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

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

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

Lecture notes and problems

IUM-lectures-2016.pdf

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.