Applications for Math+ 2024 are now open!
Math+ is a full-time 8 week program for collaborative student summer research in all areas of mathematics. Math+ 2024 is open to all current undergraduate students at Duke University and North Carolina Central University. We particularly encourage women and underrepresented minorities to apply.
Math+ 2024 runs from May 20 until July 12, 2024. The application deadline is February 15, 2024. The program consists of groups of typically 3-4 undergraduate students working together on a single project. Each team will be led by a faculty mentor assisted by a graduate student. Participants will receive a $4,000 stipend and may not accept other employment or take classes during the program.
There are 5 teams planned for Math+ 2024. As part of the application, you will list the number(s) of the projects that you would like to apply for. Information about each of the projects is given below.
Projects for Math+ 2024
- Sum-of-norms clustering with large numbers of datapoints, led by Professor Alex Dunlap
Clustering is an unsupervised learning task that asks for a division of datapoints in Euclidean space into "clusters" of nearby points. In applications, the datapoints may represent objects like pictures or videos. A good clustering may separate them by subject, color scheme, or topic. The task is unsupervised, which means that the algorithm does not receive any labels from the user about the desired clusters: it finds the clusters just using the geometry of the datapoints in Euclidean space. This means that the algorithm may find surprising structure in the data that the user did not expect!
Sum-of-norms clustering is a clustering algorithm that is simple and structured enough to be amenable to rigorous mathematical analysis. It is based on minimizing a convex functional depending on the datapoints. In the limit of a large number of datapoints, we can often think of the "local density" of the datapoints as a continuous function (or measure). It turns out that the sum-of-norms clustering algorithm makes sense when run directly on this continuous function, and that this represents the limiting behavior of the algorithm when the number of datapoints is large.
The proposed project is to develop numerical and/or analytical tools for studying this algorithm on continuous data. Numerically, we will seek to develop tools to efficiently compute the clusterings that result from continuous densities. Analytically, we will study densities with symmetries to seek rigorous results on clusterings in simple cases. We hope that both approaches will lead to new conjectures and further directions.
Ideally, students will be comfortable with multivariable calculus and linear algebra. There will be opportunities to apply programming skills and real analysis as well.
- The combinatorics of Legendrian surfaces, led by Professor James Hughes
Legendrian knots or links are knotted circles in three-dimensional space that satisfy some additional 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 weaves, evoking the way that these surfaces are constructed by weaving together a collection of sheets.
This project will use the combinatorial data of colored graphs to explore the world of Legendrian weaves. Our goal is to expand the existing set of tools for showing when two Legendrian weaves are the same and then apply these tools to a specific family of surfaces.
Participants should have a working knowledge of multivariable calculus and linear algebra. Previous experience with topology is helpful but not expected.
Computational exploration of geometric flows in G2-geometry, led by Professors Ilyas Khan and Alec Payne
Geometric flows have become a widely-used tool in the study of differential geometry. In a nutshell, a geometric flow is a solution to a “heat-type” partial differential equation on a geometric structure that (by analogy to heat) evens it out and moves it towards greater regularity and homogeneity. In short, the flow tries to transform the geometry into an “ideal” version of itself. Some problems famously resolved by flow methods are the Poincaré conjecture and the Differentiable Sphere theorem.
In this project we will investigate G2-Laplacian flow, a geometric flow introduced by Duke Professor Robert Bryant which attempts to find special geometric structures in the context of G2-geometry. G2-geometry is an exciting modern field of differential geometry that lies at the intersection of geometry, topology, representation theory, and physics. It concerns 7-dimensional manifolds with special geometric features inherited from the exceptional Lie group G2.
Specifically, we will use numerical and computational methods to study the behavior of G2-Laplacian flow, especially the behavior of solutions as they become singular. We hope to numerically model non-trivial finite-time singularities of this flow; finding such singularities is an outstanding open problem in the field.
Ideal participants would have a background in partial differential equations (ideally Math 453/557, or some experience with heat/wave/Laplace equations). Previous experience in programming or with differential geometry is desirable but not strictly necessary.
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 led to the development of software (known as “proof assistants”, or “interactive theorem provers”) to formalize mathematics. In this project we will work with the theorem prover Lean. Project members will learn how to program in Lean (about 2 weeks), and then work on a project of their choice. Options for the latter include: (1) formalizing and submitting new theorems to the Lean Mathematics Library (which is under development), and (2) expanding and polishing the Linear Algebra Game developed by the 2023 team; or developing a similar new game.
No programming experience is necessary. Participants should have working knowledge of Linear Algebra and Calculus 1.
The Howard-Curry Correspondence: https://www.quantamagazine.org/the-deep-link-equating-math-proofs-and-computer-programs-20231011/
The Lean Community: https://leanprover-community.github.io/
Modeling interactions between fluids and elastic solids, led by Professor Aric Wheeler
Understanding the interactions between fluids and structures has applications to a wide class of problems covering a large range of topics including but not limited to the aerodynamics of planes to geophysical applications related to iceberg calving and to biological applications involving e.g. blood flow in the arteries and veins in your body. In this project, we will aim to study a toy model version of the iceberg calving problem in which incoming water waves bounce off an elastic wall and cause the wall to vibrate.
We aim to study this phenomenon with a mixture of analytical approaches and coding with a particular goal of understanding how the waves propagate into the elastic material. Ideal participants would have a background in ordinary differential equations, multivariable calculus, and basic physics. Previous experience in programming is desirable, but not strictly necessary.