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

• This year we will host three projects in the fields of data science, topology, and mathematics education. Project descriptions can be found below.
• Three students will work on each project. Applicants should indicate in their application any project that is of interest to them.
• Participants will receive a stipend to live and work on the JMU campus. Students will stay in residence halls on JMU campus; this housing is provided for by the REU.
• 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 26 - August 1, 2026

Application
Please apply here using the NSF's ETAP.  Application deadline is February 20, 2026.

Students with no prior REU experience are especially encouraged to apply, as are 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).

Mentor: Dr. John Bush

Graphs are everywhere: they model road networks, social connections, molecules, patterns in the brain, etc. When viewed as metric spaces, graphs can reveal hidden topological structure, e.g., not just whether two nodes are connected, but whether larger loops or “holes” persist across scales. Persistent homology, a central tool in topological data analysis, captures these features and records them in a barcode that summarizes this topological structure.

In this project, we will investigate families of metric graphs and study their persistent homology. By computing many examples, we will look for patterns: do certain graph families always produce long-lived loops? How do barcodes change as graphs grow larger or more complex? Can these observations be explained and proved mathematically?

We will use software to experiment with many examples, then develop conjectures and seek proofs in special cases. Along the way, participants will gain experience both in the computational aspects of topological data analysis and in developing and proving theorems about the topological structure of graphs.

Required Courses: Linear Algebra

Helpful Courses: Topology, programming experience in Python

Mentor: Dr. Vladislav Kokushkin

Math Connection Theater (MCT) is the story of a unique and inspiring collaboration between STEM majors and members of JMU’s sketch comedy troupe, Maddy Night Live. Launched in 2017, the initiative has evolved into a platform for developing and performing short theatrical sketches that bring abstract mathematical concepts to life through humor, narrative, and performance. Over multiple student cohorts, this interdisciplinary partnership produced live MCT shows in 2017, 2018, 2019, and most recently in October 2025. The 2025 sketches explore a range of mathematical topics, including limits, L’Hôpital’s rule, topology, the Monty Hall problem, and others.

The beauty of MCT lies in its fusion of mathematical thinking, theatrical performance, and humor, offering a fresh, embodied approach to experiencing and communicating mathematics. In this undergraduate mathematics education project, we will analyze the video and audio recordings of seven sketches from the 2025 show to understand how mathematical ideas are physically and theatrically enacted. Specifically, we will investigate how actors experience and communicate abstract mathematics through hand gestures, body movements, spatial interactions, facial expressions, dialogue, and humor. Drawing on theories of embodied and grounded cognition, which suggest that bodily actions can support reasoning about abstract concepts, we will explore how mathematics can be “taught” from the stage through embodied performance. MCT offers a unique opportunity to examine how embodiment functions as a vehicle for mathematical experience and communication within a theatrical context.

Required Courses: None

Helpful Courses: Calculus I

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

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, 2026 (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.