Project descriptions and requirements for Fall 2025 can be found below. Applications for undergraduate team members and graduate student mentors will open on August 15 and close at 11:59 pm on August 24. All applicants will be informed of their application status by September 2.
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Approximation of State Constrained Optimal Control Problems
PDEs help us understand physical processes. More than simply understanding systems, we often want to control systems. Due to this desire, optimal control problems are frequently studied. Often these problems involve minimizing an average difference between a desired state and an actual state. However, sometimes state constraints are also added which keep the state between given bounds across the entire domain. Our goal will be to study how solutions to state constrained optimal control problems are approximated and apply what we learn to a parabolic PDE that describes how fluid flows through tissues.
Project Level: Advanced
Minimum Undergraduate Requirements: Linear algebra, analysis, experience in Matlab, PDEs.
Faculty Mentor: Sarah Strikwerda
Graduate Student: Accepting applications
Undergraduate Team Members: Accepting applications
Galois images and congruences with modular forms
This project will study Galois representations associated to certain genus 3 curves in the reducible case. The primary goal is to uncover congruences of sub-representations with Bianchi modular forms, thus producing evidence for Serre’s conjecture over imaginary quadratic fields. Another goal is to find the images of the semi-simplified Galois representations.
Project Level: Advanced
Minimum Undergraduate Requirements: Algebra 1, Algebra 2, familiarity with computer algebra packages like Magma and Sage.
Faculty Mentor: Shiva Chidambaram
Graduate Student: Ivan Aidun
Undergraduate Team Members: Accepting applications
Multipliers of closed orbits of circle maps
Let f and g be two rational maps on the complex plane that sends the unit circle to itself. If x is a point on the unit circle, and f^n(x)=x, we say it is a periodic point of period n, and the derivative of f^n at x is called its multiplier. Let f and g be two such maps, and their multipliers of periodic points of period up to some integer d are identical, are f and g conjugate by a rational map?
Project Level: Intermediate
Minimum Undergraduate Requirements: Linear algebra and knowledge of a programming language.
Faculty Mentor: Chenxi Wu
Graduate Student: Accepting applications
Undergraduate Team Members: Accepting applications
Neural Network Parameterization for PDE-Constrained Optimization
In this semester-long project, students will enforce hard constraints on control functions in PDE-constrained optimization using neural networks. By choosing suitable activation functions in the network’s output layer, we could convert a traditionally constrained problem into an unconstrained one over the network parameters. Students will implement various neural architectures to solve representative PDE control problems and benchmark them against classical adjoint-based methods.
Throughout the term, students will design and train networks that guarantee feasibility by construction, write codes to solve specific partial differential equations, analyze training dynamics and convergence, and evaluate accuracy–computational cost trade-offs. Deliverables include shareable codes demonstrating both neural and adjoint-based solutions for a specific selected problem, a written report comparing the approaches, and a final presentation outlining open challenges and potential extensions.
Project Level: Advanced
Minimum Undergraduate Requirements: Math 321, Math 322, Math 535, or equivalent courses providing foundational knowledge in partial differential equations and data science, Python.
Faculty Mentor: Yukun Yue
Graduate Student: Accepting applications
Undergraduate Team Members: Accepting applications
Quotients of Cayley Graphs and Curves on Surfaces
Fix a covering space of a compact 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 in terms of the combinatorics of paths on certain finite colored graphs.
The primary goals of this project are to (a) gain a deeper understanding of which curves on the base surface have simple (non-self-intersecting) lifts to the cover via the aforementioned representation as paths in finite colored graphs; and (b) use this understanding to determine which non-simple curves of a certain form have simple lifts. These goals ultimately derive from a desire to use these criteria to distinguish length isospectral covers arising from Sunada’s construction via their simple length spectra.
Project Level: Advanced
Minimum Undergraduate Requirements: Interested participants should have seen some group theory (for example in Math 541) and some point-set topology (for example in Math 551). Exposure to the geometry and/or topology of surfaces, graph theory, or other advanced topics is not required.
