Geometry of infinite jets and infinitely prolonged equations
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.

  1. The space J p.
  2. Jets of submanifolds.
  3. Infinitely prolonged equations.

§ 2-3. Basic differential geometric structures on infinite-dimensional manifolds.
  1. Smooth functions.
  2. Smooth mappigs, Lie transformations.
  3. Vector fields, Lie fields.
  4. Differential forms.

§ 4. Differential operators.
  1. Scalar and matrix differential operators.
  2. Prolongation of differential operators. One more example of a smooth mapping.
§ 5-6. The Cartan distribution on J p and E.
  1. Distributions on infinite-dimensional manifolds and their automorphisms.
  2. The Cartan connection.
  3. Maximal integral manifolds of the Cartan distribution.
  4. The structural element of an equation.
§ 7. Symmetries of the Cartan distribution on J p.
  1. Generating sections of symmetries of the Cartan distribution on J p.
  2. The structure of Sym (p).
  3. Evolutionary derivations.
  4. The higher Jacobi bracket.
  5. Comparison with Lie fields.
§ 8. Higher symmetries of differential equations.
  1. Exterior and interior higher symmetries.
  2. C-differential operators.
  3. Linearizations.
  4. Defining equations for higher symmetries.
§ 9. Examples of computations.

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