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-20150923.mp4 Lecture 3 (23 September 2015)] (or the [http://www.youtube.com/watch?v=w-DG6R9sczw same video on Youtube]) | * [http://video.gdeq.net/GDEq-lec-20150923.mp4 Lecture 3 (23 September 2015)] (or the [http://www.youtube.com/watch?v=w-DG6R9sczw same video on Youtube]) | ||
* [http://video.gdeq.net/GDEq-lec-20150930.mp4 Lecture 4 (30 September 2015)] (or the [http://www.youtube.com/watch?v=jzFp64xjjng same video on Youtube]) | * [http://video.gdeq.net/GDEq-lec-20150930.mp4 Lecture 4 (30 September 2015)] (or the [http://www.youtube.com/watch?v=jzFp64xjjng same video on Youtube]) | ||
* [http://video.gdeq.net/GDEq-lec-20151007.mp4 Lecture 5 (7 October 2015)] (or the [http://www.youtube.com/watch?v=VARbvG2DTkU same video on Youtube]) | |||
==Recommended literature== | ==Recommended literature== | ||
Revision as of 18:53, 12 October 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
IUM-lectures-2015-10-01-v3.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)
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)