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

Revision as of 17:16, 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

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.

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.
At'ja M., Makdonal'd I. Vvedenie v kommutativnuju algebru (Mir, 1972)(ru)(T)(160s)

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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).
 * 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)]]
+* [[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]]