A Poisson structure associated to differential and difference operators – with the Toda lattice and KdV systems as examples

I will describe a Poisson structure on the space of curves in $R^n$ from which a series of Poisson structures on several associated spaces may be obtained by Poisson symmetry arguments.
Amongst these spaces one may find differential operators and difference operators and their respective reductions.
I will present the concrete cases $n=2,3$, for which the natural examples are the KdV and the Toda lattice, as concrete examples.