Differential operators and algebra of densities

We consider the commutative algebra of densities on a manifold $M$. This algebra is endowed with the canonical scalar product and we shall consider self-adjoint operators on this algebra. We naturally come to canonical pencils of operators (acting on densities of different weights) and passing through a given operator (acting on densities of a given weight). There are singular values of weights for operators of a given order. These singular values lead to an interesting geometrical picture: 1) If $M$ is the real line we arrive at classical constructions in projective geometry. 2) If $M$ is an odd symplectic supermanifold we arrive at the Batalin-Vilkovisky groupoid.
    We also introduce a certain mapping on differential operators on algebra of densities and we relate this mapping with full symbol maps.
   The talk is based on works with Ted Voronov and on a work in progress with my student Adam Biggs.