**
***Geometry of Differential Equations*

Joseph KRASIL'SHCHIK

A program of the course at the 2-nd Italian Diffiety School,

(Forino, February - March, 1999)

and the 2-nd russian Diffiety School

(Pereslavl-Zalessky, January 26 - February 5, 1999)

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kindly use TeX, DVI or
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Smooth manifolds. Smooth locally trivial vector bundles. Sections.
The
-module structure in
.
The jet of a local section at a point . The space
. Smooth structure in
. Manifolds
and bundles
. The jet modules
. Canonical coordinates
in
associated to a local trivialization in . Dimension of . The
bundles
, . Graphs of jets.
-planes. Presentation of points of as pairs
, where
and
is an -plane.
Presentation of scalar operators as functions on . Pull-backs
and nonlinear operators
as
sections of the bundles
. Presentation of operators as
morphisms
. The universal operator
. Prolongations of nonlinear operators and their
correspondence to morphisms
. Composition of
nonlinear operators.
Differential equations as submanifolds in
. Description
of equations by nonlinear operators. The first prolongation
. Three definitions of the -the prolongation, there
equivalence. Solutions.
The Cartan plane
as the span of the set of -planes at the
point
. The distribution
.
Description of
in the form
.
Local description of
by the Cartan forms
. A local basis in
.
Involutive subspaces of the Cartan distribution. The theorem on maximal
integral manifolds. The type of a maximal integral manifold. Computation
of dimensions for maximal integral manifolds. Integral manifolds of
maximal dimension in inexceptional cases.
Lie transformations as diffeomorphisms of preserving the
Cartan distribution. Lifting of Lie transformations from to
. The case : correspondence between Lie
transformations and diffeomorphisms of . The case :
the contact structure in , correspondence between Lie
transformations and contact transformations of (inexceptional
case
and exceptional case ). Local formulas for
liftings of Lie transformations.
Lie fields. Local lifting formulas. Global nature of lifting for Lie
fields. Infinitesimal analog for the Lie-Bäcklund theorem.
One-dimensional bundles. Generating functions of Lie fields. Correspondence
between functions on and Lie fields for trivial one-dimensional
bundles. The jacobi bracket on
. Local coordinate
formulas for Lie fields and Jacobi brackets in terms of generating functions.
Bundles of higher dimensions. The element
, its definition and properties. The Spencer complexes
for a vector bundle
, their
exactness. The element
, its properties. Generating sections
as the result of construction of Lie fields with
. Jacobi brackets for generating sections. Local coordinates.
Finite and infinitesimal symmetries, definitions. ``Physical meaning''
of generating functions. Determining equations for coordinate computations.
An example: symmetries of the Burgers equation
.
The restriction
of the Cartan distribution to
. Exterior
and
interior
symmetries of an equation
. The homomorphism
.
Counterexamples.
Algebraic model. The basic constructions. Cohomological invariants.

Questions and suggestions should go to
J. S. Krasil'shchik, josephk @ diffiety.ac.ru.