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

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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

Video records of the lectures

Via http://erb-files.narod.ru and http://ium.mccme.ru/IUM-video.html

  • Lecture 9 (24 November 2010) is missing

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.)