Fall 2025

In this project, we numerically approximated solutions to the following optimal control problem which explicitly enforces state constraints.

J(u,\, p \, ; \,q) =\frac{1}{2}\|u-u_d\|_{L^2(\Omega_T)}^2+\frac{1}{2}\|p -p_d\|_{L^2(\Omega_T)}^2+\frac{\lambda}{2} \|q\|_{L^2(\Omega_T)}^2

PDE constraints:

\begin{cases} – \delta u_{xxt} – \gamma u_{xx} + p_x &amp;= F\; \mathrm{in}\; \Omega_T, \\ u_{xt} – k p_{xx} + c_0 p_t &amp;= S\; \mathrm{in}\; \Omega_T, \end{cases}

State constraints:

\begin{cases} g_1(u,p) &amp;= u_{min} – u \le 0, \\ g_2(u,p) &amp;= u – u_{max} \le 0, \\ g_3(u,p) &amp;= p_{min} – p \le 0, \\ g_4(u,p) &amp;= p – p_{max} \le 0. \end{cases}

We applied the augmented Lagrangian method (ALM). The main contribution is that, for the PVE test problem, we compare solutions of unconstrained and state-constrained optimal control formulations. The algorithm we used successfully approximated state constrained solutions. Although constrained solutions satisfy the prescribed state constraints, they yield higher values of the cost functional. This illustrates the trade-off between the unconstrained control problem. Enforcing state constraints leads to an increased objective value, indicating reduced optimality in terms of cost in exchange for constraint satisfaction.

The following image is a compilation of several graphs. Figure 3 shows a graph of the control when state constraints are not enforced. Figure 4 shows how the control changes when the state constraints are enforced. Although the maximum of the control is in a similar location, the overall shape of the graph varies. The next row of figures shows the desired u value, the optimal u in the unconstrained case, and the optimal u in the constrained case. The desired u touches the upper-bound on the state constraint, the optimal u surpasses the upper-bound, and the constrained case obeys the constraint, but the shape varies slightly from the desired u. The last row is similar to the second row, but with results for p.

Faculty Mentor: Sarah Strikwerda

Graduate Student: Hieu Thanh Nguyen

Undergraduate Team Members: Myles Jarrett, Dianhao Sun, Jason Xu, Zhixuan (Jenny) Yan

We set out to gather evidence for Serre’s modularity conjecture over the cyclotomic field containing the $3^{rd}$ roots of unity. We work with $\ell$ -torsion Galois representations arising from superelliptic curves $C$ of genus $3$ that have a $\zeta_3$ -action. The goal was to search for an irreducible $2$ -dimensional sub-quotient representation, that can then be shown to also arise from a Bianchi modular form. To limit the possible spaces of these modular forms, we worked with the Artin conductor of our representations to bound the level predicted in Serre’s modularity conjecture. However, upon analyzing these $6$ -dimensional representations, we discovered that the structure of the representation forbids such a 2-dimensional irreducible sub-quotient representation from occurring. Concretely, the semi-simplification of the $\ell$ -torsion Galois representation of the Jacobian of $C$ completely decomposes as a direct sum of $6$ characters over $\mathbb{Q}(\zeta_3)$ , if it admits a single character.

The semisimplication over Q(zeta_3) of a reducible ell-torsion representation of a Picard curve over Q is completely decomposable

Faculty Mentor: Shiva Chidambaram

Graduate Student: Eshaan Bhansali

Undergraduate Team Members: Xiyao Chen, Andrew Coursin, Erik Reed, Leo Yang

We studied the dynamics of finite Blaschke products (a family of rational maps that sends the complex unit circle to itself). In particular, we explored period-2 points of those maps, focusing on whether the multipliers (defined as $m_i = (F \circ F)'(z_i), \text {where}$ $z_i$ are the period-2 points), always get perturbed when we alter the parameters that define those maps. While symbolic computation using resultants turned out to be computationally unfeasible even for $n=2$ , numerical analysis demonstrated that the multipliers are extremely sensitive to changes in the parameters. By approximating the Jacobian of the multipliers with respect to the parameters via finite differences and estimating its rank through singular value decomposition, we found that the numerical rank equals $2n$ for all cases, suggesting that the multipliers vary locally injectively with respect to the parameters.

