Homogeneous Hamiltonian operators, projective geometry and integrable systems

First-order homogeneous Hamiltonian operators play a central role in the Hamiltonian formulation of quasilinear systems of PDEs. They have well-known differential-geometric invariance properties which find application in the theory of Frobenius manifolds. In this talk we will show that second and third order homogeneous Hamiltonian operators are invariant under reciprocal transformations of projective type, thus allowing for a projective classification of the operators. Then, we will describe how the above operators generate known and new integrable systems, and discuss the invariance properties of the systems under projective transformations.