Joseph Krasil'shchik's lectures on the cohomological invariants of nonlinear differential equations

Autumn 2016 (continuation of Spring 2016 and 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. Reminder: infinite jets, infinitely prolonged differential equations and geometric structures on them.
2. The Vinogradov ${\mathcal {C}}$-spectral sequence (variational bicomplex) on $J^{\infty }(\pi )$. One-line theorem.
3. Cohomological framework of the Lagrangian formalism
4. The ${\mathcal {C}}$-spectral sequence of an infinitely prolonged equation.
5. Compatibility complex of a ${\mathcal {C}}$-differential operator and the $p$-lines theorem.
6. Reminder: nonlocal geometry of equations and differential coverings.
7. Variational symplectic structures.
8. Two-line equations. Cotangent covering.
9. Variational Schouten bracket.
10. Hamiltonian equations. Variational Poisson bracket.
11. Compatible Poisson structures. Bi-Hamiltonian equations. Magri theorem.

Video records of the lectures

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

• Lecture 1 (14 September 2016) (or the same video on Youtube)
• Lecture 2 (21 September 2016) (or the same video on Youtube)
• Lecture 3 (28 September 2016) (or the same video on Youtube)
• Lecture 4 (5 October 2016) (or the same video on Youtube)
• Lecture 5 (19 October 2016) (or the same video on Youtube)
• Lecture 6 (26 October 2016) (or the same video on Youtube)
• Lecture 7 (2 November 2016) (or the same video on Youtube)
• Lecture 8 (9 November 2016) (or the same video on Youtube)
• Lecture 9 (16 November 2016) (or the same video on Youtube)
• Lecture 10 (23 November 2016) (or the same video on Youtube)
• Lecture 11 (30 November 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.