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, led by Professor James Hughes
Legendrian knots are knotted circles in three-dimensional space that satisfy some additional geometric constraints. We study Legendrian knots by understanding their projections, drawing them as curves in the plane that obey certain area or tangency requirements. We can define and study Legendrian surfaces analogously, albeit at the cost of considering more complicated projections. Some progress has been made in the last decade towards better understanding Legendrian surfaces by combinatorially representing them as colored graphs that encode the singularities of their projections. The family of Legendrian surfaces obtained from these graphs are known as Legendrian weaves, evoking the way that these surfaces are constructed by weaving together a collection of sheets.
Certain symmetries of Legendrian weaves contribute to our understanding of algebraic invariants associated to Legendrian knots and provide an important family of examples for related geometric constructions. However, there is currently no known algorithm for constructing a symmetric Legendrian weave, even in cases where we suspect that one might exist. This project will leverage known combinatorial constructions of Legendrian weaves to create such an algorithm. Our ultimate goal will be to use this algorithm to produce new families of symmetric Legendrian weaves with interesting algebraic invariants.
Participants should have a working knowledge of multivariable calculus and linear algebra. Previous encounters with combinatorics or topology are helpful but not expected.
- Encounter-based biological dynamics driven by stochastic processes, led by Professor Jacob Madrid
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.
Professor McPhail-Snyder's project is being postponed to a future year. Thanks for your interest and understanding!
Skein relations for the ADO invariants, led by Professor Calvin McPhail-Snyder
One way to mathematically study a knot (and also related objects in low-dimensional topology) is to use a knot invariant, a quantity computed from the knot that does not depend on how the knot is presented. Ideally these are objects (like numbers or polynomials) that are simpler to understand than the original knot but still reveal its properties. One class of knot invariants are quantum invariants named for their origin in quantum field theory. Despite this they admit an elementary definition in terms of linear algebra. Sometimes this definition can be made even simpler via a skein relation that describes how the invariant changes when we make local changes to the knot, and this can reduce the computation of the invariant entirely to pictures. The goal of this project is to find skein relations for a family of quantum knot invariants called the Akutsu-Deguchi-Ohtsuki (or ADO) invariants.
Students will need a strong background in linear algebra for this project. Some knowledge of topology (Math 411) and/or abstract algebra (Math 401/501) would be beneficial as well.Formalization of Mathematics, led by Professor Colleen Robles
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.
No programming experience is necessary. Participants should have taken a proof-based mathematics course (e.g. Math 221). This project is a good fit for anyone who enjoys programming (or would like to learn), and digging into the mechanics of mathematical proofs.
If you'd like to apply to this project, please include the following specific information (in place of the general "short explanation of why you have chosen the project and how you feel you could contribute to it" that we ask for the other Math+ projects). Tell us why you are interested in the project, and what you hope to get out of it. Which proof-based math courses did you take this year? Also please let us know if you have experience with proof assistants (e.g. Coq, Isabelle or Lean), and/or Git or GitHub (under which operating system? e.g. MacOS, Linux, et cetera); we hope to recruit one or two team members with this background.Al Powered discovery of counterexamples in combinatorics, led by Professor Fan Wei
Please note: Professor Wei's project is a joint project with Data+. It will begin at the same time as the other Math+ projects but will end two weeks later, on July 25; the stipend for this project is $\$5000$ (or $\$1000$ plus room and board for students who choose the on-campus option), rather than $\$4000$. Applications for Professor Wei's project will be handled through Math+ and not through Data+. Students interested in applying for this project should apply through the Math+ application, alongside any other Math+ projects that you may be interested in.
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, led by Professor Jiajia Yu
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.