Seminar talk, 1 April 2015

Speaker: Pavel Bibikov

Title: On geometrization of differential equation of second order quadratic in the leading derivatives

Abstract:
In 1978 V.V.Lychagin suggested a remarkable construction of geometrization of Monge-Ampère equations of second order. The essence of the construction consists in identifying of the equations to the kernels of some nonlinear differential operators that are defined by the so-called effective differential 2-forms lying on the Cartan distribution on the space of 1-jets. Advantages of this approach to studying Monge-Ampère equations are: first, a possibility to use geometric methods to study these equations, and, second, lowering the order of objects under consideration.

It is natural to generalize the Lychagin construction to other classes of differential equations. In the talk we try to do that for differential equation of second order quadratic in the leading derivatives. The first part of the talk will discuss ordinary differential equations (for which we have explicit results), and the second part of the talk will discuss partial differential equations (for which we have no explicit results). In particular, we show that point invariants of such equations are related to the symplectic invariants of quadric Grassmannians, and the contact invariants are related to quadrics on the space of effective differential 2-forms.