Alexander Verbovetsky

A preliminary program of the course at the 5-th Italian Diffiety School,

(San Stefano del Sole, July 19-31, 2002)

Introduction.I. Spaces of infinite jets.

II-III. Basic differential geometric structures on infinite-dimensional manifolds.

- The space .
- Jets of submanifolds.
- Infinitely prolonged equations.
IV. Differential operators.

- Smooth functions.
- Smooth mappigs, Lie transformations.
- Vector fields, Lie fields.
- Differential forms.
V-VI. The Cartan distribution on and .

- Scalar and matrix differential operators.
- Prolongation of differential operators. One more example of a smooth mapping.
VII. Symmetries of the Cartan distribution on .

- Distributions on infinite-dimensional manifolds and their automorphisms.
- The Cartan connection.
- Maximal integral manifolds of the Cartan distribution.
- The structural element of an equation.
VIII. Higher symmetries of differential equations.

- Generating sections of symmetries of the Cartan distribution on .
- The structure of Sym ().
- Evolutionary derivations.
- The higher Jacobi bracket.
- Comparison with Lie fields.
IX. Examples of computations.

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

- Luca Vitagliano (University of Roma)
- Gianni Manno (King's College London)

Questions and suggestions should go to school @ diffiety.ac.ru.