Starting August 2015, I am an Assistant Professor at University of Wisconsin—Stout, Wisconsin's Polytechnic University.
From 2011 until 2015, I was a Peter M. Curran Visiting Assistant Professor at Fordham University. Previously, I was a postdoctoral fellow at McGill University, mentored by Niky Kamran and funded by an NSF fellowship administered by the MSRI. I earned my PhD in 2009 at Duke University under the direction of Robert Bryant.
I study geometry in the sense of E. Cartan; this means that I examine the intrinsic geometry of PDEs using exterior differential systems, moving frames and representation theory.
In particular, I am currently studying the phenomena of involutivity and hydrodynamic integrability using Guillemin normal form and Spencer cohomology. These tools are algebraic tools developed during the 1960s to study formal integrability criteria. Nowadays these formal methods are particularly useful, because they allow us to study PDEs using computer algebra software.
Alongside my theoretical work, I am writing an open-source package for Sage to automatically build, manipulate, classify, and solve PDEs and differential ideals.
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. (Be forewarned that my travel schedule meant it was my least-prepared talk ever.)
Beyond this particular specialty, my interest in science is quite broad, and I try to stay abreast of developments in many branches of science. I try to read the weekly issues of Science and Nature, but I've also found that podcasts are an amazingly efficient way to learn new ideas. If you also have a general interest in science, 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 science reporters talk 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.|
|Inside Science||info||podcast||BBC Radio 4's new weekly science show, with Adam Rutherford. Discussions focus on the life of working scientists today.|
|Material World||blog||podcast||BBC Radio 4's old weekly general science show, with Quentin Cooper. Refreshingly sophisticated compared to a science show that airs on Fridays in the US.|
|Planet Money||blog||podcast||NPR's great 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||info||podcast||A really thoughtful project outlining the role of modern research mathematics.|
|DisasterCast||info||podcast||A fascinating series about systems engineering and safety.|
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 also started using Twitter to follow other scientists and science journalists.
Have you ever wanted to build and manipulate differential ideals and symbols of PDEs on an open-source computer algebra system?
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.)