# Joseph Krasil'shchik's lectures on the geometry of infinitely prolonged differential equations

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**Spring 2016** (continuation of Autumn 2015 lectures, but can be followed independently)

Lectures will take place at the Independent University of Moscow on Wednesday evenings in room 303 from 17:30 to 19:10

## Syllabus

- Spaces and bundles of infinite jets. The Cartan distribution on the space .
- Symmetries of the Cartan distribution on . Evolutionary derivations.
- Linearizations of nonlinear differential operators and lifts of linear ones. Cartan connection. -differential operators. Horizontal de Rham complex. Adjoint operator and the Green formula.
- Infinitely prolonged differential equations. Higher symmetries and their generating functions. Theorem about the structure of symmetries.
- Examples of computing higher symmetries. Recursion operators.
- Homological theory of recursion operators. Variational Nijenhuis bracket.
- Nonlocal geometry of infinitely prolonged equations. Differential coverings. Nonlocal symmetries and shadows. Examples of computations. Bäcklund transformations.
- Conservation laws and Abelian coverings. Generating functions of conservation lwas and the Vinogradov -spectral sequaence.
- Geometric theory of recursion operators. The tangent covering. Variational symplectic structures.
- Cotangent covering. Variational Poisson structures. (if time allows)

## Lecture notes and problems

## Video records of the lectures

*Via https://ium.mccme.ru/IUM-video.html and Math-Net.Ru*

- Lecture 1 (17 February 2016) (or the same video on Youtube)
- Lecture 2 (24 February 2016) is missing
- Lecture 3 (2 March 2016) (or the same video on Youtube)
- Lecture 4 (9 March 2016) (or the same video on Youtube)
- Lecture 5 (16 March 2016) is missing
- Lecture 6 (23 March 2016) is missing
- Lecture 7 (30 March 2016) (or the same video on Youtube)
- Lecture 8 (6 April 2016) (or the same video on Youtube)
- Lecture 9 (13 April 2016) (or the same video on Youtube)
- Lecture 10 (20 April 2016) (or the same video on Youtube)
- Lecture 11 (27 April 2016) (or the same video on Youtube)
- Lecture 12 (11 May 2016) (or the same video on Youtube)

## Recommended literature

- Виноградов А.М., Красильщик И.С., Лычагин В.В.
*Введение в геометрию нелинейных дифференциальных уравнений*. М.: Наука. Гл. ред. физ.-мат. лит., 1986. -- 336 с. - Виноградов А.М., Красильщик И.С. (Ред.)
*Симметрии и законы сохранения уравнений математической физики*. Серия: XX век. Математика и механика, Факториал, 2005, Вып. 9, Изд. 2. 380 с. - I. S. Krasil'shchik and A. M. Verbovetsky,
*Homological methods in equations of mathematical physics*, Open Education & Sciences, Opava, 1998, arXiv:math/9808130.