Seminar talk, 7 December 2022

Speaker: Valentin Lychagin

Title: On normal forms of differential operators

Abstract:
In this talk, we classify linear (as well as some special nonlinear) scalar differential operators of order $k$ on $n$-dimensional manifolds with respect to the diffeomorphism pseudogroup.

Cases, when $k=2$; $\forall n$; and $k=3$; $n=2$; were discussed before, and now we consider cases $k\geq 5$; $n=2$ and $k\geq 4$; $n=3$ and $k\geq 3$; $n\geq 4$: In all these cases, the fields of rational di fferential invariants are generated by the 0-order invariants of symbols. Thus, at first, we consider the classical problem of Gl-invariants of n-ary forms.We'll illustrate here the power of the differential algebra approach to this p roblem and show how to find the rational Gl-invariants of n-are forms in a constructive way.

After all, we apply the $n$ invariants principle in order to get (loc al as well as global) normal forms of linear operators with respect to the diffe omorphism pseudogroup.

Depending on available time, we show how to extend all these results to some cla sses of nonlinear operators.