Winter 2026 Projects
Quantum Probability via Arbitrary Functions
Faculty Mentor: Dr. Benjamin Feintzeig
Project Description: How do deterministic physical systems give rise to probabilities? In classical probability theory, one can show that some differential equations governing the dynamical evolution of physical systems lead to “universal behavior” in the limit of large times. More specifically, the system almost always approaches the same final distribution for certain physical variables, independent of the distribution of initial conditions one began with. This is called the classical method of arbitrary functions—where the arbitrary functions refer to the possible initial conditions, which turn out to be irrelevant to the long time behavior. Such results are thought to explain why physical systems display the probabilities they do, e.g., why a tossed coin lands heads with probability 1/2 or a roulette wheel has equal probability of landing on each number. In this project, we aim to extend the method of arbitrary functions to systems described by quantum physics, where both the dynamical equations and conceptual framework for probability have novel features. We will explore the behavior for concrete physical models in the limit of large times and the classical limit of small values of Planck’s constant.
Project Level: Advanced: students who have taken multiple upper-level mathematics courses
Additional Course Requirements: Some background in linear algebra, analysis, and abstract algebra will be helpful for this project. No background in physics is required.
Programming Requirements:
Faculty Mentor: Dr.
Project Description:
Project Level: Advanced: students who have taken multiple upper-level mathematics courses
Additional Course Requirements:
Programming Requirements:
Extending Low-Discrepancy Sets
Faculty Mentor: Dr. François Clément
Project Description: Low-discrepancy sets are point sets designed to be as uniformly distributed as possible in the unit hypercube. They are extensively used in sampling and numerical integration. However, there is a gap between fixed theoretical designs and applications, where the sampling budget can evolve. The aim of this project is to
1) Find methods to add points to an existing point set (for example pre-sampled points) while staying as uniform as possible.
2) Given a point budget, find in which order the sampling should be done to increase chances of not requiring the entire sampling budget.
Project Level: Intermediate: students who have taken Math 300
Additional Course Requirements:Algorithmic classes would be a plus
Programming Requirements: This will be a code-heavy project (though open-ended). Comfortable in at least one standard language is a must (C, Python, C++,…)
Wild Knot Mosaics
Faculty Mentor: Dr. Allison K. Henrich and Andrew Tawfeek
Project Description: Mosaic diagrams were developed in 2008 by Lomanoco and Kauffman to build quantum knot systems. Since then, the structure of mosaics has been widely studied by many others due to their convenient way of encoding a knot in three-dimensional space as a matrix. In 2014, it was shown by Kuriya and Shehab that mosaics are a complete invariant for tame knots, which are the concrete knots we can make with string in everyday life.
In this project, we attempt to push the capabilities of the mosaic representation further: by adapting them in order to capture knots that are not tame, namely wild knots. These knots can have infinite tangles and diverge towards infinity in various locations along the strand — but by adapting the structure of mosaic diagrams, some of these pathological objects may be represented, such as the Alexander horned sphere. We will spend this project formalizing the theory behind these wild mosaics, writing code along the way for their study and investigation. This project is not accepting new members.
Project Level: Intermediate: students who have taken Math 300
Additional Course Requirements:
Programming Requirements:
Self-Organized Criticality
Faculty Mentor: Dr. Christopher Hoffman
Project Description: Self-Organized Criticality is a concept from mathematical physics that claims to explain diverse phenomenon such as earthquakes, avalanches and forest fires. These systems all have energy that builds up slowly and then is quickly released. The amount of energy released can vary widely from very small earthquakes that can barely be felt to large earthquakes that can destroy cities.
Activated random walk is a probabilistic model that is believed to share some crucial properties with models form self-organized criticality. This model involves many particles that are performing random walks that interact with each other. We will numerically study the distribution of the amount of energy released in the activated random walk model. This project is not accepting new members.
Project Level: Intermediate: students who have taken Math 300
Additional Course Requirements: 394/5/6 would be helpful.
Programming Requirements: programming skills are important
Project Level: Advanced: students who have taken multiple upper-level mathematics courses
Additional Course Requirements:
Programming Requirements:
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Data-driven approaches to the study of macroeconomic growth
Faculty Mentor: Dr. Fanze Kong
Project Description: One of the most characteristic features of the spatial economy is the presence of diverse economic agglomerations, where firms, industries, and populations cluster within specific geographic areas, leading to increased economic activity and productivity in those regions. Extensive research has identified several contributing factors, including capital flows, technological progress, and labor migration. The form of the production function, which depends on these factors, also plays a crucial role. In practice, it is often specified as a Cobb-Douglas function. However, it is difficult to determine whether this assumption is truly compatible with empirical data.
The aim of this project is to employ data-driven methods to identify the production function that best explains macro-scale patterns of economic agglomeration, using the Solow-Swan model formulated as a partial differential equation (PDE) as the analytical framework. The analysis will incorporate several physical-law based loss formulations, including both PDE-constrained and variational (energy-based) structures.
Project Level: Intermediate: students who have taken Math 300
Additional Course Requirements: AMATH353
Programming Requirements: Python and Matlab