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Abraham D. Smith
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Peter M. Curran Visiting Assistant Professor
Mathematics Department, Fordham University 
Bronx, New York  10458-5165 USA
(Contact Information)

Mathematician (CV)

Starting September 2011, I am a 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.



Research Interests

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:

Source News Podcast Topic
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.

Additionally, aside from their good coverage of consumer technology and Internet policy, Ars Technica has a fantastic science blog known as The Scientific Method.

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:

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The classification
of connected 2,3-integrable GL(2)-structures of degree 4. which correspond to
Hessian-type integrable hydrodynamic PDEs in three variables. The classification of connected 2,3-integrable GL(2)-structures of degree 4 as well as Hessian-type integrable PDEs in three variables.

Preprints (Live)

Abraham Smith's articles on arXiv

[1]  http://arxiv.org/abs/1010.6010v1 [pdf]
A Geometry for Second-Order PDEs and their Integrability, Part I
Abraham D. Smith
Comments:25 pages
Subjects:Differential Geometry (math.DG); Analysis of PDEs (math.AP); 58A15, 37K10

For the purpose of understanding second-order scalar PDEs and their hydrodynamic integrability, we introduce G-structures that are induced on hypersurfaces of the space of symmetric matrices (interpreted as the fiber of second-order jet space) and are defined by non-degenerate scalar second-order-only (Hessian) PDEs in any number of variables. The fiber group is a conformal orthogonal group that acts on the space of independent variables, and it is a subgroup of the conformal orthogonal group for a semi-Riemannian metric that exists on the PDE. These G-structures are automatically compatible with the definition of hydrodynamic integrability, so they allow contact-invariant analysis of integrability via moving frames and the Cartan--Kaehler theorem. They directly generalize the GL(2)-structures that arise in the case of Hessian hyperbolic equations in three variables as well as several related geometries that appear in the literature on hydrodynamic integrability. Part I primarily discusses the motivation, the definition, and the solution to the equivalence problem, and Part II will discuss integrability in detail.

[2]  http://arxiv.org/abs/0912.2789v4 [pdf]
Integrable GL(2) Geometry and Hydrodynamic Partial Differential Equations
Abraham D. Smith
Comments:32 pages, 2 figures v4: typos corrected. to appear in Communications in Analysis and Geometry, Vol 18 no 4 2010
Subjects:Differential Geometry (math.DG); Analysis of PDEs (math.AP); Exactly Solvable and Integrable Systems (nlin.SI); 58A15, 37K10

This article is a local analysis of integrable GL(2)-structures of degree 4. A GL(2)-structure of degree n corresponds to a distribution of rational normal cones over a manifold M of dimension (n+1). Integrability corresponds to the existence of many submanifolds that are spanned by lines in the cones. These GL(2)-structures are important because they naturally arise from a certain family of second-order hyperbolic PDEs in three variables that are integrable via hydrodynamic reduction. Familiar examples include the wave equation, the first flow of the dKP equation, and the Boyer--Finley equation. The main results are a structure theorem for integrable GL(2)-structures, a classification for connected integrable GL(2)-structures, and an equivalence between local integrable GL(2)-structures and Hessian hydrodynamic hyperbolic PDEs in three variables. This yields natural geometric characterizations of the wave equation, the first flow of the dKP hierarchy, and several others. It also provides an intrinsic, coordinate-free infrastructure to describe a large class of hydrodynamic integrable systems in three variables.

Travel & Talk Schedule

CMS Special Session, June 2013

Francis Valiquette and I organized a special session on Pseudogroups at the 2013 Summer Meeting of the Canadian Mathematical Society in Halifax. Talk slides should be available soon.

CRM Workshop, June 2011

Francis Valiquette and I organized the workshop Moving Frames in Geometry at the Centre de Recherches Mathématiques at Université de Montréal. Talk slides are now available.
The classification
of connected 2,3-integrable GL(2)-structures of degree 4. which correspond to
Hessian-type integrable hydrodynamic PDEs in three variables.

Research Computing Cluster Administrator

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.

Linux and Open-Source Software

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.)


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