My Current Research

There has been a request for me to say something about what research I've been working on lately. Well, here we go. The first part of this is accessible; the second part rather technical. :)

First, the paper I wrote with Adam Clausen and Jim Isenberg is being prepared for publication. This means re-writing parts of the paper to suit the needs of the journal, and generally cleaning up the notation, exposition, etc. You can see the rough version of the paper on arXiv.org.

I would like to follow up this paper with a short note proving the same result using a slightly different method. This would show that the result is "nice" in a certain sense appreciated by mathy people.

Second, I'm working on various aspects of the Timelike Minimal Submanifold (TMS) problem. These objects, also called membranes, D-branes (when all fields are "turned off"), and p-branes (which sounds to me like a 3rd-grade insult), are the relativistic version of a soap bubble. By this I mean that they "minimize" area locally. I put minimize in quotes for two reasons: first, this is the space-time area we are talking about, so "maximize" might be a better word; second, these submanifolds are not actually a minimum, but rather a critical point, of an action given by only the induced volume form.

For the non-mathy people out there, this last sentence basically means the following: If you're at a mountain pass, it's true that the exact point of ground you're standing on is "level" (a ball won't roll), but not true that you can't be in a lower place on the valley floor. This is the difference between a "critical point" and a "minimum". Ok, the rest of this entry will be technical.

The TMS problem was the subject of my thesis, and also the subject of a paper written with Jim Isenberg and Lars Andersson, who is one of the senior scientists here at AEI. (To view the paper, you need a subscription to the "Journal of Hyperbolic Differential Equations", which I know you all have sitting on your coffee tables next to National Geographic and No Depression.)

Generally, one would like to view this problem as some type of non-linear wave equation and study the initial-value problem, much in the same way one studies the Einstein equations. Like the Einstein equations, there are a number of different apporaches one can take. Currently, I - along with Lars and another researcher here at AEI - are studying this problem via canonical (Hamiltonian) analysis. This analysis seems like a good way to organise variousl features of the problem, and is also useful for people who like to try to quantize things.

Like the Einstein equations, there is a diffeomorphism invariance in the equations describing TMS, which means that for most results one will have to make a gauge choice. The formalism of canonical analysis is well-suited for analysing various gauge choices one can make. In particular, we are interested in showing local wellposedness in some of the "natural" gauges one might consider, and also would like to know which gauges are good for studying either singular behaviour or good for showing global existence (in the sense of future geodesic completeness).