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

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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 bundle over a manifold and the category of projective modules over a commutative ring.
  9. Jets of locally trivial bundle over smooth manifolds. The Cartan distribution.
  10. Symmetries of the Cartan distribution and the Lie-Backlund theorem.
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