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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* [[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).
* [[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]].
* 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]].
 
* [[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)]]
[[Category:Lectures|Krasil'shchik 7984a]]
[[Category:Lectures|Krasil'shchik 7984a]]

Revision as of 17:12, 17 September 2015

Autumn 2015

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

The first lecture will meet on 9 September

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-2015-09-17.pdf

Video records of the lectures

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

Recommended literature

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