Open Education, Opava, 1998, 150 pp.
See also Diffiety Inst. Preprint Series,
DIPS 7/98.
Contents

Introduction 1 Differential calculus over commutative algebras 1.1 Linear differential operators 1.2 Multiderivations and the Diff-Spencer complex 1.3 Jets 1.4 Compatibility complex 1.5 Differential forms and the de Rham complex 1.6 Left and right differential modules 1.7 The Spencer cohomology 1.8 Geometrical modules 2 Algebraic model for Lagrangian formalism 2.1 Adjoint operators 2.2 Berezinian and integration 2.3 Green's formula 2.4 The Euler operator 2.5 Conservation laws 3 Jets and nonlinear differential equations. Symmetries 3.1 Finite jets 3.2 Nonlinear differential operators 3.3 Infinite jets 3.4 Nonlinear equations and their solutions 3.5 Cartan distribution on J^{k}(p) 3.6 Classical symmetries 3.7 Prolongations of differential equations 3.8 Basic structures on infinite prolongations 3.9 Higher symmetries 4 Coverings and nonlocal symmetries 4.1 Coverings 4.2 Nonlocal symmetries and shadows 4.3 Reconstruction theorems 5 Fröliher-Nijenhuis brackets and recursion operators 5.1 Calculus in form-valued derivations 5.2 Algebras with flat connections and cohomology 5.3 Applications to differential equations: recursion operators 5.4 Passing to nonlocalities 6 Horizontal cohomology 6.1C-modules on differential equations 6.2 The horizontal de Rham complex 6.3 Horizontal compatibility complex 6.4 Applications to computing the C-cohomology groups 6.5 Example: Evolution equations 7 Vinogradov's C-spectral sequence 7.1 Definition of the Vinogradov C-spectral sequence 7.2 The term E_{1} for J^{¥}(p) 7.3 The term E_{1} for an equation 7.4 Example: Abelian p-form theories 7.5 Conservation laws and generating functions 7.6 Generating functions from the antifield-BRST standpoint 7.7 Euler-Lagrange equations 7.8 The Hamiltonian formalism on J^{¥}(p) 7.9 On superequations 8 Appendix: Homological algebra 8.1 Complexes 8.2 Spectral sequences
References