Joseph Krasil'shchik's lectures on the linear differential operators over commutative algebras and geometry of jet spaces: Difference between revisions
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* [http://video.gdeq.net/GDEq-lec-20151028.mp4 Lecture 7 (28 October 2015)] (or the [http://www.youtube.com/watch?v=L904dSjeqX4 same video on Youtube]) | * [http://video.gdeq.net/GDEq-lec-20151028.mp4 Lecture 7 (28 October 2015)] (or the [http://www.youtube.com/watch?v=L904dSjeqX4 same video on Youtube]) | ||
* [http://video.gdeq.net/GDEq-lec-20151111.mp4 Lecture 8 (11 November 2015)] (or the [http://www.youtube.com/watch?v=callSlNSZfM same video on Youtube]) | * [http://video.gdeq.net/GDEq-lec-20151111.mp4 Lecture 8 (11 November 2015)] (or the [http://www.youtube.com/watch?v=callSlNSZfM same video on Youtube]) | ||
* Lecture 9 (18 November 2015) is missing | |||
* [http://video.gdeq.net/GDEq-lec-20151125.mp4 Lecture 10 (25 November 2015)] (or the [http://www.youtube.com/watch?v=DfVFuQLtu5M same video on Youtube]) | * [http://video.gdeq.net/GDEq-lec-20151125.mp4 Lecture 10 (25 November 2015)] (or the [http://www.youtube.com/watch?v=DfVFuQLtu5M same video on Youtube]) | ||
Revision as of 20:29, 9 December 2015
Autumn 2015
Lectures takes place at the Independent University of Moscow on Wednesday evenings in room 303 from 17:30 to 19:10
Syllabus
- Categories and functors (introduction).
- Linear differential operators with values in modules. Main properties.
- Derivations.
- Representative objects: jets and differential forms.
- Differential calculus over commutative algebras.
- Schouten-Nijenhuis brackets and related cohomologies. Algebraic model of Hamiltonian formalism.
- Frölicher-Nijenhuis brackets and related cohomologies. Algebraic model of nonlinear differential equations.
- Geometric realization. Relation between the category of vector bundles over a manifold and the category of projective modules over a commutative ring.
- Jets of locally trivial bundles over smooth manifolds. The Cartan distribution.
- Symmetries of the Cartan distribution and the Lie-Bäcklund theorem.
- Differential equations as geometric objects and their symmetries.
- Symmetries of ordinary equations and Lie-Bianchi theorem on the integration by quadratures.
Lecture notes and problems
Video records of the lectures
Via http://ium.mccme.ru/IUM-video.html and Math-Net.Ru
- 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)
- Lecture 9 (18 November 2015) is missing
- Lecture 10 (25 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.