Research
General Information
Each project will consist of a small research team consisting of typically 2-4 undergraduates, a graduate mentor, and a faculty mentor. The graduate mentor and undergraduates will meet on a weekly basis, with full team meetings every few weeks as determined by the faculty mentor.
Undergraduates who have been accepted into a project must sign up for the 3-credit "Purdue Experimental Math Lab" course (currently listed under MA 490) and must pledge that they are able to dedicate 10 hours of effort per week to the project. For information on how to apply, click on the tab 'Join PXML' at the top of the screen.
Spring 2026 Projects
Exploration of Groups using Topology
Faculty Mentor: Prof. Lvzhou Chen
Level: Advanced
Skills required: Knowledge of Group Theory or Topology (MA571). Knowing some programming language like Python would be helpful.
Description: Groups encode the symmetry of spaces. The goal of this project is to explore the notion of generalized torsion in certain explicit groups using topological ideas (surfaces built out of polygons). As part of the exploration, we may develop and implement algorithms to do computations and computer experiments. Depending on the progress and interest, we may explore other related group theoretic notions or invariants.
Solving optimization problems on the D-wave quantum computer
Faculty Mentor: Prof. Birgit Kaufmann
Level: Intermediate
Skills required: Coding experience in Python with Jupyter notebook. Elementary understanding of Quantum Mechanics is desirable, but not necessary.
Description: The D-wave quantum computer is technically a quantum annealer, simulating the Ising model in an external magnetic field. It is programmed to find its ground state energy for a given set of model parameters. It can be used to solve quadratic optimization problems that can be expressed in terms of a so-called QUBO, a quadratic unconstrained binary optimization problem. This project will explore one or several optimization problems as QUBO on the D-wave computer or on the freely available quantum simulator offered by DWave.
Fully Nonlinear Eigenvalue Problems
Faculty Mentor: Prof. Nicholas McCleerey
Level: Advanced
Skills required: Completion of MA 353 and MA 375. Some knowledge of probability and topology is strongly preferred.
Meeting times: Tuesday or Wednesday, in the afternoon or evening
Description: A common problem in partial differential equations (PDEs) is the eigenvalue problem. The original example is to consider a vibrating string, when two endpoints are fixed (e.g. on a violin). The eigenvalues correspond to the natural frequencies at which the string "prefers" to vibrate, and can be computed by solving a second-order ODE with fixed boundary values.
When you move to higher dimensions, the ODE is replaced with a PDE. For simple PDE, we can compute the eigenvalues over the unit ball by hand. For not-so-simple ones however, this is not tractable, and we need to resort to numerical methods to make progress.
In this project, we will explore eigenvalues for highly non-linear PDE in large dimensions. We will consider PDE related to the Monge-Ampere operator, which plays an important role in convex and complex geometry. On the unit ball, we can use symmetry to reduce the MA equation to a non-linear ODE, making the problem more approachable. Special focus will be given to the behaviour of the eigenvalues as the dimension of the ball goes to infinity.
Combinatorics of Cyclic and Abelian Group Actions
Faculty Mentor: Prof. Shay Phagan
Level: Intermediate/Advanced
Skills required: Completion of courses in Linear Algebra and Abstract Algebra
Description: One can recover the orbit decomposition of a cyclic group action on a finite set from knowledge of the number of orbits associated with each subgroup. This property fails for the Klein four group, so it seems reasonable that finite cyclic groups might be the only groups enjoying this "orbit decomposition reconstruction" property. We will explore this idea and possibly search also for generalizations that do hold for larger classes of Abelian groups.
Graph Energy and the Completion Number
Faculty Mentor: Prof. Thomas Sinclair
Level: Intermediate
Skills required: Completion of a proof-based course in Linear Algebra. Some experience with Python is preferred.
Meeting times: Wednesday afternoon or evening
Description: Graph energy is a linear-algebraic invariant of graphs introduced in the 1970s from theoretical chemistry and can be described as the sum of the absolute values of the eigenvalues of the adjacency matrix of the graph. The completion number of a graph was more recently introduced and gives an integer value which quantifies the defect in being able to complete a matrix which is partially positive on the edges of the graph to a positive matrix.
The project will focus on finding effective, computable bounds for the completion number of a graph and to exploring connections between the completion number of a graph, graph energy, and other invariants. Time permitting we will also seek to explore how graph energy and the completion number can be adapted to the setting of "quantum graphs" - subspaces of square matrices closed under transpose and containing the identity.