Award: DMS 1321212, Principal Investigator: Joseph M. Landsberg, Jeanne N. Clelland, Colleen Robles
This grant provides partial support for a conference entitled "New Directions in Exterior Differential Systems," to be held in Estes Park, Colorado on July 14-20, 2013. The conference is intended to disseminate new and influential applications of exterior differential systems to partial differential equations (integrable PDE, systems of hyperbolic conservation laws, and control theory), complex differential geometry (variation of Hodge structure), topology of algebraic varieties (homological rigidity of Schubert classes in compact Hermitian symmetric spaces), and parabolic geometries (including conformal and CR geometries). The conference program will include three sets of mini-courses that aim to open up exterior differential systems and some of their current applications to less experienced researchers.
Exterior differential systems are algebraic and geometric renderings of partial differential equations, rooted in ideas that go back to the nineteenth century and that untangle some of the complications of nonlinear partial differential equations. The techniques of exterior differential systems are being applied in a widening circle of problems and the mini-course components of this conference are intended to make this subject accessible to junior researchers and to those with little experience in the field. The conference web site is www.math.tamu.edu/~robles/2013EP/index.html.
Exterior Differential Systems (EDS) is a systematic approach to systems of partial differential equations that (i) determines the space of local solutions, and (ii) identifies symmetries and invariants of the system. Very recently, there have been several new and exciting applications of EDS. This meeting brought together leading geometers, graduate students and post-docs to: (a) promulgate these advances, among both experts in EDS and experts in the areas hosting the new applications, and (b) provide direction and foster collaboration for future research. In order to facilitate the substantive exchange of ideas that will lead to advances both in EDS proper and in the application areas, we portioned the lectures into two types. (1) The workshop portion of the meeting consisted of three mini-courses. Each mini-course addressed an active area of research in geometry and the role of exterior differential systems in its study. The three mini-courses were: (1.A) Hodge theory by Mark Green and Phillip Griffiths. Hodge theory has expanded well beyond its initial setting in algebraic geometry and now plays an influential role at the interface between algebraic geometry, representation theory and number theory. (1.B) Parabolic geometry by Andreas Cap, Michael Eastwood, JM Landsberg. Parabolic geometries generalize some of the most important geometries in math and physics, including CR and conformal geometries. (1.C) Integrable systems by Chuu-lian Terng. Integrable systems form a class of differential equations that arises often in engineering and differential geometry. (2) In the conference portion of the meeting participants reported on their latest research.
