SURF Mentoring
Potential projects/topics: I have a few different topics that I find interesting, that my mentee may also find interesting. The first topic is studying the mathematics behind gerrymandering. We would explore ways in which mathematicians have determined whether electoral districts are fair, which is especially relevant in the present political climate. I am especially interested in how geometry can be used to determine the fairness of maps. Ideally, we would run some code to test the fairness of various political maps and/or simulate "completely fair" maps.
For more information, see the following Quanta magazine article: https://www.quantamagazine.org/how-math-has-changed-the-shape-of-gerrymandering-20230601/
Or the following link: https://mggg.org/work
A second potential project is studying the effectiveness of hyperbolic neural networks. Usually, data fed into neural networks, which are used to make predictions, are represented as points in R^n. Hyperbolic neural nets fit data to points in hyperbolic space, a "non-Euclidean" geometry which exhibits behavior similar to that of a saddle or a Pringle chip to make predictions. As a hyperbolic geometer, I am curious about the efficacy of these hyperbolic neural networks, and how my research could be applied to understanding them. During this project, the students would learn some hyperbolic geometry, and the ultimate goal of the project would be to try to understand the following paper: https://arxiv.org/pdf/1805.09112 (and/or more recent state-of-the-art papers), and test out the effectiveness of hyperbolic neural networks on real-world data.
Potential skills gained: using computer programming to solve math problems (and vice versa)
Required qualifications: Knowledge of a programming language, especially Python. Familiarity with Python packages since as SciPy, NumPy, Pandas preferred. GitHub version control knowledge preferred; usually if the student has taken some intro CS classes, this will be satisfied. Linear Algebra (Math 355 or 354). Even better if the student has completed upper division math classes.
Direct mentor: Post-doctorate