Spring 2026

Kronecker once said: “God made the integers; all else is the work of man”. This semester we followed the theme of making K-2 children understand integers and their properties. Toddlers learn how to count, and this involves teaching the seven counting principles. From there, various methods for understanding addition and subtraction are discussed and compared. Chips, sticks, and rulers; these are just some of the visual models for integers one can use to teach children. One milestone of understanding is the Place Value System: students do arithmetic using certain rules in the base-10 numeral system. We partially discussed negative integers and commutative property and will continue from there aiming towards multiplication and fraction in the future.

Place Value Chart for 245+341Place Value Chart for 245+341 — Place Value Chart for 245+341

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

A polynomial map $f$ is postcritically finite (PCF) if for every critical point $c$ , the orbit $c, f(c), f^2(c), \ldots$ is eventually periodic, i.e. there exists <math data-latex="0 < n 0<n<m0 < n < m with $f^n(c) = f^m(c)$ . In this project, we investigated the properties of post-critically finite polynomial maps. We used python to visualize the distribution of entropy values for (sampled) quadratic polynomials of a given degree. The most notable observation being the high concentration around the lowest observed value of around 0.16, with values rising gradually after. Also using python, we mapped the galois conjugates for all postcritically finite polynomials of a given degree on the complex plane. For the quadratic case, we observed that the conjugates lie in an annulus $r_{\min} \leq |z| \leq 2$ . The inner radius is around $\sqrt{2}$ , and the outer around 2. As for the cloud itself, the boundary is mostly made up of periodic conjugates, as the eventually periodic ones are concentrated towards the interior. These observations reflect number-theoretic constraints upon what values are possible as entropies.

In each of the graphs below, N is the depth at which the program that created each figure iterated the critical orbits.

Quadratic PFC - Galois conjugates (N=12). N is the depth at which the program that created each figure iterated the critical orbits

Quadratic PCF maps - entropy spectrum (N=14). N is the depth at which the program that created each figure iterated the critical orbits

Project Mentor: Chenxi Wu

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

Given an irreducible automorphism $\varphi$ of the free group $F_N$ , work of Bestvina and Handel guarantees there is always an irreducible train track graph map representative $f:G \rightarrow G$ of $\varphi$ . Here, $G$ is a graph with fundamental group equal to $F_N$ and $f$ is graph map which has particularly nice dynamical properties. In particular, the largest eigenvalue of the transition matrix of $f$ is equal to the exponential growth rate of words in $F_N$ under iterations of $\varphi$ . This growth rate is called the stretch factor of $\varphi$ . We are interested in understanding the irreducible train track graph maps which yield automorphisms whose stretch factor is small, yet larger than 1.

In this project, we focused on determining the smallest stretch factor among irreducible train track graph maps which occur on the $N-$ rose, the graph with a single vertex and $N$ self loop edges. We conjecture that for each $N \geq 2$ , this minimum is attained by a map we call $f_{\text{izzy}}$ . The $N=4$ case is illustrated below. We have concluded that any valid graph map with stretch factor smaller than or equal to that of $f_{\text{izzy}}$ must have transition matrix $M$ with $|M|=N+1$ , and have almost completed various cases within the $|M|=N+1$ context.

The 4-rose on the left is the graph with a single vertex and 4 self loop edges. The $f_{\text{izzy}}$ map on the 4-rose is the map sending $e_1 \mapsto e_2 \mapsto e_3 \mapsto e_4 \mapsto e_1e_2$. This has the same stretch factor as the map which sends $e_3 \mapsto e_2 \mapsto e_1 \mapsto e_4 \mapsto e_3e_2$. The transition matrices of both maps as well as their largest eigenvalue, both about 1.22 are listed. The important note for the second map is that $k=3$, where $k$ is the value such that $e_i \mapsto e_{i+k}$ where subscripts are taken modulo $4$. Since $gcd(N,k)=gcd(4,3)=1$, this second graph map is also irreducible. — The 4-rose on the left is the graph with a single vertex and 4 self loop edges. The $f_{\text{izzy}}$ map on the 4-rose is the map sending $e_1 \mapsto e_2 \mapsto e_3 \mapsto e_4 \mapsto e_1e_2$ . This has the same stretch factor as the map which sends $e_3 \mapsto e_2 \mapsto e_1 \mapsto e_4 \mapsto e_3e_2$ . The transition matrices of both maps as well as their largest eigenvalue, both about 1.22 are listed. The important note for the second map is that $k=3$ , where $k$ is the value such that $e_i \mapsto e_{i+k}$ where subscripts are taken modulo 4. Since $gcd(N,k)=gcd(4,3)=1$ , this second graph map is also irreducible.

