Spring 2025

Project Level: Beginner: students who have taken MATH 234

Project Description: This is a continuation of the project “Kayles Game on Graphs” in Fall 2024. Suppose two players play a game on a certain finite graph where they take turns to either (i) remove an edge, or (ii) remove a vertex of positive degree and all edges attached to that vertex. The player wins once the opponent cannot make a valid move. This is an example of a finite combinatorial game, meaning exactly one player has a winning strategy from the start. In Fall 2024, the students analyzed the game on paths and their variations. For instance, the first player always has the winning strategy on paths, and the “style” of the game is eventually periodic with respect to the lengths of the paths. In addition to the paths, progress was made for some other graphs like “Spikes”, complete graphs, and complete bipartite graphs. This semester, the goal of the project is to investigate these three families of graphs on who has the winning strategy, and what the “style” of the game looks like for each graph.

Undergraduate Student Requirements: Knowledge of graph theory’s language is recommended (CS/Math 240 – Introduction to Discrete Mathematics is enough). Knowledge of python is required.

Faculty Mentor: Mikhail Ivanov

Graduate Student: Ian Seong

Project Level: Intermediate

Project Description:This is a continuation of the project from Fall 2024. We are aiming at investigating polynomial dynamics on p adic and other non Archimedean fields via Rueller’s transfer operator and the corresponding dynamical zeta functions.

Undergraduate Student Requirements: Familiarity with linear algebra is necessary, some knowledge of abstract algebra (541/542) or analysis (521/522) would be beneficial. Being able to program in some programming language would be beneficial.

Faculty Mentor: Chenxi Wu

Graduate Student: Accepting Applications

Project Level: Advanced: students who have taken multiple upper-level mathematics courses

Project Description: This project aims to develop advanced machine learning (ML) techniques for data assimilation (DA) to model complex dynamical systems. DA is essential for integrating observations with mathematical models, and improving prediction accuracy, particularly in systems with chaotic or nonlinear behavior. Building on previously developed reinforcement learning (RL) methods that mimic the Ensemble Kalman Filter (EnKF), this project extends these approaches to latent ML frameworks. As a next step, diffusion machine learning models will be developed, leveraging their generative and stochastic capabilities to predict system dynamics through an ensemble-like process. These innovations enable the efficient representation of high-dimensional, nonlinear systems and enhance the adaptability of DA methods to intricate dynamics.

By incorporating latent ML and diffusion models, this project addresses critical challenges such as capturing intermittent behaviors and extreme events, often inadequately represented in traditional DA frameworks or physical modeling approaches. These methods provide robust and realistic predictions by effectively modeling key observed features in nature. Integrating state-of-the-art ML tools with DA techniques is at the forefront of modern applied mathematics and computational science, advancing predictive modeling for complex dynamical systems.

Undergraduate Student Requirements: Experience with Matlab or Python. Preference will be given to students interested in applied math or interdisciplinary studies as well as students who have taken Math 415 – Applied Dynamical Systems, Chaos and Modeling and Math 443 – Applied Linear Algebra.

Faculty Mentor: Pouria Behnoudfar

Graduate Student: Zhongrui Wang

Project Level: Advanced: students who have taken multiple upper-level mathematics courses

Project Description: This project will continue a thread of research started in a past REU project [published as “A rigidity result for proper holomorphic maps between balls “, Proceedings of the AMS, 152: 1573-1585, 2024].

In this project we will study proper holomorphic maps between unit balls in complex Euclidean spaces which have lots of symmetries. This is a project in the field of several complex variables (complex analysis in higher dimensions), but the methods used will be from (geometric) group theory and linear algebra. Hence it is not necessary for participants to know any complex analysis.

I hope the project will involve a very pleasant blend of group theory, analysis, and linear algebra.

Undergraduate Student Requirements: MATH 541 – Modern Algebra and 521 – Analysis I are necessary. Other 500/600 level courses like 551 – Elementary Topology and 623 – Complex Analysis would be helpful, but are not necessary.

Faculty Mentor: Andrew Zimmer

Graduate Students: Aleksander Skenderi and Vicky Wen

Project Level: Advanced: students who have taken multiple upper-level mathematics courses

Project Description: The first half of the project aims to explore the linear algebra of quaternion valued matrices, and the properties of quaternion valued Gaussian random variables and random matrices. In the second half of the project we will check whether the results of the recent preprint https://arxiv.org/abs/2412.04579 can be extended to quaternion valued random matrices.

Undergraduate Student Requirements: Linear algebra (341 or 375 preferred), probability (431 or 531 preferred), at least one proof based course, experience with Mathematica would be useful, but not required.

Faculty Mentor: Benedek Valko

Graduate Student: Yahui Qu

Project Level: Advanced: students who have taken multiple upper-level mathematics courses

Project Description: Learning from data to understand the physical world is ubiquitous in computational and engineering science. Indeed data-driven discoveries have showed emerging success for physical systems. However, collecting experimental data can still be expensive and time-consuming. When given a tight financial and labor budget, scientists need to first identify the valuable measurements and plan for the experiments accordingly. This gives rise to optimal experimental design (OED). The goal of OED is to “design” an optimal way to query data, so that these strategically chosen measurements meet the minimal experimental cost.

One of the main challenges in data science is that the measurements we have in hand are polluted with random noise perturbation, in either benign or adversarial fashion. With such inaccurate data, scientific machine learning tasks would never be fully accomplished, sometimes could fail miserably. In order to address the noisy learning bottleneck, the goal of our project will be looking into the open questions:

1. Can experimental design automatically identify the accurate measurements over the noisy ones?
2. Can experimental design be further tailored to prune noisy inputs?

Application examples are broad, including imagining science, physical models, biological systems and so on, depending on the mentees’ academic background and interests. Students who are interested in applied math, numerical analysis and probability theory areencouraged to join the project.

Undergraduate Student Requirements: Any courses from Math 514 – Numerical Analysis, 521 – Analysis I, 531 – Probability Theory, 535 – Mathematical Methods in Data Science, 704 – Methods of Applied Mathematics-2, 715 – Methods of Computational Mathematics II, 717 – Stochastic Computational Methods. Students should be comfortable with Python or Matlab.

Faculty Mentor: Ruhui Jin

Graduate Student: Accepting Applications

Project Level: Advanced: students who have taken multiple upper-level mathematics courses

Project Description: The project aims to study the Galois representations associated to certain genus 3 curves in the reducible case. Concretely, one goal is to find the semi-simplied images using L-polynomials and Chebotarev density theorem. Another goal is to uncover congruences with Bianchi modular forms by identifying sub-representations.

Undergraduate Student Requirements: Algebra 1 (MATH 541), Algebra 2 (MATH 542), Algebraic number theory (MATH 567). Familiarity with computer algebra packages like Magma and Sage would be helpful.

Faculty Mentor: Shiva Chidambaram

Graduate Student: Accepting Applications