# 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

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

## Recommended literature

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