The goal of this talk is to give a unified approach to the solutions of a class of isoperimetric problems by relating them to the exterior differential systems (EDS) studied by R. Bryant and P. Griffiths. It stayed largely unnoticed that one and the only PDE (but with different initial data) lies in the base of many classical isoperimetric inequalities. But more importantly, by a funny change of variables (provided, for example, via EDS) this PDE can be made linear: namely, the inverse heat equation. Then such classical inequalities as log-Sobolev, Poincare, Beckner--Sobolev, Bobkov's inequality become particular solutions of this linear inverse heat equation. This observation allows us to invent many new isoperimetric inequalities (sharp of course) that are ``MORE sharp" than the sharp classical ones. For example, we improved the Beckner--Sobolev inequality, and the talk will contain such an improvement for a particular exponent (where the new inequality looks especially nice). There are also other new isoperimetric inequalities in the paper having as CONSEQUENCES the classical ones. Our understanding is that the method of exterior differential systems (EDS) has been never used in any harmonic or geometric analysis inside the scope of our knowledge .
(Joint work with Paata Ivanisvili). Speaker(s): Alexander Volberg (Michigan State University)