Joseph Krasil'shchik's lectures on the linear differential operators over commutative algebras and geometry of jet spaces

Autumn 2015

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

Syllabus

1. Categories and functors (introduction).
2. Linear differential operators with values in modules. Main properties.
3. Derivations.
4. Representative objects: jets and differential forms.
5. Differential calculus over commutative algebras.
6. Schouten-Nijenhuis brackets and related cohomologies. Algebraic model of Hamiltonian formalism.
7. Frölicher-Nijenhuis brackets and related cohomologies. Algebraic model of nonlinear differential equations.
8. Geometric realization. Relation between the category of vector bundles over a manifold and the category of projective modules over a commutative ring.
9. Jets of locally trivial bundles over smooth manifolds. The Cartan distribution.
10. Symmetries of the Cartan distribution and the Lie-Bäcklund theorem.
11. Differential equations as geometric objects and their symmetries.
12. Symmetries of ordinary equations and Lie-Bianchi theorem on the integration by quadratures.

Lecture notes and problems

IUM-lectures-2015-11-18.pdf

Video records of the lectures

Via http://ium.mccme.ru/IUM-video.html, Math-Net.Ru, and YouTube

• Lecture 1 (9 September 2015) (or the same video on Youtube)
• Lecture 2 (16 September 2015) (or the same video on Youtube)
• Lecture 3 (23 September 2015) (or the same video on Youtube)
• Lecture 4 (30 September 2015) (or the same video on Youtube)
• Lecture 5 (7 October 2015) (or the same video on Youtube)
• Lecture 6 (14 October 2015) (or the same video on Youtube)
• Lecture 7 (28 October 2015) (or the same video on Youtube)
• Lecture 8 (11 November 2015) (or the same video on Youtube)

Recommended literature

• Dzhet Nestruev (M.M.Vinogradov i dr.). Gladkie mnogoobrazija i nabljudaemye (2e izd., MCNMO, 2002)(ru)(600dpi)(T)(317s), (English translation: Nestruev J. Smooth manifolds and observables, Springer, 2003).
• Joseph Krasil'shchik in collaboration with Barbara Prinari, Lectures on linear differential operators over commutative algebras. (The 1st Italian diffiety school, July, 1998), Diffiety Inst. Preprint Series 1 (1999), DIPS 1/99, local copy.
• I. S. Krasil'shchik and A. M. Verbovetsky, Homological methods in equations of mathematical physics, Open Education & Sciences, Opava, 1998, arXiv:math/9808130.
• At'ja M., Makdonal'd I. Vvedenie v kommutativnuju algebru (Mir, 1972)(ru)(T)(160s)
• Maklejn S. ( Mac Lane S. ) Kategorii dlya rabotayushchego matematika (FML, 2004)(ISBN 5922104004)(ru)(600dpi)(T)(O)(353s)
• 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.