Mathematics

Session I

Rayhan Ahmed, "Linear Preservers of Spectrum and Spectral Radius on the Space of Toeplitz Matrices"

A Toeplitz matrix is a rectangular array of numbers with constant values along each descending diagonal, and a linear map is a function that satisfies the properties of additivity and homogeneity. This project characterizes the forms of linear maps on the space of Toeplitz matrices which preserve the set of eigenvalues as well as linear maps that preserve matrix spectrum and spectral radius. Building on previous results about determinant preservers, we show that eigenvalue and spectrum preservers always take one of two forms. Furthermore, by using families of nilpotent matrices to restrict linear maps, we find similar results for spectral radius preservers in low-dimensional cases. The information gained from this research will add to the body of knowledge in linear algebra and could be useful in algorithms seeking to compute some of the properties considered.

Student Major: Mathematics Major

Advisor: Dr. Chi-Kwong Li

Brooke Anderson, "Identifying Oyster Reef Sites for Restoration using Optimization Models"

Oyster populations in the Chesapeake Bay have declined due to pollution, disease, and long-term over-harvesting, while oyster reef restoration remains costly. Optimization models can help identify reef sites that maximize long-term oyster biomass under a limited budget. In this study, optimization models are used to select 25 out of 49 candidate reefs in the Tangier Sound-Pocomoke Sound system for restoration that result in the most population growth. The model is tested under several different constraints, including community group quotas and variations in reef sizes. Seven backbone sites were identified as optimal reefs to restore across every scenario. The model was then modified to investigate the effects of harvesting a percentage of oysters from a subset of selected reefs. These results help identify optimal sites that are both productive and resilient to harvesting pressure when constrained to restoration limits.

Student Major: Computational & Applied Mathematics & Statistics Major

Advisor: Dr. Rex Kincaid

Artur Badalyan, "Predicting Wildfire-Driven Search Interest Across 11 California Media Markets Using Spatiotemporal Deep Learning"

SmokePM serves as a harmful byproduct of Wildfires for the health of the general public. Search interest in key features related to wildfires, such as "air pollution" and "air purifier," can be modeled from the trends in smokePM and area-specific Social Vulnerability Index, among other important factors. In this paper, we aim to predict the wildfire-driven search interest across all 11 California media markets using spatiotemporal deep learning. Our architecture combines a Long Short-Term Memory model in concert with a Multilayer Perceptron to predict such search interests. We model all 11 areas jointly within our architecture to allow for cross-regional learning and output all regions at once instead of obtaining per-region results across all 11 regions. Preliminary results of our model suggest that overall patterns and trends are being recognized; however, on a large scale, the model underperforms in predicting extreme spikes in search events.

Student Major: Mathematics Major

Advisor: Dr. Yuming Sun

Session II

Luke Anders, "Isoperiodic Foliations of the Strata 𝛺ℳ (1,𝑡,−1−𝑡)"

Supposing one has a billiards table shaped like a regular polygon, it is a classical problem to analyze the trajectory of a billiard ball which rolls on it ad infinitum. When doing so, it becomes useful to identify edges of the polygon with each other. Naturally arising out of this is what is known as a translation surface. It is of interest to study translation surfaces in further abstraction. Particularly important is the moduli space, the space parametrizing all translation surfaces of a given family. Motivated by the philosophy that the geometry of a stratum of the moduli space reveals properties of the individual surfaces and vice versa, this project focuses on the class of strata ΩM(1,t,-1-t) closely related to the work of Faraco-Tahar-Zhang and aims to comprehensively describe its geometry. Studied by Fields Medalists Mirzakhani and McMullen, this philosophy has historically been fruitful.

Student Major: Mathematics Major

Advisor: Dr. Yongquan Zhang

Ian Fabris, "Forecasting Bifurcations in Flow-Kick Systems of Differential Equations"

Differential equations are crucial for modeling "tipping points," where systems like forests or climates undergo sudden, irreversible shifts. Although early warning signals (EWS) exist for continuous systems, they have not yet been applied to "flow-kick" systems, which are models that combine continuous processes with periodic impulses, such as regular wildfires in a forest with continuous vegetation growth. We identified predictable signals that precede catastrophic transitions in these hybrid systems. Using Python-based numerical simulations and analytical modeling, we found unique trends in the spectral domain that often precede an impending bifurcation, such as reddening in the power spectrum produced by the Fourier-transform of time-series data. We also analyzed various detrending methods for separating environmental noise from the true data and constructed a new algorithm meant to remove the periodicities from the data to allow better analysis. The effects of highly subsampled datasets on prediction capabilities were also evaluated. Ultimately, identifying these warnings has the potential to aid preparedness by enhancing our ability to predict bifurcations and potentially mitigate or slow down catastrophic transitions in meteorological systems, population systems, and other natural systems. 

