Normal forms for differential operators

In my talk I'll give an overview of the results obtained by me, as well as jointly with co-authors,  related to the problem of classifying commuting (scalar) differential, or more generally, differential-difference or  integral-differential operators in several variables.
Considering such rings as subrings of a certain complete non-commutative ring $\hat{D}_n^{sym}$ (not the known  ring of formal pseudo-differential operators!), the normal forms of differential operators mentioned in the title are obtained after conjugation by some invertible operator ("Schur operator"), calculated with the help of one of the operators in a ring. Normal forms of commuting operators  are polynomials with constant coefficients in the differentiation, integration and shift operators, which have a finite order in each variable, and can be effectively calculated for any given commuting operators.
I'll talk about some recent applications of the theory of normal forms:  an effective parametrisation of torsion free sheaves with vanishing cohomologies on a projective curve, and a correspondence between solutions to the string equation and pairs of commuting ordinary differential operators  of rank one.