Changes in multipliers upon perturbation in parameters.

Faculty Mentor: Chenxi Wu

Graduate Student: Yourong Zang

Undergraduate Team Members: Xiangyi Pan, Xinyi Shi, Rohan Sohoni

In this project, 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. Future directions include extending the framework to nonlinear and higher-dimensional PDEs, handling more complex constraints, and exploring RKHS-based reformulations for improved constraint enforcement.

The figure shows two side-by-side cases. In each case, the top panels plot ‘True’ vs ‘Learned’ curves for forcing at several times, and the bottom panels compare heatmaps of u (true, learned) and the error, with MSE and training settings shown

Faculty Mentor: Yukun Yue

Graduate Student: Martin Guerra

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

In this project, we studied quotients by the actions of almost-conjugate subgroups induced by the Galois correspondence of covering spaces on Cayley graphs of surfaces of genus 2 or greater. By creating a surjective homomorphism from the fundamental group to a finite group to study the differences between the 2 graphs by counting the number of simple lifts of twinless paths in the given almost conjugate subgroups. We wrote methods to check for these conditions, which we will test in the future, and generated examples of all almost-conjugate subgroups of the Symmetric groups S6 through S10. We will likely continue this project in the next semester and hope to better utilize parallelization and supercomputing clusters to expand these examples.

A:=\left\{a \in S L_3\left(\mathbb{F}_2\right): a_{2,1}=a_{3,1}=0\right\} \text { and } B:=A^{\top}\\

The figure in question depicts two finite directed graphs constructed as quotients of the Cayley graph of a finite group by a pair of almost conjugate (but not actually conjugate) subgroups. Vertices of the quotient graphs are enumerated by cosets of powers of the commutator b = [a_1, a_2], and edges are colored according to the generating set: Edges of color a_1 and a_2 are represented by solid and dashed lines, respectively. Self-loops are not depicted and can be inferred because each vertex is adjacent to exactly one incoming and outgoing edge of each color.

Faculty Mentor: Max Lahn

Graduate Student: Rachel Hanger

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

The goal of this project is to explore the subconstituent algebra of a graph and determine the irreducible T-modules of the 13 distance-regular graphs of valency three. For said graphs, we developed a workflow for determining the standard module decomposition into irreducible T-modules using orthogonality, the eigenvalues of the adjacency matrix using the action of the matrix on the irreducible T-modules, and the dimension of the Terwilliger algebra using Wedderburn’s Theorem.

Examples of distance-regular graphs with valency 3

Faculty Mentor: Paul Terwilliger

Graduate Student: Jimmy Vineyard

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

Hyperbolic geometry studies spaces of constant negative curvature. In this project, we combine geometric tilings and algebraic group actions to study hyperbolic surfaces such as the Klein quartic and the modular surface. We visualize these surfaces using fundamental domains and computer-generated tilings. We learn how different fundamental domains and discrete groups influence the tiling results. We also construct physical models using crochet, where stitch increases naturally produce negative curvature.

Faculty Mentor: Grace Work

Graduate Student: Kendra Ebke

Undergraduate Team Members: Kane Haoru Li, Pengrui Song, Emma Trask, Wanchen Zhao

Fall 2025

Expanded accordion Approximation of State Constrained Optimal Control Problems

Expanded accordion Galois images and congruences with modular forms

Expanded accordion Multipliers of closed orbits of circle maps

Expanded accordion Neural Network Parameterization for PDE-Constrained Optimization

Expanded accordion Quotients of Cayley Graphs and Curves on Surfaces

Expanded accordion The subconstituent algebra of a graph

Expanded accordion Exploring Hyperbolic Geometry Through Crochet