Alexandre VINOGRADOV

A program of the course at the 1st Italian Diffiety School,

(Forino, July 17 - August 1, 1998)

and at the 1st Russian Diffiety School

(Pereslavl-Zalessky, January 26 - February 5, 1998)

- Algebraic approach to manifolds. Interpretation of functions as measuring means.
- Spectrum of a K-algebra. Zarissky topology. Maps of spectrums. Examples.
- The spectrum theorem.
- Smooth hull of a
**R**-algebra. Examples. - Algebraic definition of smooth manifolds.
- Manifolds with boundary.
- Submanifolds, quotient manifolds and Cartesian product of manifolds.
- Coordinate definition of smooth manifolds.
- Examples. Configuration manifolds.
- Smooth maps.
- Equivalence of two definitions of manifolds.
- Diffeomorphisms, embeddings, immersions. The Sard theorem.
- Tangent vectors as differentiations at a point.
- Vector fields as differentiations of algebras.
- Restrictions of vector fields to submanifolds and differentiations from algebras to modules.
- Tangent space. Tangent map. Tangent bundle.
- Bundles. Product bundles. Vector bundles. Subbundles. Modules of sections.
- Examples. Principle bundles. Shtifel manifolds. Grassmann manifolds.
- Morphisms of bundles. The category of bundles.
- Whitney product of bundles and tensor product of modules of sections. Examples.
- Restrictions of bundles and induced bundles.
- Connectedness in a principle bundle. Form of connectedness.
- Cotangent bundle. Vector fields and differential forms as sections. Tensor bundles.
- Commutator of vector fields. Lie algebras of vector fields.
- Phase flow of a vector field. Lie derivative.

- Sternberg, S.: Lectures on Differential Geometry, Prentice Hall, Inc. Englewood Cliffs, New York, 1964, chap. 3, par. 5.
- Bocharov, A.V., Verbovetsky, A.M., Vinogradov, A.M., and
other: Symmetries and conservation laws of mathematical
physics equations,
*Faktorial*, Moscow, 1997 (in Russian) and AMS, 1999 (in English), chap. 1, par. 3.

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