed Environments: Inference for 3D Imaging Mean Functions in Medical Research"

The challenge of analyzing 3D images in medical research areas has long been a difficult problem due to the complexities involved in developing statistical methods that effectively handle irregularly shaped data. Our brain has a fascinating structure that cannot be easily statistically modeled. Therefore, this project, instead of using a classic linear or curvature model, will explore functional data analysis as a framework for high-dimensional models, overcoming challenges related to irregular domains, computational efficiency, and scalability for large datasets. The proposed methodology builds upon advanced statistical methods so that it will ideally identify regions with strong signals or differences between subject groups. The significance of this research extends across scientific, mathematical, and statistical perspectives by providing a generalizable solution for complex object analysis. This work has broad applications in medical imaging, particularly for studying neurodegenerative diseases like Alzheimer’s, where improved statistical models can enhance diagnostic accuracy and treatment for debilitating conditions.

Student Major(s)/Minor: Mathematics Major, Computer Science Minor

Advisor: Dr. Guannan Wang

Shang Li, "Turing Instability in Reaction–Diffusion Systems on Isotropic Metric Graphs"

Reaction–diffusion systems provide a unified framework for understanding how local reaction kinetics coupled with spatial diffusion can generate a rich variety of spatiotemporal patterns in physical, chemical, and biological contexts. Since Turing’s seminal insight that diffusion-driven instabilities can destabilize homogeneous equilibria, much attention has focused on continuous domains; more recently, however, network-like domains such as  metric graphs have drawn interest for their ability to model filamentous structures in neuroscience, vascular biology, and engineered sensor arrays.  In this study, we investigate Turing pattern formation in the classical Brusselator model on two prototypical metric-graph architectures: isotropic n-star and j-k-bridge graphs.  By combining analytical criteria derived from the interplay between the reaction–diffusion Jacobian and the graph Laplacian spectrum with numerical simulations on 3-star graph and 2-2-bridge graph, we demonstrate the emergence of symmetric, asymmetric, and mixed modes as the system parameter crosses its Turing threshold.  Our results highlight the critical role of network topology and parameter variation in selecting among competing eigenmodes, thus extending classical Turing theory to complex networks and providing a basis for future studies of pattern selection in natural and engineered graph-structured systems.

Student Major(s): Mathematics and Economics

Advisor: Dr. Junping Shi

Emma Williams, "Analyzing Variance Patterns in Convolutional Neural Network–Derived Image Features and Their Role in Mechanism-of-Action Identification"

The Cell Painting CNN, a convolutional neural network developed by the Broad Institute and trained on fluorescently stained cell microscopy images, produces numeric ‘feature’ outputs that group unknown compounds by mechanisms of action (MOA) more accurately than the features produced by current industry-standard, CellProfiler. While CellProfiler features are formulated with clear, measurable properties (e.g., cell area, stain intensity) that are generally easy to interpret in 3D space, the CNN learns unitless “hidden features” through pixel-based optimization processes. We aim to determine what aspects of the CNN’s variance structure—how variation is extracted and how treatments cluster in standardized, interpretable principal component (PC) space—drive this advantage. Relying primarily on PCA and related dimensionality reduction methods, we align outputs from both approaches to identify variance differences that enhance MOA grouping, as well as any shared variance patterns that arise independent of how "features" are calculated. While conclusions are still in progress, future work will explore autoencoder-based PCA-like methods to better address the nonlinear nature of image data while retaining interpretability.

Student Major(s): CAMS and Economics

Advisor: Dr. Gregory Hunt