Applications for Math+ 2026 are now open!
Click here to apply to Math+ 2026 on MathPrograms
Math+ is a full-time 8 week program for collaborative student summer research in all areas of mathematics. Math+ 2026 is open to all current undergraduate students at Duke University and North Carolina Central University. We particularly encourage women and underrepresented minorities to apply.
Math+ 2026 runs from May 18 until July 10, 2026. The application deadline is February 15, 2026. The program consists of groups of typically 3-4 undergraduate students working together on a single project. Each team will be led by a faculty mentor assisted by a graduate student. Participants will receive a $4000 stipend; we expect to offer an option for on-campus housing, and students who choose this option will instead receive an $800 stipend in addition to room and board. Participants may not accept other employment or take classes during the program.
There are 5 teams planned for Math+ 2026. As part of the application, you will list the number(s) of the projects that you would like to apply for. Information about each of the projects is given below.
If you have any questions about Math+ 2026, please email the program co-directors, Professors Heekyoung Hahn (heekyoung.hahn@duke.edu) and Lenny Ng (lenhard.ng@duke.edu).
Projects for Math+ 2026
Large deviations of means on stratified metric spaces
Project leader: Professor Nicholas Cook
The sample mean provides a natural notion of the center of a set of N points in a vector space. A suitable generalization of the sample mean for general metric spaces (that may lack operations of addition and/or scalar multiplication) is provided by the Fréchet mean, computed by minimizing the sum of squared distances to the N points. Recently there has been interest in the behavior of Fréchet means for random sets of points in metric spaces, due to applications in areas such as medical imaging and phylogenetics. Generalizations of the law of large numbers and central limit theorem have revealed curious phenomena not seen in the vector space setting, such as a tendency in some cases for the Fréchet mean to remain fixed at a certain point with very high probability, a phenomenon called “stickiness”.
This project will investigate the behavior of Fréchet means at the level of large deviation principles, which give accurate estimates for probabilities of rare events. A general aim is to understand the stickiness phenomenon and its link to generalized notions of negative curvature for metric spaces.
The project provides a concrete and elementary setting in which students will gain exposure to concepts such as abstract metric spaces, curvature, large deviations, entropy, and optimization.
Recommended background: a course in probability (230/340) and multivariable calculus. Much of the math will boil down to multivariate optimization. Background in real analysis (431/531) will likely be beneficial but is not assumed.
Stochastic Burgers equations on compact manifolds
Project leader: Professor Alex Dunlap
This project will explore questions at the intersection of dynamics, probability, PDE, and geometry. We will consider the Burgers equation on a compact manifold (e.g. a torus or a Klein bottle), forced by the derivative of a Poisson point process. Our goal will be to understand the space of statistically-stationary solutions to this equation. The form of the forcing means that this problem can be posed as an optimization problem over paths minimizing a certain energy functional. Students will investigate the geometry of solutions to this optimization problem, and in particular how the solutions interact with the topology of the manifold. Along the way, we will draw lots of pictures, both by hand and with the computer.
Students should have a strong facility with multivariable calculus. Exposure to at least one of basic real analysis or optimization would be helpful but is not strictly required. Students who know something about manifolds or elementary topology would be able to use this knowledge, but the questions are already very interesting for very simple manifolds and so it is not even necessary to know what a manifold is. Similarly, it is not necessary to have studied the Burgers equation or to have even taken a PDE class before (although again, we will be able to use that knowledge if students have it).
Diffusion maps to understand redistricting (and graph partitions)
Project leaders: Professors Gregory Herschlag and Biji Wong
Eigenvalues of graph diffusion operators are useful for capturing geometric and structural properties of data, such as intrinsic dimensionality and clusters. Roughly, the larger eigenvalues capture the essential structure of the data, while the smaller eigenvalues correspond to noise. Recently, diffusion operators, and resulting spectral analyses, have been extended to act both within and across bones as part of the bone segmentation effort led by Ingrid Daubechies. Specifically, Tingran Gao (along with others) developed fibered diffusion methods to register evolutions/differences of substructures in bones as well as refine sub-regions via spectral analyses on bones.
We have a novel application for these fibered diffusive operators in the context of redistricting and graph partitioning: Currently, not much is known about the mutual information of generating political districts. For example, does making certain district choices in one part of the state have any surprising downstream influence in another (potentially distant) part of the state? Our goal is to understand the space of redistricting plans (more generically graph partitions) under various probability distributions. Potential questions we can answer are: Can we identify regions that allow for continuous deformation of districts? Alternatively, are there natural “breaks” in the space that allow for redistricting problems to be treated as a collection of sub-regions that we can district?
In terms of prerequisites, there will be a strong coding component to this project so some background with algorithms and/or software development is encouraged. We will be working with (discrete) diffusion operators so it will have been helpful to have had some introduction to either the heat equation (e.g. a PDE course), a discretization of the heat equation (e.g. a numerical PDE course), or random walks on graphs (e.g. some course that included spectral graph theory and graph Laplacians).
Computational algebraic number theory with geometric applications
Project leader: Professors Farid HosseiniJafari and Colleen Robles
This project explores a beautiful interaction between geometry and number theory. The geometric motivation of the project is to understand extendability properties of certain line bundles that are invariant (or homogeneous) under the action of an arithmetic group. The general question is somewhat subtle and mysterious, and the goal of this project is to develop a robust collection of illuminating examples. Much of the technical work of the project is to analyze the units in the ring of integers of an algebraic number field. Students will gain hands-on experience with key concepts in algebraic number theory and computational tools such as Sagemath.
This is a good project for any student with a strong background in linear algebra (Math 221), and an interest in algebraic number theory. Curious students are welcome to contact Prof. HosseiniJafari and/or Prof. Robles to discuss the project.
Comparing distances between hierarchical clustering structures
Project leader: Professor Ling Zhou
Ultrametric spaces are metric spaces that can be represented by rooted trees, where distances reflect how clusters merge. Finite ultrametric spaces arise in hierarchical clustering, phylogenetic analysis, and other applications where data are organized into nested groups. In practice, one often wants to compare different ultrametric structures arising from data sets, which requires defining and understanding distances between ultrametric spaces.
In this project, we will study two ways of measuring distance between hierarchical structures. One approach comes from metric geometry and compares structures directly through their metric information, while the second approach comes from topology and summarizes hierarchical structure using barcode representations that record merge events across scales. Although these distances are related by general inequalities, they capture different structural features and need not coincide. Students will explore explicit families of ultrametric spaces, work out examples and counterexamples, and investigate conditions under which the two notions of distance agree or differ. A background in linear algebra (Math 221) is sufficient; programming experience is helpful but not required.