Joseph Krasil'shchik's lectures on the linear differential operators over commutative algebras and geometry of jet spaces

Autumn 2015

Syllabus

1. Category and functors (introduction).
2. Linear differential operators with values in modules. Main properties.
3. Derivations.
4. Representative objects: jets and differential forms.
5. Differential calculus over commutative algebras.
6. Frölicher-Nijenhuis brackets and related cohomologies. Algebraic model of Hamiltonian formalism.
7. Frölicher-Nijenhuis brackets and related cohomologies. Algebraic model of nonlinear differential equations.
8. Geometric realization. Relation between the category of vector bundles over a manifold and the category of projective modules over a commutative ring.
9. Jets of locally trivial bundles over smooth manifolds. The Cartan distribution.
10. Symmetries of the Cartan distribution and the Lie-Bäcklund theorem.
11. Differential equations as geometric objects and their symmetries.
12. Symmetries of ordinary equations and Lie-Bianchi theorem on the integration by quadratures.