The course is continuation of the course by Joseph Krasil'shchik taken in the Spring semester. The participants are assumed to be acquainted with the basics in the geometry of differential equations, though the first lectures will include a short introduction. See also Symmetries and Conservation Laws for Equations of Mathematical Physics, Moscow, The Factorial Publ. House, 1997.

1. Introduction
   • The basic geometric structures on infinitely prolonged differential equations (reminder)
   • Algebra of linear differential operators over commutative algebras (reminder and continuation)

2. Cohomological theory of symmetries and recursion operators

   • An algebraic model of differential equations: algebras with connections
   • Algebra of vector-valued differential forms and the Frölicher-Nijenhuis bracket
   • A complex associated to a flat connection
   • C-cohomology of differential equations
   • Geometric meaning of the groups H_Cⁱ, i = 0,1,2

3. Recursion operators and nonlocal theory

   • Coverings in the category of differential equations
   • Examples and applications
   • Nonlocal symmetries and shadows
   • Action of recursion operators in algebras of nonlocal symmetries

4. Cohomological theory of conservation laws and Lagrangian formalism with constraints

   • Algebraic theory of the adjoint operator and Green's formula
   • The C-spectral sequence
   • The terms E₀ and E₁
   • The Noether theorem, the inverse problem of variational calculus

5. Horizontal cohomology of differential equations

   • Horizontal cohomology with coefficients in linear coverings
   • The horizontal compatibility complex
   • Applications to the C-cohomology and to the C-spectral sequence: k-line theorem