# Seminar talk, 7 December 2022

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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 ${\displaystyle k}$ on ${\displaystyle n}$-dimensional manifolds with respect to the diffeomorphism pseudogroup.

Cases, when ${\displaystyle k=2}$, ${\displaystyle \forall n}$, and ${\displaystyle k=3}$, ${\displaystyle n=2}$, were discussed before, and now we consider cases ${\displaystyle k\geq 5}$, ${\displaystyle n=2}$ and ${\displaystyle k\geq 4}$, ${\displaystyle n=3}$ and ${\displaystyle k\geq 3}$, ${\displaystyle n\geq 4}$. In all these cases, the fields of rational differential invariants are generated by the 0-order invariants of symbols.

Thus, at first, we consider the classical problem of Gl-invariants of ${\displaystyle n}$-ary forms. We'll illustrate here the power of the differential algebra approach to this problem and show how to find the rational Gl-invariants of ${\displaystyle n}$-are forms in a constructive way.

After all, we apply the ${\displaystyle n}$ invariants principle in order to get (local as well as global) normal forms of linear operators with respect to the diffeomorphism pseudogroup.

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

Video