Here are a couple of possible projects to give you an idea of the scope of a project.
Irrationality and Transcendence Required Background: Calculus 1 and 2 Book: Calculus, by Michael Spivak. Prospectus: "We will be investigating the ideas of transcendence and irrationality of real numbers. We will begin by studying continued fractions, building up tools to prove the irrationality of numbers like e and pi. After further analytical preparation, we'll turn our attention to our main goal, Liouville's theorem, which provides an explicit transcendental number. Time permitting, we'll try to show that e is transcendental as well."
Linear Dimension Reduction Required Background: Linear Algebra Suggested Background: Intro Analysis 1 Book: Machine Learning by Tom Mitchell. Prospectus: "After a brief introduction to some foundational ideas in linear algebra, we will begin to study the mathematical foundations behind PCA (principle component analysis) and other types of linear dimension reduction. We will then shift gears slightly and develop the ideas behind SVM (support vector machines) and other foundational ideas in classification. The goal of the project is to begin working with a real world data set (facial data set, sensors, etc.) and try to do meaningful dimension reduction and classification on the data."
Differential Geometry Background: (Project suggested by student) Prospectus: We want to learn some Riemannian geometry -- our plan is to work through Chapters 5 and 6 of Callahan's Geometry of Spacetime. Chapter 5 is a computation-and-picture-heavy description of the metric and curvature on a surface embedded in Euclidean space, and Chapter 6 discusses intrinsic definitions -- the theorem egregium, geodesics, and tensors. This overlaps somewhat with a standard intro Riemannian course, but those have a tendency to be overwhelmingly formal -- the plan here is to build a good collection of concrete examples. If we have more time at the end of the semester, we can of course talk a bit about connections.
Here are some projects that were done at University of Maryland to give you and idea of the range of possible topics.
"Time Integration Schemes for the 1D Heat Equation" "Cohomology of an Annulus" "Option Pricing Using Fast Fourier Transform" "The Fundamental Group and Seifert-van Kampen Theorem" "Optimal Control and Applications to Biomedical Problems" "Fibonacci Numbers" "Continuity in Topology" "The Classical Construction Problem and Constructable Numbers"
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