3. Postdoctoral Program - MSRI

Your Name: Abraham David Smith
Year of Ph.D: 2009
Institution of Ph.D.: Duke University, Department of Mathematics
Ph.D. advisor: Robert L. Bryant
Institution where you held your MSRI PD fellowship: McGill University, Department of
Mathematics and Statistics
Mentor at that institution: Niky Kamran
Institution (or company) where you are going next year: Fordham University, Mathematics
Department
Position: Visiting Assistant Professor
Anticipated lenght: (if it is a tenure track position just write tenure-track) 4 years
My annual report is here:
This is my report of academic activities as an MSRI NSF All-Institutes postdoctoral fellow for
the 2010-11 academic year under the mentorship of Niky Kamran at McGill University.
I received positive referee reports on my paper “Integrable GL(2) Geometry and Hydrodynamic
Partial Differential Equations” arXiv:0912.2789 during the summer of 2010, and the final
version of that paper was accepted by Communications in Analysis and Geometry and published
in the October 2010 volume.
My research since the Spring of 2010 has been focused on a geometric study of hyperbolic
partial differential equations [PDEs] in N>2 independent variables and one dependent variable.
This began as an extension of the paper listed above, but it is now more accurate to say that the
paper above covers a peculiar sub-case of this larger program. This research program has split
into two parts: First, I sought an intrinsic (coordinate-free) classification of second-order PDEs
using techniques from conformal differential geometry and moving frames. This involved the
development of a new geometric structure and the corresponding invariant theory. I finished an
early preprint [arXiv:1010.6010] in October 2010 that covers this material. Second, I am
continuing to try to understand the geometric criteria on these PDE structures that correspond to
“hydrodynamic integrability”, a popular notion of integrability arising from the analysis of semi-
Hamiltonian (a.k.a. "rich") systems of conservation laws. I hope that the geometric approach
will clarify which aspects of the analysis are contact invariant (independent of the coordinate
description of the PDE) and which are not. If luck prevails, it will provide a complete list of
all such integrable PDEs in the physically interesting case of N=4 (space+time).
While developing the second part of this research program, there have been several unexpected
technical hurdles that have slowed me from my original 1-year timeline for this program, now in
its 15th month. However, I expect to have a preprint on the "Part 2" material before the