Autumn 2025 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:
Matching Points in the Concave Setting
Faculty Mentor: Dr. Stefan Steinerberger
Project Description: We study the problem of matching N red points with N blue points so that some function is minimized. If the function is a convex function of the distance, then we are in the classical setting of Optimal Transport and much is known. We will be interested in the case where the function is a concave function of the distance. There, very little is known and new phenomena are waiting to be discovered.
Simultaneously, we will try to solve a very concrete problem of how to match TAs with sections that they will be assigned to teach. This is a complex real-valued matching problem with a number of constraints; currently, it is all done by hand — surely, with a little bit of help from mathematics, we can do better than that!
Project Level: Advanced: students who have taken multiple upper-level mathematics courses
Additional Course Requirements:
Programming Requirements: Some programming skills would be useful.
Solving games mathematically
Faculty Mentor: Dr. Francois Clement
Project Description: Ever wondered why you never win as second player in tic-tac-toe? Hesitating to trade a resource to a rival in Settlers of Catan because they could finish ahead of you? Fascinated by papers showing Mario Kart and the Legend of Zelda are NP-hard? This project aims to look at whatever games you are interested in and attempt to “solve” them mathematically.
Depending on the type of game and its complexity, solving it can take many forms. It could be determining positions from which there is an optimal strategy that guarantees victory, or a probabilistic analysis of the game situation and the future plays you should make and their outcomes. For more complicated games, this could also involve showing that finding the optimal strategy cannot be done in polynomial time.
Project Level: Intermediate: students who have taken Math 300
Additional Course Requirements:
Programming Requirements: