Vladimir CHETVERIKOV

A program of the course at the Join Russian-Italian Diffiety School,

Pereslavl-Zalessky (Russia), August 17 - 30, 1999

Introduction.§ 1. Spaces of infinite jets.

§ 2-3. Basic differential geometric structures on infinite-dimensional manifolds.

- The space J
^{¥}p.- Jets of submanifolds.
- Infinitely prolonged equations.
§ 4. Differential operators.

- Smooth functions.
- Smooth mappigs, Lie transformations.
- Vector fields, Lie fields.
- Differential forms.
§ 5-6. The Cartan distribution on J

- Scalar and matrix differential operators.
- Prolongation of differential operators. One more example of a smooth mapping.
^{¥}p andE^{¥}.§ 7. Symmetries of the Cartan distribution on J

- Distributions on infinite-dimensional manifolds and their automorphisms.
- The Cartan connection.
- Maximal integral manifolds of the Cartan distribution.
- The structural element of an equation.
^{¥}p.§ 8. Higher symmetries of differential equations.

- Generating sections of symmetries of the Cartan distribution on J
^{¥}p.- The structure of Sym (p).
- Evolutionary derivations.
- The higher Jacobi bracket.
- Comparison with Lie fields.
§ 9. Examples of computations.

- Exterior and interior higher symmetries.
C-differential operators.- Linearizations.
- Defining equations for higher symmetries.

Questions and suggestions should go to Jet Nestruev, jet @ diffiety.ac.ru.