Math+ 2024 ran from May 20 until July 12, 2024. The 2024 program featured 5 projects and 19 undergraduate student researchers.
See here for a Duke news story about one of these projects: Through Math+, Undergraduate Math Students Push the Frontier of Geometry.
Projects for Math+ 2024
Sum-of-norms clustering with large numbers of datapoints
Project leader: Professor Alex Dunlap
Project manager: Haotian Gu
Team members: Ben Greene, Kaden McLaughlin, Shuhuai Yu
Clustering, the task of dividing a set of points into different groups, has broad applications in machine learning, pattern recognition, and statistics. A well-known approach, k-means clustering, involves clustering points into k clusters and is a non-convex optimization problem. Since k-means clustering is a non-convex problem, finding the global minimum solution can be challenging. In this project, we investigated an alternative clustering scheme known as Sum-of-Norms (SON), which formulates the problem as a convex optimization problem. One key advantage of SON is that it does not require specifying the number of clusters in advance. We have obtained several interesting results regarding the application of SON in both discrete and continuous configurations. Read more
The combinatorics of Legendrian surfaces
Project leader: Professor James Hughes
Project manager: Zijun Li
Team members: Holly Keegan, Helen Pertsemlidis, June Wood
Take a shoelace, knot it with itself, and glue the ends of the shoelace together. The object that you have created is a topological knot. In our project, we consider two knots that are isotopic to each other, which means that you can wiggle one knot around without breaking it to get it to look like the other. Tracing the process of wiggling the knot produces a cylinder Φ that can be thought of as living in five-dimensional space. Our two knots are the boundaries of two surfaces, Σ1 and Σ2, that also live in five-dimensional space. We can glue Φ to Σ2 to create a new surface Σ2+Φ. The goal of our project is to show that Σ2+Φ is equivalent to Σ1. Read more
Computational exploration of geometric flows in G2-geometry
Project leaders: Professor Ilyas Kahn and Professor Alec Payne
Project manager: George Daccache
Team members: Matthew Chen, Jack Qian, Paul Rosu
Our project explores a differential equation related to a special type of seven-dimensional space known as a $G_2$ manifold, which plays a significant role in both Riemannian geometry and theoretical physics. This equation describes a mathematical process called the Laplacian flow, which transforms a space that is "close" to a $G_2$ manifold to be "more like" an actual $G_2$ manifold. Imagine the manifold as an object made of clay that gradually changes shape when gently tugged. The Laplacian flow is like the tugging, guiding the transformation of the manifold into a desired form. During this transformation, we sometimes encounter singularities, which are points where the process stops making sense, like a sudden crack in the clay. These singularities can reveal important details about the behavior of the flow and understanding them is necessary for any geometric or topological applications of the flow. In our research, we have successfully numerically simulated new flows that form singularities in finite time, and studied how different changes in their initial conditions can lead to different behavior. These results help us better understand the fundamental nature of this complicated process. Read more
Formalization of mathematics
Project leader: Professor Colleen Robles
Project managers: Stavan Jain, Ricardo Prado Cunha, Anoushka Sinha
Team members: Will Harris, Clara Henne, William Ho, Adam Kern, Dominic King, Arim Lim, Justin Morrill
A mathematical proof is a logical explanation that establishes why a mathematical idea is true. These proofs are traditionally done using natural language and intuition, meaning they rely on the reader's understanding and cannot be easily checked by a computer. However, formal proofs use precise rules and syntax, allowing every step to be verified by a computer, ensuring they are more rigorous. To create and verify these formal proofs, mathematicians use proof assistants, special software tools designed to help develop and check mathematical proofs. One such tool is Lean 4, an open-source software that allows users to write formal arguments in a special programming language. Lean is particularly powerful because it not only checks the proofs but also helps guide users in constructing them. In this project, we used Lean to create an interactive problem set for students learning abstract algebra. Over six weeks, we completed four sections of this teaching aid, aiming to provide more insight into mathematical proofs and Lean. We hope future teams will continue building on our work, expanding the problem set further. Read more
Interactions between fluids and elastic solids
Project leader: Professor Aric Wheeler
Project manager: Kevin Dembski
Team members: Zachary Robers, Michael Thomas, Olly Yang
We aim to study a simpler version of the iceberg calving problem. In our model, we simulate waves approaching an iceberg using the shallow water equations and deploy the linear wave equation to model the iceberg's compression. This setup allows us to study how waves interact with the iceberg through a series of equations and conditions designed to preserve energy and mass. To analyze this system, we use two main approaches. First, we make a linear approximation of the system, which helps us understand the conditions under which the system is stable. Stability here means that small perturbations to the system don’t evolve into large changes which could lead to behavior such as iceberg calving. Second, we program a simulation to study the problem in more detail. By using classical numerical methods, we can track the fluid and solid velocities, the height of the waves, the compression of the iceberg, and the position of the boundary between the water and the iceberg. Together, these approaches give us valuable insights into how the system evolves under different conditions, helping us better understand the processes involved in fluid-structure interaction. While this work remains a simplified means to represent and analyze iceberg calving, extending this work to incorporate more realistic physical conditions could lead to insights on how and why icebergs calve. Read more