Math+ 2026 ran from May 18 until July 10, 2026. The 2026 program featured 4 projects and 19 student researchers.
Projects for Math+ 2026
Large deviations of means on stratified metric spaces
Project leader: Professor Nicholas Cook
Project manager: Sofia Poinelli
Team members: Akshat Basannavar, Andy Jiang, Nhooja Roy
Imagine averaging a large number of data points. As you collect more data,the mean settles down near some fixed “true” value. But occasionally, it strays far from the target. Large deviation theory quantifies just how unlikely these deviations are. It assigns a measure of rarity to each possible location of the average, describing how quickly the probability of ending up there decreases as the sample size grows.
We study this question on spaces with a twist: instead of an ordinary flat plane or line, our data lives on shapes stitched together out of simpler pieces, like flat sheets joined along their edges. In these spaces, there is no natural way to add points together and divide by their number, so the usual notion of an average no longer applies. Instead, we use the Fréchet mean, the point that is, on average, closest to all the data.
These stitched-together spaces have corners, junctions, and other singular features where the geometry changes abruptly. These singularities make the Fréchet mean much harder to analyze than in ordinary flat spaces, where the theory is well understood. Our goal is to understand how the geometry of the space influences the rarity of large deviations of the Fréchet mean. READ More
Stochastic Burgers equations on compact manifolds
Project leader: Professor Alexander Dunlap
Project manager: David Koralov
Team members: Owen Banks, Zimo Li, Jack Qian, Yifei Wang
Stochastic PDEs describe relationships between partial derivatives of functions and random noise. In this project, we study a stochastic version of Burgers equation over a closed line segment, with constant-in-time Dirichlet boundary conditions and bulk random forcing given by a Poisson point process. This equation is a relatively simple case useful for understanding a more general family of stochastic PDEs, called the KPZ equations, which are highly relevant to theoretical physics and mathematical modeling. Using PDE theory, we can reduce the problem of solving the stochastic Burgers equation to the problem of finding the path that connects two points that minimizes a certain “action” function. The action function rewards paths for passing through Poisson forcing points and for staying on boundaries, but discourages paths from traveling through space. Our project studies this optimization-style problem. In particular, we characterize how the solution to the equation evolves over time, and prove several theorems about how it changes as the boundary parameters vary. READ More
Diffusion maps to understand redistricting (and graph partitions)
Project leader: Professor Gregory Herschlag and Professor Biji Wong
Project manager: Spencer Whitehead
Team members: Jenny Chan, Peakay Clifford, Eileen Santana, Felix Sesin, Leo Yang
Past attempts at quantifying redistricting have often relied on usual statistical methods; for instance, one may generate a large sample, or ensemble, of districting plans on the map of precincts in a US state (such as CT) and compute the number of congressional districts won by republicans and democrats according to real election data. Such methods identify outlier districting plans whose partisan outcomes are statistically unlikely, providing evidence of possible gerrymandering. This is how gerrymandering may be detected at the global level, that is, across a districting plan. In this project, we seek to develop tools which enable this sort of statistical analysis and detection of gerrymandering on a more local level – at the level of individual districts in a districting plan. The essential idea is the following: to determine whether a district votes atypically, we must first learn how districts similar to the enacted district vote, and this requires learning exactly which districts are similar to the enacted district in the first place. READ More
Computational algebraic number theory with geometric applications
Project leader: Professor Farid HosseiniJafari and Professor Colleen Robles
Project manager: Jin Lee
Team members: Aryan Agarwal, William Anderson, George Hall, Michael Klausner, Lauren Schwartz, Austin Setzler, Benjamin Wang
Manifolds are generalizations of Euclidean space. Some examples (like a sphere) are quite familiar, though others are more exotic. There are meaningful ways to attach additional geometric structures called line bundles to the manifold. For example, one could attach a line through the origin to each point of a sphere. Unlike the sphere, the manifolds naturally occurring from Hodge Theory do not necessarily satisfy a property known as compactness, meaning they may wander off the manifold or in an infinite direction. It is possible to compactify these spaces by gluing on ``boundary components at infinity''. However, the line bundle need not admit a well-defined extension to the compactification. Degenerate behavior may exist around these boundary components, or ``cusps'', which twist the line bundles. READ More