Project Mentor: Paige Hillen

Graduate Student Mentor: Rachel Heikkinen

Undergraduate Team Members: Isabella Levinthal, Vahe Ohihoin, Xinyi Shi, John Tsonis

Our project’s goal, working with UW’s School of Medicine, was to mathematically model the dynamics of HIV infection and immune response. We wanted to focus on CD4 and CD8 T-cells and how the CD4/CD8 ratio impacted patient health outcomes. Since HIV directly infects CD4 cells, this became quite biologically complex. Following the model described in Banks et. al., we reduced the ODE system and ran attraction basin simulations to uncover three equilibrium states based off of viral load. Further global stability analysis revealed that CD4 activation rate is a primary negative effector of the CD4/CD8 ratio, proving it as an interesting target for future research.

Project Mentor: Rob Striker & Amy Cochran

Graduate Student Mentor: Emma Hayes

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

Our project extends recent work by our advisors on the Sine-beta process, a one-parameter family of point processes arising in random matrix theory. In their preprint, they derive the pair correlation function (which captures the likelihood of finding points in two given regions) for $\beta = 2n$ as a $2n$ -th order ODE. We pursued two extensions of this result. On the computational side, we wrote a Mathematica program that outputs these ODEs for arbitrary $n$ . On the mathematical side, we focused on the case $\beta = 6$ , where we aimed to replace the classical formula, in terms of a sixfold integral, with a nicer expression. We did this by decomposing the third-order complex ODE as a first-order ODE composed with a second-order one, then solving them in sequence. This part of the project is ongoing.

Simulation of Wigner's semicircle law for n by n Gaussian Orthogonal Ensemble with n=50, 1000, and 10000. — Simulation of Wigner’s semicircle law for n by n Gaussian Orthogonal Ensemble with n=50, 1000, and 10000.

Project Mentor: Benedek Valko

Graduate Student Mentor: Yahui Qu

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

Dynamical systems model many real-world phenomena, such as weather conditions and the nervous system. However, many operational models that emulate these systems are caught up in complexity and fail to model basic statistical quantities such as regional average temperatures, seasonal variabilities, energy spectra, or correlations between activating neural tissue and their recovering inhibitor. We developed a latent data assimilation methodology that integrates statistical measurements with forecasts to model complex dynamics while keeping predictions physically grounded. In this framework, we train an autoencoder that maps a high-dimensional physical state into a lower-dimensional latent space, and augment the latent state with coarse statistical observations about the system before refining the predictions with an ensemble Kalman filter. We applied this framework to two separate models. First, the stochastic Lorenz ’96 model, a geophysical model of atmospheric energy, where we achieved accurate latent dynamics. Second, the stochastically coupled FitzHugh–Nagumo model, where we obtained accurate activation–recovery behavior by using spike and fast–slow correlation statistics as the observed statistical measurements.

Schematic diagram showing a framework for forecasting high-dimensional complex systems using statistical measurements. The high-dimensional physical state is passed through an encoder to obtain a low-dimensional latent representation, which is augmented by coarse statistical measurements such as estimated means, variances, or energy spectra. The augmented latent state is evolved by a long short-term memory unit, with the statistical observations being assimilated using an ensemble Kalman filter. The refined latent state is decoded back to the physical analysis state. A crossed-out branch indicates that direct assimilation in the full high-dimensional system is too computationally expensive. — Schematic diagram of the proposed framework. A high-dimensional dynamical state is encoded into a low-dimensional latent representation, augmented with coarse statistical observations, advanced with an LSTM forecast model, and refined through an ensemble Kalman filter before being decoded back to the physical analysis state. This avoids computationally expensive full-state assimilation while preserving key statistical and physical structure.

