Spring 2026

The goal of this project is to develop materials, methods and activities to introduce basic natural approaches of teaching mathematics into the communities of parents, children and teachers with a focus on elementary school (K-5).

The approach will focus on building and developing the following:

1. Games to introduce intuition.
2. Motivation for mathematical ideas.
3. Visualizations of mathematical concepts.
4. Introduction of mathematical language through formal definitions.
5. Introduction of mathematical models – rules for the games.
6. Formulation of mathematical problems – how you “win” a game.
7. Instruction on algorithmic systematic solutions for the mathematical problem.

Project Mentors: Shamgar Gurevich & Ohhoon Kwan

Graduate Student Members: Junyi Cheng, Rahul Panda, Taiwo Felicia Taiwo

Undergraduate Team Members: Quintin Hammi, Yifan Mo, Arham Shahzad

Motivated by the classical theory of complex multiplication, Kaneko defined values of the modular \(j\)-function at real quadratic irrationalities as its integral over the corresponding closed geodesics. He conjectured that these values are explicitly bounded. Part of this conjecture has been proved recently in arxiv:2505.14500. In this project, we will explore analogues of this conjecture for modular forms other than the \(j\)-function.

Students will study the basics of modular forms and the conjecture of Kaneko, then use the computer program SAGE to experimentally tabulate cycle integrals of modular forms. From these, they will try to formulate conjectures analogous to that of Kaneko, and attempt to prove them following the strategy for the \(j\)-function.

Project Mentor: Yingkun Li

Graduate Student Mentor: Yiwen Bai

Undergraduate Team Members: Mindeok Seo, Leo Yang, Lang Zhou, Daniel Zikel

We say a polynomial map \(f\) to be postcritically finite, if for every critical points \(c\), there are \(0<n<m\) such that \(f^n(c)=f^m(c)\). For post critically finite maps of a given degree on \(\mathbb{R}\) and on non Archimedean local fields, there are known algorithms to find all their topological entropies. We will look at the results of these algorithms and investigate the properties of these numbers.

Project Mentor: Chenxi Wu

Undergraduate Team Members: Eleanor SomsakHein, Youran Wang, Haochun Zhang, Mirana Zhang

Let \(G\) be a graph. That is, \(G\) is a set of vertices with edges joining some of the vertices. We allow for multiple edges between vertices, as well as for edges from a vertex to itself. When a graph map \(f : G \to G\) is “irreducible,” there is a choice of length for each edge \(e\) so that \(f(e)\) is a path in \(G\) with length equal to the length of \(e\) times some value \(k\). This value \(k\) is called the stretch factor of \(f\).

The fundamental group of \(G\) is the free group of some rank \(N\), and we say the rank of \(G\) is \(N\). In this project, we will investigate which rank 4 graphs admit irreducible graph maps with small stretch factors. We will work on writing code to list each of these graphs up to isomorphism. After moving on to ranks 5 and 6, we will look for patterns, and try to determine a formula for the number of rank \(N\) graphs admitting small stretch factor graph maps.

Project Mentor: Paige Hillen

Graduate Student Mentor: Rachel Heikkinen

Undergraduate Team Members: Isabella Levinthal, Vahe Ohihoin, xinyi shi, John Tsonis

There is a long and “storied” history of viral dynamic models of interactions between the virus HIV and the immune system. Several different models drastically altered our understanding of how HIV causes diseases, and changing the classic “slow virus” into a rapidly changing virus that wears the immune system down in a war of attrition. In the last 5 years though it has become clear that a new biomarker, the CD4/CD8 ratio is the best predictor of mortality and morbidity, but how this marker changes over time on therapy is not understood. In this project we will look at many earlier iterations of HIV models and develop a new model that will hopefully explain the current poorly understood and variable rise of the ratio on therapy. We have access to key data from the Multi Center AIDS Cohort Study as well as the Women’s Interagency HIV study and other data.

Project Mentor: Rob Striker & Amy Cochran

Graduate Student Mentor: Emma Hayes

Undergraduate Team Members: Emma Mayhew, Tuan Tai Nguyen, Tyler Seils, Yilin Shi

The project will build on the results of the recent preprint https://arxiv.org/abs/2509.15446, which studied the pair correlation function of point processes arising from the scaling limits of random matrices. The primary goal is to solve one of the proposed open problems (Problem 6 in Section 7): showing directly that two different representations of a particular correlation function are the same. If time permits we could also explore other questions.

