Spring 2025

The Kayles Game is a combinatorial two-player game where players alternately knock down one or two adjacent pins from a sequence of bowling pins. The objective is to make it impossible for the opponent to make a legal move. The game can be thought of as a graph made up of paths, where each player removes either (i) an edge, or (ii) a vertex of positive degree and all the edges attached to it. In this context, a graph is called an N-position if the next player has a winning strategy, and a P-position if the previous player does. This semester, we focused on studying the game on complete bipartite graphs, represented as matrix with 1s indicating edges. We developed Python and Rust programs to determine P- and N-positions for complete bipartite graphs, using symmetry-based strategies and automorphisms to identify winning conditions.

Faculty Mentor: Mikhail Ivanov

Graduate Student: Ian Seong

Undergraduate Team Members: John Fitzpatrick, Huaiyuan Jing, Jason Kolbly, Jihong Min, Chenxi Xia

In this project, we explored the dynamics of polynomials over the p-adic integers by investigating their topological entropy, a measure of complexity in dynamical systems. We used the concept of scaling disks to model the behavior of these polynomials and wrote a Python program to construct directed graphs representing their dynamics. Our computations consistently showed zero entropy, meaning the systems do not exhibit exponential growth in complexity. The result stems from the fact that p-adic polynomials tend to be non-expanding, which prevents the kind of chaotic behavior seen in other settings. While this doesn’t rule out all complexity, it highlights how topology and number theory shape the behavior of dynamical systems in surprising ways.

Faculty Mentor: Chenxi Wu

Graduate Student: Eiki Norizuki

Undergraduate Team Members: PJ Clementson, Lexiang Liu, Aiden Styers, FengYing Wang

While supervised learning is commonly used in data assimilation, reinforcement learning (RL)
offers a natural advantage through its sequential decision-making, aligning well with the iterative
structure of assimilation. We develop a novel RL-based approach that enhances assimilation by
incorporating an ensemble-based uncertainty. This framework treats ensemble members as
agents, enabling parallel, while also bounding actions to maintain physical feasibility. Applied
Lorenz 63 and 96, the results match the performance of ensemble Kalman filter with significantly
lower computational cost. In the next step, We will extend our method to multimodal data
assimilation. As a next step, we are preparing to extend this method to multimodal data assimilation.

Faculty Mentor: Pouria Behnoudfar

Graduate Student: Zhongrui Wang

Undergraduate Team Members: Samanyu Arora, Aidan Dooley, KK Thuwajit, Dylan Wallace

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.

Faculty Mentor: Andrew Zimmer

Graduate Students: Aleksander Skenderi and Vicky Wen

Undergraduate Team Members: Kyle Huang, Jinwoo Park, Jaan Amla Srimurthy

The project consisted of two parts. In the first part we reviewed properties of quaternion valued square matrices and complex and quaternion valued normal random variables. In the second part we studied the eigenvalue distribution of a random quaternion tridiagonal block matrix. We reduced the evaluation of the eigenvalue density to the problem of determining the second moment of a Vandermonde-like structured matrix built from quaternion random variables. We further transformed the problem into the study of the determinant of a complex valued random matrix which led to the analysis signs of certain structured permutations.

Faculty Mentor: Benedek Valko

Graduate Student: Yahui Qu

Undergraduate Team Members: Jack Grosskreuz, Pritam Kayal, Vivek Saravanan, Sanan Sharifzade

Optimal experimental design (OED) plays a critical role in computational and engineering sciences. It strategically identifies key experimental variables prior to data collection, guiding the experimental process and helping to reduce operational costs. The development of OED research spans decades, with well-established design approaches demonstrating empirical success. 

However in many practical scenarios, data samples are contaminated by noise of various magnitudes, a.k.a. the heteroscedastic noise. This is in sharp contrast to the classical design study where the fundamental assumption is that all the measurements contain random noise in the same variance. Such heterogeneity in the noise pollution poses substantial challenges to experimental design. And classical design approach can fail miserably. (See Figure (a).) 

Our project focuses on addressing computational and statistical obstacles of heteroscedastic experimental design. In this noisy regime, our key message is that estimating the design value of each measurement must leverage both its accuracy and its feature-wise importance. We demonstrate the numerical success in Figure (b). 

Faculty Mentor: Ruhui Jin

Graduate Student: Martin Guerra

Undergraduate Team Members: Phoebe Kuang, Tate Waugh

Galois representations are an important tool used in algebraic number theory to study the structure of number fields. In 1975, Jean-Pierre Serre conjectured that every odd, irreducible two-dimensional Galois representation over a finite field arises from a modular form in a certain way. In 2009, Chandrashekhar Khare and Jean-Pierre Wintenberger proved Serre’s conjecture for Galois representations of the field Q, but the case of a general number field, that is a finite extension of Q, is still an open problem. We investigated the case of Q(ζ_3) where ζ_3 is a primitive third root of unity. We used Magma, a computational algebra system, to examine the mod l reduction of two-dimensional Galois representations arising from genus three curves with an order-3 automorphism with the purpose of finding a modular form which corresponds to the representation in the manner described by Serre.

Faculty Mentor: Shiva Chidambaram

Graduate Student: Ivan Aidun

Undergraduate Team Members: Xiyao Chen, Lin Ha, Hampe Meng, Erik Fuller Reed