Project Mentor: Pouria Behnoudfar

Graduate Student Mentor: Marios Andreou

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

The Schreier graph associated to a subgroup $A$ of a finite group $G$ is the quotient of the Cayley graph of $G$ by the action of $A$ by left multiplication. This graph is a directed graph whose edges are colored by a fixed generating set of $G$ , and every vertex is adjacent to exactly one incoming edge of each color and exactly one outgoing edge of each color. When $G$ is the codomain of a surjective group homomorphism $\rho \colon \pi_{ 1 } \left( S , s \right) \to G$ of the fundamental group $\pi_{ 1 } \left( S , s \right)$ of a closed, orientable surface $S$ of negative Euler characteristic, paths in the Schreier graph of $A$ encodes the lifting behavior of closed curves on $S$ to a finite degree cover. Our group was interested in comparing the Schreier graphs associated to a pair of almost conjugate subgroups of $G$ , where two subgroups $A$ and $B$ are called almost conjugate if each contains the same number of elements from each conjugacy class as does the other.

In this project, we investigated the degree to which the Schreier graphs associated to two almost conjugate subgroups $A$ and $B$ of a finite group $G$ can be distinguished by their graph-theoretical properties. We proved that two such Schreier graphs are isomorphic (by an isomorphism which preserves edge orientation and color) if and only if the subgroups $A$ and $B$ are conjugate, paving the way for the desired categorization. We developed criteria for determining when paths in such Schreier graphs correspond to simple lifts in the associated surface covers. Concretely, we surveyed all pairs of almost conjugate subgroups of symmetric groups of low rank, catalogued in a previous MXM project, and determined that distinguishing the associated Schreier graphs is computationally infeasible for certain generating sets.

Project Mentor: Max Lahn

Graduate Student: Rachel Hanger

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

Continuing from Fall 2025, our project this semester was to complete decompositions of the standard module into an orthogonal direct sum of irreducible T-modules for each of the 13 distance-regular graphs of valency three. Using the workflow we developed last fall, we have completed the process for the final five graphs and now have results for all 13 graphs

Project Mentor: Paul Terwillliger

Graduate Student: Jimmy Vineyard

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

Our goal is to solve PDE constrained optimization problems using Neural Network parameterization. Traditionally, we solve $\min_f\mathcal{J}(u;f)$ subject to PDEs $\mathcal{L}(u;f)=0\mathcal{L}(u;f)=0$ with boundary conditions $\mathcal{B}(u)=0$ by discretizing the domain of the functions $u,f$ , randomly initializing the value of $f$ over said discretized mesh, and using gradient descent to find the optimum. A neural network parameterization allows potential benefits like a smoother optimization landscape and easy implementation of constraints like $f\geq 0$ by using a non-negative activation in the last layer like $\text{ReLU}$ or $\text{softplus}$ , eliminating the need for projected gradient descent. In particular, we find positive results in problems involving the heat equation and the Vlasov-Poisson equation.

Faculty Mentor: Yukun Yue

Graduate Student: Martin Guerra

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

Spring 2026

Exposition to Mathematics: Bringing mathematical thinking to the community

Bounds of cycle integrals of modular forms

Entropy of post critically finite polynomial maps

Irreducible graph maps with small stretch factor

Developing model of HIV infection that correlates with treatment induced immune recovery

Identities from random matrices

Learning Statistically Accurate Dynamics of Complex Systems

Isospectral Surfaces and Twinless Paths in Cayley Graph Quotients (Continuing Project)

Distance-regular graphs with valency 3 (Continuing Project)

Neural Network Parameterization for PDE-Constrained Optimization (Continuing Project)