Student Major/Minor: Physics Major, Mathematics Minor

Advisor: Dr. Leah Shaw

John Farrell, "Two-Patch Population Model with Density-Dependent Motility"

Differential equations express the rate of change of a value over time as a function of multiple variables. A logistic equation is a type of differential equation modeling population growth, in which the population levels off at some defined value N. This project examines systems of two logistic equations, modeling two interdependent populations: members of one population are allowed to travel, or “disperse” to the other. In some cases, combining two populations, or “patches,” allows them to grow to a greater combined size than they would have otherwise. My findings may have applications in conservation, allowing ecologists to determine when combining two patches of a species increases their equilibrium population size, helping them survive. With the guidance of Professor Junping Shi, I will use techniques learned in MATH 302 (Ordinary Differential Equations) to determine when allowing dispersal allows the equilibrium size of populations to increase.

Student Major: Mathematics Major

Advisor: Dr. Junping Shi

Mahmut Inanoglu, "Crushed, Not Just Caught: How Reef Compaction Sets the Limit on Sustainable Oyster Harvest"

Oyster reefs are built from the shells of past generations, which hold living oysters above the sediment that would otherwise bury them. Harvesting therefore removes oysters from a reef that is also their habitat, and compacts that reef, crushing its structure into a lower, denser layer. We asked how much can be harvested annually without driving the reef to collapse. Beginning from JARS, a four-equation model of oyster population dynamics, we introduced harvesting in increasingly realistic forms, culminating in a six-equation model in which crushed oysters and crushed shell are tracked in their own compartments rather than simply removed. For each we determined the largest annual harvest a reef could sustain. Compaction, not the size of the catch, set the limit: damage reduced sustainable harvest to a fraction of its undamaged level, and eliminated it entirely on a fully compacted reef. Damaged reefs decline slowly, so simulations that stop early can certify a harvest that eventually fails. Sustainable harvest therefore depends on limiting damage to reef structure, not only on limiting the catch.

Student Major(s): Mathematics and Economics Major

Advisor: Dr. Leah Shaw

Nathan Klintworth, "Flexible Partitions of Sparse Graphs"

A proper coloring of a graph is a partition of its vertex set into independent sets, and the flexibility of proper (list) colorings asks whether a constant fraction of vertex preferences can be honored by some proper coloring. We initiate the study of flexibility for the much larger family of vertex partitions into prescribed degeneracy classes and prove some results about the types of graphs that are guaranteed to be flexible and for a specific family of graphs we determine the exact flexibility bound by constructing partitions in careful ways and using probability to guarantee enough vertex preferences are satisfied.

Student Major(s): Mathematics and Theatre Major

Advisor: Dr. Gexin Yu

Georgia Wensell, "A Four-Season Blue Crab Model with Hematodinium"

Earlier research on the Bitter Crab Disease suggests that the Chesapeake Bay’s blue crab population experiences outbreaks with the temperature impacting transmission and mortality rates. These outbreaks have caused economic losses in the fishery industry in Virginia and Maryland, even in non-epidemic years. This project builds upon the mathematical model of the blue crab population that was built in previous years using differential equations. This mathematical technique is used to predict the amount of blue crabs that would be susceptible or infected by the disease. The goal of this research is expanding from a three-season model to four seasons. This addition more accurately represents the timing of reproduction, fishing, and disease transmission. The model is currently a closed system, but new research allows for an open system, taking into account crab migration and simplifying the reproduction function to a time independent variable. Alongside reproduction, this research explores the possibilities for disease recovery and what that looks like for blue crabs.

Student Major/Minor: Mathematics Major, Data Science Minor

Advisor: Dr. Leah Shaw

Zhuoyan Zheng, "How Do Insects Smell in the Wind - A Computational Model of Insect Brain"

Animals often experience multiple sensory signals at the same time. For insects, odors are usually carried by wind, meaning that smell and airflow information arrive together during odor tracking. This project investigates how airflow signals influence odor processing in the insect brain. Using a computational model of the insect antennal lobe, an early brain region involved in odor processing, this study compares cases where the same airflow signal is shared across neural populations with cases where each population receives an independent signal. The results show that shared airflow increases coordinated neural firing and improves the reliability of odor representations read by downstream brain regions. These findings provide insight into how the brain combines information from different senses and may guide the development of bio-inspired sensory systems.

Student Major: Mathematics Major

Advisor: Dr. Mainak Patel

Video Presentations