# Math+ 2024

Math+ 2024 ran from May 20 until July 12, 2024. The 2024 program featured 5 projects and 19 undergraduate student researchers.

### Projects for Math+ 2024

##### Sum-of-norms clustering with large numbers of datapoints

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 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