Math+ 2025 runs from May 20 until July 11, 2025. The 2025 program featured 5 projects and 21 undergraduate student researchers.
Projects for Math+ 2025
Symmetries of Legendrian surfaces
Project leader: Professor James Hughes
Project manager: Jason Ma
Team members: Vincent Chen, Patton Galloway, Luciana Wei
If you take several strands, cross adjacent strands arbitrarily, we get what’s called a braid. Joining the ends of the strands creates a knot or a link, represented as a braid closure. Then, we can use several operations on them to create a surface whose cross sections are braids, called weaves. We can encode the crossing data combinatorially simply by associating them with colors, creating a colored graph. Our project is focused on when these graphs exhibit certain rotational symmetry. To accomplish this, we employ a combinatorial strategy that focuses on plabic tilings. Generally speaking, plabic tilings are tilings of n-gons generated by creating cliques from certain kinds of sets that are unchanged under addition. These tilings in turn create rotationally symmetric weaves through a process called T-shift, detailed by Casals, Le, Sherman-Bennett, and Weng. The end goal of our project is to create an algorithm that can encode this combinatorial data, given specific numeric conditions. READ More
Encounter-based biological dynamics driven by stochastic processes
Project leader : Professor Jacob Madrid
Project manager: Joel Greenfield
Team members: Kanishk Agarwal, Jinmu Liang, Arina Pinaeva, Marvin Tai
Many biological processes are characterized by the interaction between a randomly evolving process within a defined state space and a localized target set. Such targets may represent renewable or nonrenewable natural resources on a global scale, environmental resources critical for survival on a local scale, or chemically reactive surfaces on a microscopic scale. Recent studies have investigated these encounter-based phenomena by analyzing the joint distribution of the search process's position and the number of encounters with the target set. This framework has predominantly been applied to random walks on discrete state spaces and Brownian motion in continuous state spaces.
This project aims to build upon this foundational work by introducing new processes that are dynamically driven by encounters with the target set. Specifically, we seek to define processes where the interaction with the target modifies the evolution of the search process itself. Furthermore, we aim to optimize the underlying search process to achieve specific desired outcomes, such as maximizing the efficiency of resource collection or minimizing the time required for a given reaction.
Potential students should have experience with multivariable calculus, linear algebra, and basic probability theory. Other relevant subjects include real analysis, differential equations, and stochastic processes; however, a background in these subjects is not required.
Formalization of Mathematics
Project leader: Professor Colleen Robles
Project manager: Daniel Zhou
Team members: Huiyu Chen, Adam Kern, Justin Morrill, Letian Yang
The Linear Algebra Game is an innovative educational tool that teaches the Lean 4 proof assistant through the familiar context of linear algebra concepts. Rather than simply learning abstract mathematical theorems, students engage with an interactive game where they construct formal proofs step-by-step, building both their understanding of rigorous mathematical reasoning and proficiency in Lean 4's proof language. The game transforms traditional linear algebra topics—from vector spaces to linear transformations—into engaging puzzles that require students to think precisely about mathematical logic while mastering the syntax and techniques of modern theorem proving software. READ More
Al powered discovery of counterexamples in combinatorics
Project leader: Professor Fan Wei
Project manager: Raymond Sun
Team members: Yasir Alhasaniyyah, Iurii Beliaev, Thomas Kidane, Kate Newbold, Robert Sun, Kevin Wang
Our research leverages cutting-edge machine learning and computational search techniques to tackle long-standing open problems in mathematics. By reframing abstract conjectures as optimization or search tasks, we use AI to explore vast possibility spaces and hunt for elusive counterexamples—specific instances that would disprove a long-held belief. READ More
Translation-invariant optimal transport distance
Project leader: Professor Jiajia Yu
Project manager: Shrikant Chand
Team members: Peilin He, Zakk Heile, Jayson Tran, Alice Wang
Many modern datasets represent the same object in different orientations, making direct comparison challenging. For example, two adjacent slices of tissue from the same region can appear rotated or reflected relative to each other after preparation. Standard tools for comparing data distributions, like the Wasserstein distance from optimal transport, are popular because they account for geometric structure. However, they incorrectly treat rigid motions like rotations, reflections, and translations as meaningful differences. Other methods, such as Gromov-Wasserstein, are invariant to such motions but scale poorly with sample size and dimension. Our project developed Rigid-Invariant Sliced Wasserstein via Independent Embeddings (RISWIE), a new pseudometric that efficiently compares high-dimensional probability measures while ignoring rigid misalignments. READ More