differential operators and

introduction to geometry of jet spaces

Joseph KRASIL'SHCHIK

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)

- 1.
- Rings. Rings with unit. Associative rings. Commutative rings. Fields. Examples.
- 2.
- Homomorphisms of rings. Kernels and images. Epi-, mono-, and isomorphisms. Algebras. Examples.
- 3.
- Ideals. Quotient rings. Prime and maximal ideals.
- 4.
- Modules. Generators. Submodules and quotient modules. Direct sums. Examples.
- 5.
- Homomorphisms of modules. Kernels and images. Epi-, mono-, and isomorphisms. Exact sequences. Compositions of homomorphisms. Commutative diagrams.
- 6.
- Homomorphisms of modules over commutative algebras. Bimodule structure.
- 7.
- Free and projective modules.
- 8.
- Tensor products. The tensor, symmetric and skew-symmetric algebras of a module.
- 9.
- Categories and functors. Natural transformations of functors. Representable functors and representative objects.

- 10.
- Vector fields on manifolds as derivations of . The Leibniz rule.
- 11.
- Derivations of commutative algebras. Module and Lie algebra structures.
- 12.
- Module-valued derivations. The modules . The commutator identity . First order differential operators and their description.
- 13.
- The algebraic definition of linear differential operators . Equivalence of the algebraic and analytical definitions for the case .
- 14.
- The bimodules . The tautological operators of order and . The bimodule embeddings , the module . The bimodules .
- 15.
- Composition of linear differential operators. The structure of an associative algebra in .
- 16.
- The exact sequences
- 17.
- The operators , their universal property. The natural isomorphism . Representability of .
- 18.
- The universal composition as a natural transformation of functors. Associativity of the universal composition.

- 19.
- The submodules . The jet modules . The operators .
- 20.
- The universal property of operators . The natural isomorphism of -modules . Representability of .
- 21.
- Functorial nature of
. The exact sequences
- 22.
- The module of infinite jets .
- 23.
- The universal cocomposition as a natural transformation of functors. Coassociativity of the universal cocomposition.

- 24.
- The embedding , , . The module of 1-forms . The de Rham operator , . The splitting .
- 25.
- The universal property of . The natural isomorphism of -modules . Representability of .
- 26.
- Description of by generators and relations.
- 27.
- The modules , the module . Wedge product in .
- 28.
- The de Rham differential , its properties. The de Rham complex and the de Rham cohomology.
- 29.
- The Spencer jet complex

- 30.
- The modules . The functors .
- 31.
- Elements of as skew-symmetric -valued multiderivations.
- 32.
- Wedge product , its properties. Structure of superalgebra in .
- 33.
- Inner product (contraction) , its properties.
- 34.
- The formula

- 35.
- Action of derivations on differential forms (Lie derivative), its properties.
- 36.
- The formula

- 37.
- Smooth algebras.

- 38.
- Motivations to consider fiber bundles: how to extend algebras of observables? First examples: Cartesian products of manifolds, the Moebius band, tangent and cotangent bundles.
- 39.
- Locally trivial fiber bundles. Morphisms. Sections. Subbundles.
- 40.
- Pull-backs. Restriction of a bundle to a submanifold. Whitney products.
- 41.
- Vector bundles. Morphisms. Kernels and images. Modules of sections, the functor .
- 42.
- Direct sums and tensor products of vector bundles.
- 43.
- Evaluation of -modules at points of . Isomorphism between the category of vector bundles over and the category of finitely generated -modules.

- 44.
- Example: evaluation of at points of . Ghosts. The geometrization functor and geometrical modules. Geometrization of algebraic Calculus over .
- 45.
- Evaluation of at point of . Identification with Taylor series of order .
- 46.
- General geometric definition of the manifolds . Points of as classes of tangent sections. Jets of sections. Local coordinates.
- 47.
- Fiber bundles and . Vector bundle structure in . Fibers of the bundles .

- 48.
- Scalar nonlinear differential operators as functions on . General nonlinear differential operators .
- 49.
- Nonlinear differential equations as submanifolds of . Solutions.
- 50.
- How to diminish the order of a differential equation using the universal cocomposition?
- 51.
- The Cartan distribution on , its maximal integral manifolds.
- 52.
- Lie transformations and Lie fields. Symmetries.
- 53.
- Passing to infinity.

- M. F. Atiyah and I. G. MacDonald,
*Introduction to commutative algebra*, Addison--Wesley Publ. Comp., Reading, Massachusetts, 1969. - S. Lang,
*Algebra*, Addison--Wesley Publ. Comp., Reading, Massachusetts, 1965. - I. S. Krasil'shchik, V. V. Lychagin, and A. M. Vinogradov,
*Geometry of jet spaces and nonlinear partial differential equations*, Gordon and Breach, New York, 1986. -
I. S. Krasil'shchik,
*Calculus over commutative algebras: A concise user guide*, Acta Appl. Math.**49**(1997), 235--248,

URL: http://diffiety.ac.ru/preprint/96/01_96abs.htm. -
Jet Nestrujev,
*Smooth manifolds and observables*,*IUM*, Moscow, 2000 (in Russian). English translation to appear.

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