Last updated January 14, 1999 |

- Calculus on smooth manifolds: vector fields, differential forms, distributions.
- Locally-trivial and vector bundles.
- Category theory: categories, functors, diagrams.
- Commutative algebra: rings, algebras, modules, spectres.

- Jet manifolds and jet bundles. Canonical coordinates.
- Linear differential operators and liftings.
- Nonlinear operators. Compositions and prolongations.
- Equations. Solutions. Coni and types.
- The Cartan distribution. Structure of integral manifolds.
- Symmetries and generating sections. Applications.
- Internal and external symmetries.

- Elements of calculus in commutative algebras.
- Infinite jets. Infinite prolongations of nonlinear operators.
- The Cartan distribution on infinite jets. Integrability and structure of integral manifolds.
- Infinite prolongations of differential equations. Formal integrability.
- Symmetries of the Cartan distribution on infinite jets. Evolution derivations and linearizations.
- Higher symmetries of nonlinear equations. The structural theorem.
- Examples.

- The category of differential equations (CDE): objects and morphisms.
- Coverings in CDE. Examples.
- Backlund transformations. Recursion operators.
- Nonlocal symmetries and shadows. Examples.
- The universal Abelian covering. The reconstruction theorem.

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