3. Postdoctoral Program - MSRI

POSTDOCTORAL FELLOW REPORT
June 2010
-------- Original Message --------
Subject: MSRI NSF postdoc yearly report
Date: Tue, 1 Jun 2010 10:41:51 -0400
From: Abraham Smith
To: Niky Kamran , hbarcelo@msri.org
This is my report of academic activities as an MSRI NSF All-Institutes postdoctoral
fellow for the 2009--2010 academic year.
From August to December 2009, my main activity was to continue my project on the
geometry of hydrodynamic PDEs in 3 variables, producing the paper ``Integrable GL(2)
Geometry and Hydrodynamic Partial Differential Equations'' arXiv:0912.2789. In
September, I gave a talk on this paper at the CIRGET Geometry and Topology Seminar
at UQàM. In January 2010, this paper was submitted to Communications in Analysis and
Geometry; as of June 1, I am still waiting for the referee's report.
Since the submission of that paper, I have been working on a broad generalization of that
theory to PDEs in any number of variables. This theory involves the development of a
new geometric structure and the corresponding invariant theory. I expect this theory to
allow characterization and classification of large families of integrable hyperbolic PDEs.
The existing theories of integrable hyperbolic systems are usually limited to 2 (1 space +
1 time) or 3 (2 space + 1 time), but my theory will yield results in the most physically
appropriate case, 4 variables. Such PDEs frequently arise in fluid mechanics, general
relativity, and various other wave equations, so this theory should have a broad impact on
the understanding of many systems that are important to a variety of scientists and
engineers.
Progress on this project has been steady, and I am optimistic that this work will result in
an article ready for submission around September, tentatively titled ``A Geometric
Framework for Hydrodynamic Integrability.'' In April 2010, I presented an outline of this
project in a special session at the AMS Sectional Meeting in St. Paul.
Aside from my own research, I have attended many seminars in analysis and geometry
and participated in weekly meetings with my mentor, Niky Kamran, and his other
postdocs and graduate students. In December 2009, I attended the ``Careers in
Academia'' workshop at AIM. During the Winter session, I gave two talks at McGill
directed at graduate students and postdocs in geometry and analysis.
Upcoming plans include the ``Differential Geometry and Application'' conference in
Brno in August, a lecture at Texas A&M Univ in late October, and the Texas Geometry
and Topology Conference in November. At all three events, I will continue discussions
with potential collaborators, most notably Dennis The, who share interest in the
geometric approach of partial differential equations. It is also likely that I will visit the