Giovanni Moreno

The geometry of the space of Cauchy data of nonlinear PDEs

First-order jet bundles can be put at the foundations of the modern geometric approach to nonlinear PDEs, since higher-order jet bundles can be seen as constrained iterated jet bundles. The definition of first-order jet bundles can be given in many equivalent ways — for instance, by means of Grassmann bundles. In this paper we generalize it by means of flag bundles, and develop the corresponding theory for higher-oder and infinite-order jet bundles. We show that this is a natural geometric framework for the space of Cauchy data for nonlinear PDEs. As an example, we derive a general notion of transversality conditions in the Calculus of Variations.

Completely exceptional 2nd order PDEs via conformal geometry and BGG resolution

Abstract By studying the development of shock waves out of discontinuity waves, in 1954 P. Lax discovered a class of PDEs, which he called “completely exceptional”, where such a transition does not occur after a finite time. A straightforward integration of the completely exceptional conditions allowed Boillat to show that such PDEs are actually of Monge–Ampere type. In this paper, we first recast these conditions in terms of characteristics, and then we show that the completely exceptional PDEs, with 2 or 3 independent variables, can be described in terms of the conformal geometry of the Lagrangian Grassmannian, where they are naturally embedded. Moreover, for an arbitrary number of independent variables, we show that the space of r th degree sections of the Lagrangian Grassmannian can be resolved via a BGG operator. In the particular case of 1st degree sections, i.e., hyperplane sections or, equivalently, Monge–Ampere equations, such operator is a close analogue of the trace-free second fundamental form.

Lowest degree invariant second-order PDEs over rational homogeneous contact manifolds

For each simple Lie algebra

On a bicomplex induced by the variational sequence

\mathfrak{g}

Contact manifolds, Lagrangian Grassmannians and PDEs

(excluding, for trivial reasons, type

Natural Boundary Conditions in Geometric Calculus of Variations

{\sf C}

ON THE CANONICAL CONNECTION FOR SMOOTH ENVELOPES

) we find the lowest possible degree of an invariant second-order PDE over the adjoint variety in

ON FAMILIES IN DIFFERENTIAL GEOMETRY

\mathbb{P}\mathfrak{g}

On a geometric framework for Lagrangian supermechanics

, a homogeneous contact manifold. Here a PDE

On the vertex-to-edge duality between the Cayley graph and the coset geometry of von Dyck groups

F(x^i,u,u_i,u_{ij})=0