Diierential Invariants and Invariant Diierential Equations

This paper surveys recent results on the classi cation of di erential invari ants of transformation groups and their applications to invariant di erential equations and variational problems Consider a group of transformations acting on a jet space coordinatized by the inde pendent variables the dependent variables and their derivatives Scalar functions which are not a ected by the group transformations are known as di erential invariants Their importance was emphasized by Sophus Lie who showed that every invariant sys tem of di erential equations and every invariant variational problem could be directly expressed in terms of the di erential invariants As such they form the basic building blocks of many physical theories where one begins by postulating the invariance of the equations or the variational principle under a prescribed symmetry group Lie also demonstrated how di erential invariants could be used to integrate invariant ordinary di erential equations and succeeded in completely classifying all the di erential invariants for all possible nite dimensional Lie groups of point transformations in the case of one independent and one dependent variable Lie s results were pursued by Tresse and much later Ovsiannikov In this paper I will survey some recent results extending the earlier classi cation theorems and then discuss recent applications to the study of invariant evolution equations which is of great interest in image processing April y Supported in part by NSF Grants DMS and DMS and computer vision cf Space considerations preclude the inclusion of proofs and signi cant examples here I shall assume the reader is familiar with the fundamentals of the Lie theory of sym metry groups of di erential equations as discussed for instance in my book I shall employ the same basic notation here For simplicity I shall deal with complex valued functions here although most of the results are equally in the real case Let G be an r dimensional local transformation group acting on the space M X U C p C q co ordinatized by p independent and q dependent variables In the single dependent variable case q we allow G to be a group of rst order contact transformations B acklund s Theorem implies there are no other contact transformation groups Let G n denote the associated prolonged group action on the jet space J whose coordinates are denoted by x u n The space of in nitesimal generators of G its Lie algebra will be denoted by g with associated prolongation g n In order to properly study the di erential invariants of a transformation group we must understand the geometry of its prolongations Let sn denote the maximal orbit dimension of the prolonged action G n so that G n acts semi regularly on the open subset V n fz J j dim g n jz sng J consisting of all points contained in orbits of maximal dimension We shall in fact assume that G n acts regularly on V n although all our results suitably interpreted are valid in the semi regular case The orbit dimensions satisfy the elementary inequalities sn sn sn q p n n In particular they form a nondecreasing sequence