Seminar talk, 7 December 2022: Difference between revisions

Revision as of 13:20, 4 November 2022

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 differential 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 problem 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 (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.

@@ Line 4: / Line 4: @@
 | abstract = In this talk, we classify linear (as well as some special nonlinear) scalar differential operators of order <math>k</math> on <math>n</math>-dimensional manifolds with respect to the diffeomorphism pseudogroup.
-Cases, when <math>k = 2</math>; <math>\forall n</math>; and <math>k = 3</math>; <math>n = 2</math>; were discussed before, and now we consider cases <math>k\ge5</math>; <math>n = 2</math> and <math>k\ge4</math>; <math>n = 3</math> and <math
+Cases, when <math>k = 2</math>; <math>\forall n</math>; and <math>k = 3</math>; <math>n = 2</math>; were discussed before, and now we consider cases <math>k\ge5</math>; <math>n = 2</math> and <math>k\ge4</math>; <math>n = 3</math> and <math>k\ge3</math>; <math>n\ge4</math>: In all these cases, the fields of rational differential invariants are generated by the 0-order invariants of symbols.
->k\ge3</math>; <math>n\ge4</math>: 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 <math>n</math> invariants principle in order to get (loc
+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 problem and show how to find the rational Gl-invariants of n-are forms in a constructive way.
-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
+After all, we apply the <math>n</math> invariants principle in order to get (local as well as global) normal forms of linear operators with respect to the diffeomorphism pseudogroup.
-sses of nonlinear operators.
+Depending on available time, we show how to extend all these results to some classes of nonlinear operators.
 | video =
 | slides =

Seminar talk, 7 December 2022: Difference between revisions

Revision as of 13:20, 4 November 2022

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