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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 such as complex, Riemannian, symplectic, contact, etc. all with their own interesting flavor. This project would focus on one of these flavors of manifolds.

Topic: Knots
Suggested Text: Why Knot?: An Introduction to the Mathematical Theory of Knots with Tangle, Colin Adams and The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, Colin Adams
Suggested Background: N/A
Description: Whether we like it or (k)not, knots are an everyday occurrence. Most of our knots can be untangled since their ends aren’t joined. But what about knots whose ends are joined, like celtic knots? Can we prove that they cannot be untangled? Knot Theory addresses the following generalization: How can we determine if two knots are the same or different? One way to tell that two knots are different are by using invariants, e.g., the Jones polynomial (discovered by Vaughan Jones, a professor and Fields medalist here at Vanderbilt).

Topic: Space-Filling Curves
Suggested Text: Space-Filling Curves, Hans Sagan
Suggested Background: MATH 3200 (Introduction to Topology)
Description: In the 1800s, Georg Cantor showed that the unit interval and unit square are in a one-to-one correspondence, astounding the mathematical community. Such a bijection, however, cannot be continuous. Weakening the hypothesis that such a function from the unit interval to the unit square is only surjective, there are in fact continuous such functions. These are ‘space-filling curves’. More generally, the Hahn-Mazurkiewicz Theorem states that a non-empty Hausdorff topological space is a continuous image of the unit interval if and only if it is compact, connected, locally-connected, and second-countable.

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