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

'''Autumn 2015'''

This course will be continued in [[Joseph Krasil'shchik's lectures on the geometry of infinitely prolonged differential equations|Spring 2016]]

Lectures takes place at [https://mccme.ru/en/nmu/ 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==
[[Media:IUM-lectures-2015.pdf|IUM-lectures-2015.pdf]]

==Video records of the lectures==

''Via [http://ium.mccme.ru/IUM-video.html http://ium.mccme.ru/IUM-video.html] and [http://www.mathnet.ru/php/conference.phtml?eventID=25&confid=763 Math-Net.Ru]''

* [http://video.gdeq.net/GDEq-lec-20150909.mp4 Lecture 1 (9 September 2015)] (or the [http://www.youtube.com/watch?v=vGI_VTSH-Kg same video on Youtube])
* [http://video.gdeq.net/GDEq-lec-20150916.mp4 Lecture 2 (16 September 2015)] (or the [http://www.youtube.com/watch?v=kxQR-GnXYyo 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-20151007.mp4 Lecture 5 (7 October 2015)] (or the [http://www.youtube.com/watch?v=VARbvG2DTkU same video on Youtube])
* [http://video.gdeq.net/GDEq-lec-20151014.mp4 Lecture 6 (14 October 2015)] (or the [http://www.youtube.com/watch?v=fHfRcDVLhAM 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])
* 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])

==Recommended literature==

*[[Media:Dzhet Nestruev (M.M.Vinogradov i dr.). Gladkie mnogoobrazija i nabljudaemye (2e izd., MCNMO, 2002)(ru)(600dpi)(T)(317s).djvu|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), [http://diffiety.ac.ru/preprint/99/01_99.pdf DIPS 1/99], [[Media:Joseph_Krasil%27shchik_in_collaboration_with_Barbara_Prinari%2C_Lectures_on_linear_differential_operators_over_commutative_algebras_%28The_1st_Italian_diffiety_school%2C_July%2C_1998%29%2C_Diffiety_Inst._Preprint_Series_1_%281999%29%2C_DIPS_1_99.pdf|local copy]].
* I. S. Krasil'shchik and A. M. Verbovetsky, ''Homological methods in equations of mathematical physics'', Open Education &amp; Sciences, Opava, 1998, {{arXiv|math/9808130}}.
*[[Media:At'ja M., Makdonal'd I. Vvedenie v kommutativnuju algebru (Mir, 1972)(ru)(T)(160s).djvu|At'ja M., Makdonal'd I. Vvedenie v kommutativnuju algebru (Mir, 1972)(ru)(T)(160s)]]
*[[Media:Maklejn S. ( Mac Lane S. ) Kategorii dlya rabotayushchego matematika (FML, 2004)(ISBN 5922104004)(ru)(600dpi)(T)(O)(353s) MAct .djvu|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.
[[Category:Lectures|Krasil'shchik 7984a]]