Geometry and Topology
This page contains a list of ideas for DRP projects, but is by no means exhaustive.
Topic: Smooth Manifolds
Suggested Text: Introduction to Smooth Manifolds, John M Lee (text may vary based on specific subject matter)
Suggested Background: MATH 3200 (Introduction to Topology) and some Linear Algebra are useful
Description: The word manifold is thrown around a lot in math and physics, but what is a manifold? Informally, manifolds are spaces where at each point, it looks like Euclidean space. For instance, the Earth is a sphere, but to people standing on the surface, it looks flat like the plane. This notion can be made rigorous using ideas from topology, and it doesn’t stop there! You can then study manifolds with additional structures. This project would focus on topics surrounding smooth manifolds.
Topic: Symplectic Geometry
Suggested Text: Introduction to Symplectic Topology, McDuff and Salamon OR Lectures on Symplectic Geometry, Ana Cannas de Silva (pdf)
Suggested Background: MATH 3200 (Introduction to Topology) and MATH 2600 (Linear Algebra)
Description: One of the additional structures one could place on a smooth manifold is a symplectic structure. This consists of a smoothly varying symplectic form on the tangent space at the point. While it might not be immediately apparent, this structure gives rise to many interesting mathematical phenomena (the Principle of the Symplectic Camel for instance) as well as being adjacent to Hamiltonian mechanics. This project would explore basic properties of symplectic manifolds with the end goal of connecting symplectic manifolds to contact/Kahler manifolds or Hamiltonian dynamics.
Topic: de Rham Cohomology
Suggested Text: From Calculus to Cohomology: De Rham Cohomology and Characteristic Classes, Ib Madsen & Jxrgen Tornehave
Suggested Background: MATH 2300 (Multivariable Calculus) and MATH 2600 (Linear Algebra), though MATH 3100 (Introduction to Analysis) and MATH 3200 (Introduction to Topology) may be helpful
Description: “The integration on forms concept is of fundamental importance in differential topology, geometry, and physics, and also yields one of the most important examples of cohomology, namely de Rham cohomology, which (roughly speaking) measures precisely the extent to which the fundamental theorem of calculus fails in higher dimensions and on general manifolds.” — Terence Tao, Differential Forms and Integration