Undergraduate research projects in mathematics are a great way for young mathematicians working toward their bachelor degrees to experience what real research in mathematics is like, and gives opportunities for interaction with faculty outside of the classroom setting.
If you are interested in this project, or perhaps a pseudo-reading course in the area of topology/geometry (probably when I'm teaching at most one course), please contact me.
A list of projects so far:
Summer 2016 - Invariants of Quotients by Circle Actions
My paper "The differential structure of an orbifold" shows that the underlying (local) semi-algebraic variety of an orbifold, equipped with its ring of smooth functions, contains a complete set of invariants of the orbifold; that is, an entire orbifold atlas can be reconstructed from the variety with its differential structure. Can something similar be done for spaces that (locally) are quotients of manifolds by circle actions? If the differential structure is not sufficient, what other pieces of information are required? This was an REU project with three undergraduate students (Naiche Downey, Lucas Goad, and Michael Mahoney) and one graduate student (Suzanne Craig). The results have appeared in a paper submitted for publication.
- arXiv version of the paper
- A poster for an REU poster session
- Mathematica program for computing invariant polynomials of circle actions (torus actions, in fact)
Spring 2015 - Evolution of Curves and Surfaces
A recent project that I have been interested in is using the power of Mathematica to explore evolutionary techniques for (so-far simple) optimisation problems. This was a project under the umbrella of the Illinois Geometry Lab in Spring 2015.
- Project Website (developed by Yingqiu Huang)