I study geometry in the sense of E. Cartan; this means that I examine local geometry (especially the geometry of differential equations in jet space) using exterior differential systems, moving frames and representation theory. In particular, I am currently studying second-order partial differential equations in three (or more) variables and their associated conformal Spin geometry. Additionally, I am interested in the related frameworks for studying algebroids, pseudo-groups, and distributions. Control theory, mathematical relativity and Finsler geometry are neat, too.
On Dec 9, 2013, I gave a talk about my ongoing work at the Fields Institute in Toronto, as part of the Focused Research Workshop on Exterior Differential Systems and Lie Theory. You can watch a video of the talk to get a sense of my current project.
Beyond this particular specialty, my interest in science is quite broad, and I try to stay abreast of developments in many branches of science. If you also have interest in these topics, I recommend these resources, which I follow religiously:
|RadioLab||blog||podcast||Awesome presentation of contemporary science and the boundaries of human knowledge, for a general audience.|
|The Guardian's Science Weekly||news||podcast||The Guardian's Alok Jha talks with experts in science and science policy. This is a nice median between the popular discussion of RadioLab and the technical intricacies of the Nature and AAAS/Science podcasts.|
|Nature News||news||podcast||Newest articles from probably the most prestigious journal in the world. Excellent podcast!|
|Science/AAAS||news||podcast||Newest articles from probably the second most prestigious journal in the world. Excellent podcast and good coverage of AAAS meetings.|
|Material World||blog||podcast||BBC Radio 4's weekly general science show. Refreshingly sophisticated compared to a science show that airs on Fridays in the US.|
|Planet Money||blog||podcast||NPR's great new production analyzing modern global economics in a thoughtful and reasonably scientific way. Unfortunately, most of the content never reaches the airwaves; you have to follow the podcast.|
|Relatively Prime||summary||podcast||A really thoughtful new project outlining the role of modern research mathematics.|
The most insightful commentaries I have seen about research are The Importance of Stupidity in Scientific Research by Martin A. Schwartz and You and Your Research by Richard Hamming. Regarding mathematics as a social activity, everyone should read On Proof and Progress in Mathematics and this short essay by William Thurston. On a lighter note, a fantastic description of how it feels to do mathematics was recently written by Yasha Berchenko-Kogan.
Recently, I've started using Twitter to follow other scientists and science journalists. I'm not yet convinced that it is a useful technology, but it's worth a try. Here are some recent tweets:
I am the administrator for the Fordham University Research Computing Cluster, a service offered through the Academic Computing Environment. This is a modern Linux environment, provided and maintained to support the ongoing research of Fordham faculty.
If you would like an account, please fill out this questionairre to make the process as fast as possible. I am happy to meet and talk with anyone about how the Research Computing Cluster can serve your research.
Over time, I have spent an unreasonable number of hours learning the intricacies of Linux system administration. I have submitted patches and (properly documented) bug reports to various open-source programs, and I see the world of open-source software as another branch of the academic discipline. We all build on each-others' work, and it is scientifically unethical to rely on tools that you cannot ultimately deconstruct, understand, and modify.
Currently, for my own research, I am developing an Exterior Differential Systems package for Sage, the python-based, open-source math engine that is quickly taking over the world. My package will be modeled on Jeanne Clelland's excellent Cartan package for Maple, whose only problem is that it is a package for Maple.
My current and recent undergraduate research students have worked on problems in computer vision, including analyzing Spline curves in Randers geometry and developing moving-frame methods for image recognition and signal analysis.
Regarding Linux, I have used all the major distributions at one time or another, going back a long way, but for the moment I am settled on Debian. I recently found this, containing this disc and more, while going through an old box of books. I guess I really have been using linux a long time! (Actually, I think my brother bought that CD, but I do clearly recall running "make config" to build my kernel in RedHat 4.2 (Biltmore) in 1997.)
I have tinkered with things like IPv6 and other networking technologies. (I am curious what mathematical modeling and control theory can reveal about network catastrophes due to the Bufferbloat problem that has recently been brought to light.)