Faculty Mentor: Max Lahn
Graduate Student: Accepting applications
Undergraduate Team Members: Accepting applications
The subconstituent algebra of a graph
This project is in the area of Algebraic Combinatorics. We will consider a situation in graph theory that involves combinatorics on one side and algebra on the other.
The project has to do with the subconstituent algebra of a graph. This topic might be unfamiliar, so let me briefly describe it. Start with a finite, undirected, connected graph G with vertex set X. The (0,1)-adjacency matrix A of G is a familiar object of study. Indeed the eigenvalues of A tell us something about the combinatorial structure of G. The matrix A generates a commutative subalgebra M of Mat_X(C) called the adjacency algebra of G. Next, we enlarge M as follows. Recall the path-length distance function partial. Fix a vertex x of G. For a nonnegative integer i the ith subconstituent of G with respect to x consists of the vertices of G that are at distance i from x. To this subconstituent we associate a diagonal matrix E*_i in Mat_X(C) that has (y,y)-entry 1 if partial(x,y)=i, and (y,y)-entry 0 if partial(x,y)≠i. Intuitively, E*_i is the projection onto the ith subconstituent of G with respect to x. Call E*_i the ith dual primitive idempotent of G with respect to x. The nonzero dual primitive idempotents form a basis for a commutative subalgebra M*=M*(x) of Mat_X(C), called the dual adjacency algebra of G with respect to x.
The subconstituent algebra T is the subalgebra of Mat_X(C) generated by M and M*. By construction T is finite-dimensional and noncommutative. For a given graph G, the general goal is to work out the irreducible T-modules and relate the results to the combinatorial structure of G.
The specific project is the following: there is a famous family of nicely behaved graphs said to be distance-transitive. The trivalent distance-transitive graphs are classified, and there are 12 examples. For each example, we seek to work out the irreducible T-modules. Once this result is accomplished, the group members write up their findings and get a nice publication with multiple coauthors.
The general prerequisites for this project are (i) a good understanding of Math 340 linear algebra; (ii) enthusiasm for mathematical exploration. Experience with Math 475 Combinatorics and Math 541 Algebra is helpful but not required. If the above description of the algebras T, M, M* is not meaningful to you, then that is not a problem; prior experience is not expected and we will go over everything from first principles.
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Bibliography
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On trivalent graphs
Biggs, N. L.; Smith, D. H.
Bull. London Math. Soc. 3 (1971), 155–158.
Bipartite distance-regular graphs of valency three
Ito, Tatsuro
Linear Algebra Appl. 46 (1982), 195–213.
The Terwilliger algebra of the hypercube
Go, Junie T.
European J. Combin. 23 (2002), no. 4, 399–429.
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Project Level: Advanced
Minimum Undergraduate Requirements: Linear Algebra (320 or 340 or 341 etc) is required. Combinatorics Math 475 and Algebra Math 541 is helpful but not required.
Faculty Mentor: Paul Terwilliger
Graduate Student: Jimmy Vineyard
Undergraduate Team Members: Accepting applications
Exploring Hyperbolic Geometry Through Crochet
One of the foundations of Euclidean geometry is the idea that given a line and a point not on the line there exists exactly one line passing through the given point and parallel to the given line. Hyperbolic geometry is an example of a non-Euclidean geometry where this property of parallel lines does not hold. In this exploratory project we will examine the basics of hyperbolic geometry through constructing physical models of hyperbolic space using crochet. Students will be encouraged to follow their interests and pursue questions to deepen their understanding and delve into various applications and aspects of hyperbolic geometry as the semester progresses.
Project Level: Beginner
Minimum Undergraduate Requirements: A basic understanding of crochet or an eagerness to learn will be helpful.
Faculty Mentor: Grace Work
Graduate Student: Accepting applications
Undergraduate Team Members: Accepting applications