Last updated January 14, 1999
Geometry of nonlinear differential equations


Necessary background:


  1. Calculus on smooth manifolds: vector fields, differential forms, distributions.
  2. Locally-trivial and vector bundles.
  3. Category theory: categories, functors, diagrams.
  4. Commutative algebra: rings, algebras, modules, spectres.
A course program


Jets and the classical theory of symmetries.
  1. Jet manifolds and jet bundles. Canonical coordinates.
  2. Linear differential operators and liftings.
  3. Nonlinear operators. Compositions and prolongations.
  4. Equations. Solutions. Coni and types.
  5. The Cartan distribution. Structure of integral manifolds.
  6. Symmetries and generating sections. Applications.
  7. Internal and external symmetries.
Infinite jets and the theory of higher symmetries

  1. Elements of calculus in commutative algebras.
  2. Infinite jets. Infinite prolongations of nonlinear operators.
  3. The Cartan distribution on infinite jets. Integrability and structure of integral manifolds.
  4. Infinite prolongations of differential equations. Formal integrability.
  5. Symmetries of the Cartan distribution on infinite jets. Evolution derivations and linearizations.
  6. Higher symmetries of nonlinear equations. The structural theorem.
  7. Examples.
Coverings and nonlocal theory

  1. The category of differential equations (CDE): objects and morphisms.
  2. Coverings in CDE. Examples.
  3. Backlund transformations. Recursion operators.
  4. Nonlocal symmetries and shadows. Examples.
  5. The universal Abelian covering. The reconstruction theorem.
Relations to homology algebra and perspectives


Problems: tex source, PostScript, dvi
Questions and suggestions should go to J. S. Krasil'shchik, josephk @ diffiety.ac.ru.