Math+ 2025 runs from May 20 until July 11, 2025. The 2025 program featured 5 projects and 21 undergraduate student researchers.
This page will be updated soon with project reports.
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
In the last 20 years there has been a great deal of progress formalizing mathematics, thanks to the Howard-Curry correspondence between types and programs (in computer science) and propositions and proofs (in mathematics). This has lead to the development of software (known as “proof assistants”, or “interactive theorem provers”) to formalize mathematics. In this project we will work with the proof assistant Lean. Project members will learn how to program in Lean (about 2 weeks), and then work on a project of their choice (six weeks). Options for the latter include: (1) formalizing and submitting new theorems to the Lean Mathematics Library (which is underdevelopment); (2) porting to Lean 4, and expanding and polishing the Linear Algebra Game developed by the 2023 Math+ team, or developing a similar new game; and (3) continuing the development of the Algebra in Lean tutorial workbook developed by the 2024 Math+ team. For more information about the general project, please see https://canvas.duke.edu/courses/32612.
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
This is an innovative project that explores the intersection of artificial intelligence and mathematics. This initiative aims to leverage AI's capabilities in pattern recognition and exhaustive search to tackle complex problems in discrete mathematics, such as finding counterexamples to open conjectures. By framing these mathematical challenges as computational problems, students will utilize machine learning models, including reinforcement learning and graph neural networks, to efficiently explore vast mathematical spaces. This approach not only accelerates mathematical discovery by potentially uncovering new patterns and insights but also complements human intuition and creativity, driving forward the resolution of long-standing open questions in fields like graph theory and game theory.
This project offers a unique opportunity for students to engage with cutting-edge developments in AI and its applications to abstract mathematical domains. Participants will collaboratively develop a framework that mathematicians can use to explore conjectures and gain insights, even when formal proofs remain elusive. By working on this project, students will contribute to the development of AI techniques while gaining valuable experience in both theoretical and applied aspects of these fields.
Students interested in participating in this project should have a strong foundation in mathematics, particularly in discrete mathematics and graph theory. Familiarity with machine learning concepts and experience with programming languages such as Python are essential. Knowledge of deep learning frameworks like PyTorch or TensorFlow will be beneficial, as the project involves implementing machine learning models to test various conjectures. Additionally, students should be comfortable with exploring new AI techniques and willing to engage in collaborative research efforts that bridge multiple disciplines. Enthusiasm for mathematical discovery and problem-solving using computational methods will be key to successfully contributing to this groundbreaking initiative.
Translation-invariant optimal transport distance
Project leader: Professor Jiajia Yu
Project manager: Shrikant Chand
Team members: Peilin He, Zakk Heile, Jayson Tran, Alice Wang
Measuring differences between data points is fundamental to many machine learning tasks, including classification, clustering, registration, and dimensionality reduction. The choice of measurement can significantly influence the outcomes of these tasks. While the Euclidean distance is the most widely used metric, it often fails to capture the underlying geometry of data in applications where the data points represent complex objects, such as images or networks, or where the Euclidean distance is ill-defined.
In recent years, the rapid growth of machine learning has positioned optimal transport (OT) as a powerful tool in diverse areas such as image classification, domain adaptation, generative modeling, and clustering. OT addresses the problem of minimizing the cost of moving a distribution of mass (e.g., a pile of earth) to match another distribution (e.g., a hole). The minimal cost, known as the Wasserstein distance, provides a meaningful metric between probability distributions and imposes a rich geometric structure on the space of these distributions, reflecting the geometry of the underlying "ground" space on which they are defined.
Despite its many advantages, the optimal transport distance has a notable limitation: it is not invariant under rigid transformations. For instance, two images depicting the same object in different orientations may be considered far apart under OT, despite their similarity. This limitation motivates the need for alternative metrics. One such approach is the Gromov-Wasserstein distance, which generalizes OT to compare distributions defined on different metric spaces. However, this metric presents significant computational challenges.
The proposed project aims to address these challenges by developing efficient computational tools for the Gromov-Wasserstein distance or designing novel metrics that are invariant under rigid transformations while retaining the advantages of optimal transport.
Students are expected to be familiar with multivariate calculus, linear algebra and probability. While prior exposure to differential geometry, ODE, PDE, or numerical analysis, numerical linear algebra would be beneficial, it is not required. Programming experience with MATLAB or Python is also helpful but not essential.