Joseph Krasil'shchik's introductory lectures on the Geometry of Differential Equations

Autumn 2010 - Spring 2011

Short syllabus

• Jets of bundles
  The Cartan distribution
  Symmetries of the Cartan distribution
  The Lie-Bäcklund theorem

• Nonlinear differential equations and operators
  Classical symmetries of differential equations, their computation and application
  The Bianchi-Lie theorem

• Infinite jets and geometric structures on them
  The Cartan distribution on infinite jets and its symmetries
  Linearizations and evolutionary vector fields

• Prolongations of differential equations
  Infinitely prolonged equations
  Higher symmetries of differential equations, their computation and application

• Recursion operators for symmetries
  Nijenhuis brackets

• Differential coverings and nonlocal symmetries

• The Vinogradov spectral sequence
  Homological foundations of Lagrangian formalism

• Schouten brackets and Hamiltonian structures

• Natural coverings in the category of differential equations and integrability

Video records of the lectures

Via http://erb-files.narod.ru

• Lecture 1 (15 September 2010)

• Lecture 2 (22 September 2010)

• Lecture 3 (6 October 2010)

• Lecture 4 (20 October 2010)

• Lecture 5 (27 October 2010)

• Lecture 6 (3 November 2010)

• Lecture 7 (10 November 2010)

• Lecture 8 (17 November 2010)

• Lecture 9 (24 November 2010) is missing

• Lecture 10 (1 December 2010), audio is missing

• Lecture 11 (8 December 2010)

• Lecture 12 (15 December 2010)

• Lecture 13 (9 February 2011)

• Lecture 14 (16 February 2011)

• Lecture 15 (2 March 2011)

• Lecture 16 (9 March 2011)

• Lecture 17 (16 March 2011)

• Lecture 18 (23 March 2011)

• Lecture 19 (30 March 2011)

• Lecture 20 (6 April 2011)

• Lecture 21 (13 April 2011)

• Lecture 22 (20 April 2011)

• Lecture 23 (27 April 2011)

• Lecture 24 (4 May 2011)

• Lecture 25 (11 May 2011)

• Lecture 26 (18 May 2011)

Recommended literature

A. V. Bocharov, A. M. Verbovetsky, A. M. Vinogradov (editor), S. V. Duzhin, I. S. Krasil'shchik (editor), A. V. Samokhin, Yu. N. Torkhov, N. G. Khor'kova, and V. N. Chetverikov Simmetrii i zakony sokhraneniya uravnenij matematicheskoj fiziki, Factorial, Moscow, 2nd edition, 2005.

(English translation: Symmetries and conservation laws for differential equations of mathematical physics, AMS, 1999.)