In previous centuries mathematicians sought solutions of differential equations in formula as a matter of course. This approach to the study of differential systems has fallen out of fashion (and possibly favor) as a result of the vast revolutions in analysis and the types of problems requiring solution in the current epoch. Nevertheless a tremendous heritage in this area has been bequeathed to us by previous generations which we have begun to explore only in the last 4 decades or so. This resurgence has produced a fascinating body of mathematics and its applications, such solitons, Poisson geometry, infinite dimensional hamiltonian systems, Backlund transformations; to name a few. One such 19th century topic, Darboux integrability, has been the subject of a number of recent studies which reveal it to have strong geometric content and utility for constructing explicit solutions in formula of certain partial differential equations. However, because Darboux integrability is a comparatively rare phenomenon it is natural to ask if some weaker form of it may exist so as to be of wider applicability while retaining some useful aspects of the original. In this talk I will briefly recall the basic ideas of this form of integrability and then propose a weaked version which will be explored via a few elementary examples. Time permitting we will also explore its application to certain harmonic maps.
Hyperbolic Reduction of Partial Differential Equations Sponsored by the Meyer Fund