REU Site: Appalachian Mathematics and Physics Site

Location

Participants will be housed in a dormitory at Marshall University and be provided a campus meal plan. Housing and the meal plan is 100% paid for by the grant (i.e. it does not come out of the stipend!).

Marshall University was recognized as an R2 research institution in 2019. The faculty in the Department of Mathematics and Physics have a history of involvement in REUs, having hosted an NSA-funded REU in combinatorics in 2016 and being involved in a computational REU that ran in 2010--2014. Marshall University runs a Summer Undergraduate Research Experience (SURE) program and was recently awarded a different REU program in civil and environmental engineering.

All participants will have access to classroom space and computational hardware as-needed, and physics participants will have access to the necessary laboratory equipment to conduct their research. Participants will live in the dormitories of Marshall University and will be provided with a meal plan for the duration of the program. The dormitories are approximately a five minute walk from the two buildings in which research will occur. Marshall University has two libraries that can provide academic resources for participants, as well as additional collaboration spaces. The Memorial Student Center serves as a center point of campus with a welcoming foyer as well as nationally-franchised fast food and coffee shops.

Marshall University is located in Huntington, the second largest city in West Virginia. Its downtown area connects to the western edge of campus, and the city and its surrounding towns are host to numerous artistic and musical events each summer. Local transportation options exist for participants without a car. The Tri-State Transit Authority provides a bus service that connects the university to to local recreation, shopping locations, and restaurants in Huntington. Greyhound provides bus services to other nearby cities such as Charleston, WV and Ashland, KY. Huntington also hosts an Amtrak station with service to distant cities such as Chicago, Cincinnati, Philadelphia, and New York City.

West Virginia hosts numerous public recreation areas -- for example Beech Fork State park lies ten miles to the south of Huntington while the New River Gorge National Park, the United States' newest national park, lies near Fayetteville, WV, about a two hour drive from Huntington.

Projects

Binary neutron star mergers (advisor: Dr. Maria Babiuc-Hamilton)

One of the pivotal theoretical challenges faced by Multi-Messenger Astrophysics (MMA) is the accurate modeling of the effect of tides on the gravitational waves (GW) during binary neutron star (BNS) mergers and the mechanisms behind their electromagnetic counterparts, such as the emission of Fast Radio Burst (FRB) during the pre-merger and Gamma Ray Bursts (GRB) after the merger. We will address these three challenges while simultaneously nurturing the next generation of MMA researchers. This is achieved by engaging undergraduate students in the development of innovative techniques to demystify BNS mergers and their counterparts. By combining rigorous training with efforts in developing innovative approaches to the unresolved theoretical problems related to understanding BNS mergers, we ensure that future researchers are equipped to bridge the gap between observational data and theoretical predictions. Three well-defined projects can be explored:

Project 1: Analytical modeling of the influence of tides on the gravitational wave signal during BNS mergers
Project 2: The role of the magnetic field in the emission of Fast Radio Bursts during BNS mergers
Project 3: Analytical modeling of Gamma Ray Bursts

Our analytical models are appropriate for undergraduates, requiring no knowledge of General Relativity, advanced mathematical methods or computer software. We will develop our software in Python, guided by a user-friendly Jupyter Notebook, or using Mathematica computational platform.
Recommended prerequisites: Differential equations, electricity and magnetism, mathematical analysis, experience with programming

Computation of analytic capacity (advisor: Dr. Stephen Deterding)

A function is said to be analytic on a domain if it is differentiable at every point of the domain. An entire function is a function which is differentiable at every point in the complex plane. Unlike the real variable case a bounded entire function must be a constant. The analytic capacity of a set is a measure of how large a bounded function that is analytic outside the set can become. Sets with analytic capacity zero are removable in the following sense: every function that is analytic outside a set with analytic capacity zero can be extended to an entire function. For a set that is not removable, the analytic capacity provides a measure of how close it comes to being removable.

Analytic capacity can be directly computed for simple sets, such as a disk, but for a more complicated set, even something as simple as the union of three disjoint disks, its analytic capacity must be approximated by numerical methods. One algorithm for computing analytic capacity is the Least Squares Method, which selects a minimal element from a span of approximating functions to determine a sequence of decreasing upper bounds for analytic capacity. A similar procedure creates a sequence of increasing lower bounds which combine to estimate analytic capacity. Another algorithm is the Boundary Integral Equation Method which uses numerical methods to solve an integral equation from whose solution the analytic capacity can be determined directly.

Participants will study the accuracy of these methods by implementing them using Matlab. By using these methods to approximate the analytic capacity of sets where the exact value is known, participants can evaluate the accuracy of both methods. Participants can then use one or both methods to approximate the analytic capacity of sets for which this value is not known. Since these sets may contain features such as non-smooth boundaries some modification of the algorithms may be required in order for them to produce the correct answers.
Prerequisites: linear algebra; experience with programming is helpful but not required

Filtration of textile dyes (advisor: Dr. Sean McBride)

The purpose of this project will be to collect data for several new dyes during the functionalization process of porous polycarbonate filters, referred to as hysteresis data. During the functionalization process, the concentration of dyes run through the filter is increased from $50 \mu{M}$ up to $1000 \mu{M}$ and then back down to $50 \mu{M}$. If there was no hysteresis in the data during this cycle, then the rejection as a function of concentration for increasing and decreasing concentration would be the same and we would see no functionalization of the filter. However, for some anionic azo dyes, we see a large hysteretic behavior after ramping up and decreasing the concentration of the dyes run through the filter. The end goal of this research is to determine what variables or environmental conditions are responsible for this hysteric behavior for certain azo dyes. Once this is known, such attributes or combination of environmental parameters can be engineered in later research, leading to advanced dye removal systems.
Recommended prerequisite: basic laboratory skills

Statistical machine learning to create predictive models (advisor: Dr. Raid Al-Aqtash)

Participants will begin by learning about statistical learning (e.g. supervised vs non-supervised), the k-nearest neighbors algorithm, and the R programming language. Participants will learn about finding data sets, cleaning them, methods to handle missing information, and partitioning the data into training, validation, and testing sets. Comparisons between the knn algorithm with various types of regression as well as variable selection and model tuning will be explored during the creation of a predictive model.
Recommended prerequisite: an introductory statistics course, some experience with programming

REU Site : Appalachian Mathematics and Physics Site

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