James Madison University (JMU) has been chosen for a National Science Foundation Research Experiences for Undergraduates (REU) site in mathematics.  We are looking for exceptional students (who are US citizens or permanent residents) who want to explore mathematical research to help them decide whether to pursue graduate study in the mathematical sciences. A commitment to graduate study is not a prerequisite for this program, but rather a desired outcome.

Program Features

• Participants will be divided into project groups of size 3.  This year’s projects are in the fields of biomath, data science, topology, and analysis. (See below for project descriptions.)
• Participants will receive a stipend to live and work on the JMU campus.  They will stay in the Apartments on Grace.
• Participants will have access to university amenities.
• We will have weekly visitors/presenters to the program.
• Participants will visit graduate schools to learn about the experiences of graduate students in different types of institutions.
• We will have occasional weekend social excursions.  Previous trips included visits to nearby Shenandoah National Park and Washington, D.C.
• Participants have travel funding to support attending and presenting at the Joint Mathematics Meetings the following January.
  

Stipend
$7,000

Program Dates
May 21 - July 30, 2025

Application
Please apply here using math programs.org.  Application deadline is February 24, 2025.

Students from groups underrepresented within their disciplines (e.g., women, underrepresented minorities, students with disabilities), veterans of the U.S. Armed Forces, first-generation college students, and students from socioeconomically depressed regions (e.g., Appalachia) are especially encouraged to apply.

Mentor: Dr. Alex Capaldi
In the production of tequila, mezcal, and other agave-based alcohols, farmers typically cut the flowering part of the plant, the inflorescence.  Unfortunately, this makes the plant unavailable for pollinators, such as many bat species.  A new conservation initiative is to have bat-friendly tequila plantations where farmers allow a portion of their agave crop to flower.  However, the timing and spatial placement of these flowering agave plants that result in the healthiest bat populations are not known.  The goal of this project is to build an agent-based model, a type of computer simulation, to determine an optimal strategy of the spatial arrangement of agaves on a plantation and harvesting times that balances both economic and conservation goals.

Required Courses: a differential equations course OR a mathematical modeling course OR a statistics course beyond intro stats.

Helpful Courses: General Biology/Ecology, Programming in MATLAB or Python

Mentor: Dr. Behnaz Moradi-Jamei
This project investigates the potential of network-based neural architectures and data-driven methods to enhance community detection in complex networks. Community detection, a fundamental problem in network analysis, involves identifying clusters of nodes that are more densely connected within the network. Traditional methods often fall short when applied to large, complex networks due to computational limitations and scalability issues in addition to lacking contextual relevance.

Utilizing the power of these neural architectures—an emerging approach in machine learning that applies neural network principles to graph data—combined with data-driven methods, this project aims to overcome these challenges. By integrating these approaches with advanced graph preprocessing techniques, we seek to improve the detection of communities in various network contexts, such as social networks and collaborative environments.

Students participating in this project will explore the intersection of machine learning, neural networks, data-driven methods, and network science, contributing to cutting-edge research with broad applications in data science and beyond.

Required Courses: None

Helpful Courses: Programming in Python

Mentor: Dr. David Duncan
Representation varieties are geometric objects that arise in many branches of mathematics including abstract algebra, low-dimensional topology, and mathematical physics. To specify a representation variety R(π, G), one generally specifies a finitely-presented group π as well as a matrix group G. Then R(π, G) can be viewed as consisting of tuples of matrices in G satisfying certain polynomial equations coming from π. In many cases, the variety R(π, G) will highlight properties of π that are otherwise difficult to observe, thus providing a tool for studying the complexities of π.

This project will primarily focus on the case where π is the fundamental group of a three-manifold Y. Many of the research questions involve seeking to better-understand the topological properties of R(π, G), and the extent to which these reflect properties of Y. For example, are there general conditions on Y that guarantee R(π, G) is connected? Conversely, if R(π, G) is connected, what does this mean for Y? To what extent do these answers depend on the choice of matrix group G? There are also interesting connections with Chern-Simons theory that can be pursued.

Required Courses: Linear Algebra

Helpful Courses: Abstract Algebra (group theory), Topology

Mentor: Dr. Meric Augat
The L'vov-Kaplansky Conjecture asks whether any multilinear polynomial acting on the space of n x n matrices must have a linear subspace as its image. This has been proven true for 2 x 2 matrices over certain fields (for example, the complex numbers), but it remains open for larger matrix sizes.

The L'vov-Kaplansky Conjecture is surely very difficult to prove (or disprove), however this project asks a weaker question: starting with a multilinear polynomial, then will its image over the n x n matrices be a linear subspace if we choose n large enough?

A multilinear polynomial (just like matrices, our variables do not commute: XY is not equal to YX) is linear in each of its variables. Here, linear is in the sense of a linear transformation. Examples include  xy - yx,  xyz + 2yxz – 5zxy,  and  xyzw + wzyx. The polynomials x² + yz,  and  xy + xz are not multilinear.

Formally the project's conjecture is the following: if P is a multilinear polynomial with complex coefficients, then there exists a positive integer N, so that for any n bigger than N the range of P on the n x n complex matrices is a subspace of the n x n matrices.

For a multilinear polynomial, some of its behavior may not be captured when we evaluate it on small matrix sizes. A trifling example: the multilinear polynomial  xy – yx  is always equal to zero when evaluated on 1 x 1 matrices -- since 1 x 1 matrices all commute. The expectation is that any irregularities in the image of the polynomial will disappear for large enough sizes.

The main applications of this project are to the field of noncommutative functional analysis; a subfield of complex analysis that studies noncommutative functions through their evaluations on matrices of any size. A feature of the field is that we can always evaluate our functions on larger matrix sizes.

Required Courses: Linear Algebra

Helpful Courses: Complex Analysis, Programming in Mathematica or similar

This REU participates in the Mathematical Sciences REU Common Reply Date agreement.  Any student who receives an offer from an REU that participates in the Common Reply Date agreement, including this REU, is not required to accept or decline the offer until March 8, 2025 (or later).  The goal of the Common Reply Date agreement is to help students make informed decisions, particularly when faced with the potential of receiving offers from multiple REUs.

Please write to mathreu@jmu.edu with any questions.

This material is based upon work supported by the National Science Foundation under Grant Number 2349593. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

The Department of Mathematics and Statistics is committed to creating learning environments that support and are improved by a diversity of thought, perspective, and experience. We affirm that the lives and experiences of Black, Indigenous, and People of Color matter. We recognize that within the study and culture of mathematics and statistics there are deep-rooted and systemic inequalities, racism, and sexism that have disproportionately affected some members of our community. We strive to recognize and reverse these inequities. We embrace all backgrounds, identities, names, and pronouns. We see you, we hear you, and we stand with you. You are welcome in our department.