While the problem itself can be explained with tools only from calculus, it has connections to a number of beautiful areas of mathematics: probability and stochastic processes, complex analysis, special functions. We will review some of the required background at the beginning of the semester.

Project Mentor: Benedek Valko

Graduate Student Mentor: Yahui Qu

Undergraduate Team Members: Xilei Li, Shengqi Qiu, Spencer Venancio, Lingfang Yuan

Many physical models, machine learning techniques, or data assimilation methods produce detailed state trajectories of complex systems; yet, they often fail to capture the correct statistical behavior, such as variability, correlations, or the frequency of rare events. In many applications, however, statistical fidelity is more important than pointwise accuracy. This project explores how statistical information alone can be utilized to guide and enhance dynamical models.

The central idea is to use statistical supervision, summary statistics such as moments, correlations, or distributional measures, as the primary observations. Rather than imposing these constraints directly on the full high-dimensional state, the project applies them indirectly through a data assimilation over a latent representation that acts as a low-dimensional control space. Adjustments are made in latent space so that, after decoding, the full-space dynamics exhibit the desired statistical properties. Importantly, the goal is not to learn or interpret the latent variables themselves, but to achieve statistically accurate behavior in the original variables.

Students will implement and compare methods that update latent variables based on discrepancies between predicted and target statistics, using ensemble-based approaches. They will then evaluate how these statistical corrections affect the behavior of the full-space dynamics. The project emphasizes interpretability, computational efficiency, and principled use of weak supervision, and provides hands-on experience at the intersection of machine learning, statistics, and dynamical systems.

Project Mentor: Pouria Behnoudfar

Graduate Student Mentor: Marios Andreou

Undergraduate Team Members: Tan Bui, Cody McKenna, Singer Xing, Daniel Youngberg

Fix a covering space of a closed, orientable surface. The homotopy lifting property provides a means to lift closed loops on the base to paths in the cover, and one might ask about topological properties of these lifts: Do they close up? Do they self-intersect? It turns out that these questions have answers which can be characterized completely in terms of the combinatorics of paths on certain finite edge-colored directed graphs which arise as quotients of the Cayley graph of a finite group by a particular subgroup.

The primary goal of this project is to determine whether or not two non-isomorphic covers of a given surface corresponding to almost conjugate subgroups of a finite group can be distinguished by the simple (non-self-intersecting) lifts of certain curves. This goal ultimately derives from a desire to use these criteria to distinguish isospectral covers arising from Sunada’s construction via their simple length spectra.

In a previous semester, our undergraduate research team (a) wrote SageMath code to provide a plethora of examples of such pairs of subgroups; (b) developed simple criteria for determining whether or not a given curve on the base surface has simple lifts to a given cover via the aforementioned representation as paths in finite edge-colored directed graphs; and (c) used this understanding to develop SageMath code to distinguish pairs of almost conjugate subgroups. This semester, we will use this code to verify or reject the distinguishability conjecture in the found examples; and (b) depending on the results, attempt to prove or disprove the conjecture in general.

Project Mentor: Max Lahn

Graduate Student: Rachel Hanger

Undergraduate Team Members: Benjamin Janke, Emmanuel Josiah Zhagui-Quito, Simon Bjorn Kellum, Jaan Amla Srimurthy

This is a continuation of the Terwilliger MXM project from Fall 2025.

We continue to investigate the subconstituent algebra T of a distance-regular graph with valency three. The project goal is to work out the irreducible T-modules. We did many examples in Fall 2025. We hope to do the remaining examples in Spring 2026.
After all the examples are worked out, then we hope to publish our results in a journal article with multiple coauthors.

Project Mentor: Paul Terwillliger

Graduate Student: Jimmy Vineyard

Undergraduate Team Members: Kevin Kauflin, Barnabas Valko, Hanyi Wu

In Fall 2025, we investigated neural network parameterization as an approach to solving PDE-constrained inverse problems. Given observations of a state governed by the heat equation, our goal was to recover the unknown source term by minimizing a Tikhonov-regularized objective. We compared three methods: the classical adjoint-state approach, vanilla gradient descent with automatic differentiation, and a neural network parameterization of the control variable. We tested these methods on benchmark problems spanning a range of settings, including highly oscillatory target states and problems requiring hard constraints such as non-negativity. This semester, we hope to extend the framework to nonlinear and higher-dimensional PDEs, handle more complex constraints, and explore RKHS-based reformulations for improved constraint enforcement.

Faculty Mentor: Yukun Yue

Graduate Student: Martin Guerra

Undergraduate Team Members: Pritam Kayal, Mark Miller, Yifan Yang