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Representation Varieties & the Topology of Three-Manifolds - 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

If you are a JMU student who is interested in this project, reach out to David Duncan at duncandl@jmu.edu. 

Time series analysis studies observations collected over time to identify patterns, understand relationships, and make forecasts. Students will explore research questions motivated by applications in health, energy, and finance, using real data and computational experiments.

Depending on the project, tools may range from conventional time series models to more advanced approaches using machine learning, large language models (LLMs), and deep learning, or a combination of them. Students will use Python or R for data preparation, visualization, modeling, and evaluating forecast accuracy and uncertainty.

Possible projects include:

• Health: Using continuous glucose monitoring (CGM) data to predict glucose levels and the risk of low or high blood sugar in people with diabetes, with potential applications in guiding automated insulin delivery in artificial pancreas systems.
• Energy: Analyzing and forecasting energy consumption across seasons, including cooling demand in summer and heating demand in winter, to understand how weather and usage patterns affect demand and support more efficient energy planning and management.

Interested students should contact Ali Tavasoli at hqyq4k@jmu.edu with a brief description of their interests, relevant coursework, and programming experience.

How can an algorithm improve its decisions as new information arrives? Online learning studies algorithms that learn and update their decisions as they receive data and feedback. Students will explore how these methods interact with optimization and mathematical models of systems that evolve over time.

Possible projects include comparing learning and optimization algorithms, investigating how noisy observations affect decisions, and examining how learning rates or the frequency of updates influence performance and stability.

Tools may include first-order methods in optimization, differential equation models, and computer simulations in Python. Projects can emphasize mathematical analysis, computational experiments, or a combination of both.

Interested students should contact Ali Tavasoli at hqyq4k@jmu.edu with a brief description of their interests, relevant coursework, and programming experience.

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