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.
Fall 2025 Projects
Modeling Opinion Dynamics and Consensus Formation
Faculty Mentor: Prof. Yue Liu and Prof. Alexandria Volkening
Level: Intermediate
Description: This project involves modeling the dynamics of opinions in a population using differential equations, network theory, and simulations. The goal is to understand how the structure of a social network and the ways that individuals interact may impact whether there is agreement or polarization in a population.
Variations of Pick’s theorem and Piecewise Linear Geometry
Faculty Mentor: Prof. Sam Nariman
Level: Advanced
Skills required: Completion of MA 353 and MA 375. Some knowledge of probability and topology is strongly preferred.
Meeting times: This project is a continuation of a summer REU and is not taking new students for the fall.
Description: Pick's theorem gives a formula for the area of a planar integral polygon (where all vertices lie in Z2 in terms of the number of integral points in the polygon. Peter Greenberg developed Piecewise Linear SL2(Z) geometry which in particular generalizes Pick’s theorem. Greenberg observed relations to different parts of mathematics including Algebraic K-theory, foliations, etc. But our main goal is to explore this piecewise area preserving geometry based on Greenberg’s works
Learning Bridge Numbers of Knots
Faculty Mentor: Prof. Thi Hahn Vo
Level: Beginner/Intermediate
Skills required: Basic background in linear algebra or topology is helpful, but not required. Familiarity with Python (or willingness to learn quickly).
Meeting times: Negotiable
Description: Traditionally, knot theory is the study of embeddings of circles in 3-space, and it models a wide range of real-world entanglements such as vortex knots and linked vortex rings in fluids. A central problem in knot theory is the computation of knot invariants, quantities that capture how “complicated” a knot is. One such invariant is the bridge number, defined by minimizing a count over all possible diagrams of a knot.
Despite its importance, there is currently no algorithm to compute the exact bridge number for all knots, since one must in principle consider infinitely many diagrams. However, the availability of large labeled datasets — for example, exact bridge numbers for over one million classical knots up to 16 crossings — opens the door to machine learning methods as powerful predictive tools.
This project will explore the use of supervised learning techniques (e.g., neural networks, k-nearest neighbors, and decision trees) to classify 3-bridge and 4-bridge knots among classical knots of up to 16 crossings. Students will gain experience both in the mathematical aspects of knot invariants and in modern applied machine learning, including feature engineering, training, and model evaluation.
Symmetries of Fractals
Faculty Mentor: Prof. Shuyi Weng
Level: Intermediate
Skills required: A proof-based course that contains elementary group theory. Taking or having taken complex analysis or general topology is useful but not required. Coding experience is preferred.
Meeting times: Negotiable
Description: A fractal is a geometric shape containing detailed structure at arbitrarily small scales. Many fractals appear similar at various scales, which is called the self-similar property of fractals. The self-similar property opens up symmetries beyond the familiar rotations and reflections, often leading to rich structures that are particularly of interest to geometric group theorists. In this project, students will use computing applications to visualize different types of fractals and analyze their geometric and combinatorial properties. We will review current literature on the subject and combine existing theorems with observations from the project to draw conclusions about these fractals.