Research

Note:

This page is outdated and "under construction." For a statement about my current research, see here.

Main Interests:

My main mathematical interests are in geometry and topology. In particular, I study the geometry of smooth manifolds and fiber bundles endowed with an Ehresmann connection. A connection describes how elements in the fibers of a bundle E change with respect to infinitesimal changes in the base manifold M. Connections represent the minimal amount of additional structure that one must add to a manifold or bundle in order to obtain any meaningful geometric properties.

If E = TM is the tangent bundle to the base manifold, then a connection determines a second-order differential equation on the manifold. Just as second derivatives are used to define the curvature of a space curve in Calculus 3, connections determine the curvature and "shape" of a manifold. The notion of curvature extends to connections on fiber bundles, but now one must think of it as happening in the bundle E rather than on the base manifold M.

Connections also generalize the notion of parallel translation in a vector space to parallel transport along a manifold. A connection determines a system of diffeomorphisms between the fibers of a bundle, one for each path connecting the fibers in the base manifold. This "connecting" of fibers via parallel transport explains why connections are so-named.

This statement gives a brief overview of my work to date.

Here is a preprint paper.

Here are slides from a talk I gave to fellow grad students in August 2013.

And here are slides from my PhD defense.

I am also interested in pseudo-Riemannian 2-step nilpotent Lie groups. More on this to come very soon. (Hopefully.)

Other Activities:

I am also involved in the following seminars. All of these take place in the Seminar Room, JB 353.

Connection Geometry Seminar (CGS)
Organized by Professor Parker
Mondays, 3:30 - 5:00 pm

This is the continuation of a long-running (4 yrs) seminar on manifolds, bundles, Lie groups, and general connections. You can read more about it here.

Splitting Tangent Bundles (STB) Seminar
Organized by Professors Parker and Walsh
Wednesdays, 3:30 - 4:30 pm

This is the continuation of seminar that began in the fall semester of 2013. Parker's interest is in the (non-)existence of pseudoRiemannian metrics of prescribed signature on a given manifold. Walsh's interest is in tangent bundles to even-dimensional spheres. Again, you can read more about this here.

Bordism and K-Theory (BKT) Seminar
Organized by Professor Walsh
Mondays, 2:30 - 3:30 pm

This seminar is an introduction to Bordism and K-theory. It began in the fall semester of 2013 as the follow-up to a seminar on Differential Topology offered the previous semester. Last semester Prof. Walsh gave most of the talks, but this semester the veteran grad students *should* do most of the presenting.

Positive Curvature Seminar (PCS)
Organized by Professor Walsh and myself
Wednesdays, 2:30 - 3:30 pm

This seminar will be an introduction to positive scalar and positive Ricci curvature metrics. The goal is to cover the fundamental results of the theories. I will give all of the presentations, following an outline provided by Prof. Walsh.

Graduate Research Seminar (GRS)
???days, ??:?? - ??:??

This seminar provides an opportunity for graduate students to present their work to peers. In the past, students have used this seminar to practice thesis defenses, prepare for presentations in other seminars or classes, and to simply bounce ideas off of other students. It also provides a forum for graduate students to learn a bit about each other's interests. Students interested in presenting should email me.