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

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Autumn 2015

This course will be continued in Spring 2016

Lectures takes place at the Independent University of Moscow on Wednesday evenings in room 303 from 17:30 to 19:10

Syllabus

  1. Categories 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. Schouten-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.

Lecture notes and problems

IUM-lectures-2015.pdf

Video records of the lectures

Via http://ium.mccme.ru/IUM-video.html and Math-Net.Ru

Recommended literature