Kostroma, Russia,

February 1 - February 12, 2007

This school is organized in cooperation with

- the "
**Bol'shaya peremena**" programm of Michael Batin (Kostroma, Russia); - Istituto Italiano per gli Studi Filosofici, (Naples, Italy);

- Secondary Calculus
- Courses
- Prerequisites for beginers
- List of participants
- Seminars and Diplomas
- Organizing committee
- School photos. (last updated April, 2007.)
- Previous schools

**Introduction to Differential Calculus over Commutative Algebras**(by A. M. Vinogradov):- the course aimed to show that the natural language of classical physics is differential calculus over commutative algebras and that this fact is a consequence of the classical observability mechanism. As a key example, calculus over smooth manifolds was developed according to this philosophy, i.e., "algebraically". For instance, it was shown that differential geometry can be developed over an arbitrary commutative algebra.
- Tangent vectors (coordinate description). Controvariance. Differential of a smooth map. Standard basis of T_pM. Matrix representation of the differential. Tangent manifold. Covectors (coordinate description). Covariance.
- Smooth bundles. Examples. Morphisms. Classification of bundles over S^1. Bundles with Stiefel and Grassmann manifolds. Hopf fibration. Vector bundles. Morphisms. Induced bundles. Sections. Vector fields and differential forms as smooth sections. Tensor fields. Vector fields along a smooth map.
- Local basis of vector fields. Action of diffeomorphisms on vector fields. Relative and compatible vector fields (geometrical and algebraic interpretation). Trajectories of a vector field. Introduction to transformations and infinitesimal transformations. One-parameter groups of transformations. Flow of a vector field. Local groups. Infinitesimal transformations of vector fields and Lie derivatives. Commutator of vector fields.
- Differential forms as multilinear skew-symmetric maps on the module of vector fields. Wedge product. Graded algebra structure. Differentials of functions and exact 1-forms. Differentials of forms. De Rham complex. Pull-back of forms associated with a smooth map. Infinitesimal transformations of forms. Operators of insertion of vector fields. Lie derivatives of differential forms. Cartan Formula.
- Formula for the commutator of a Lie derivative and an insertion operator. Formula for the Lie derivative of a k-form evaluated on k vector fields. Lie derivatives of tensors. Symmetries of geometric objects. Complexes, differential algebras and their cohomologies. Morphisms of complexes and induced morphisms in cohomology. Algebraic and geometrical homotopies. Infinitesimal homotopies, associated Lie derivatives and Cartan formula. Homotopy formula. Poincare Lemma and its generalization to vector bundles. Long exact sequence in cohomology associated with a short exact sequence of complexes.
- Cohomological interpretation of the Newton-Leibniz formula. Cohomological definition of the integration over [0,1]. Invariance under orientation- preserving diffeomorphisms. Sketch of how to define integration over any orientable manifold in a purely cohomological way.
- Distributions, their Symmetries and Characteristics
- Theorem of Frobenius
- Lemma of Darboux closed for 2-forms
- Morse Lemma for isolated singularities of functions
- Lemma of Darboux for 1-forms
- Local classification of distributions of co-dimension 1
- Contact manifolds and the Jacobi bracket
- Symplectic manifolds and the Poisson bracket
- Finite Jets
- Jets of submanifolds.

Jets of sections.

Canonical coordinates on jet spaces associated with adapted coordinate systems.

Basic constructions with jets spaces. - Geometry of Jet Spaces
- R- planes.

The Cartan distribution, Cartan fields, Cartan forms.

Structure of the Cartan Distribution.

Ray submanifolds.

Structure of maximal integral manifolds of the Cartan distribution. - Geometry of PDE’s
- Differential equations.

Multivalued solutions of PDE’s.

Classical symmetries of a differential equation.

Higher Order Contact Trasformations

- Contact transformations.
- Lie transformations, point transformations.

Lie-Bæcklund theorem.

Lie Fields.

Lifting of a Lie Fields.

Genereting section of a Lie field.

Jacobi brackets. **Classical Mechanics and Symplectic and Poisson Geometry**(by A. M. Vinogradov):- The course aimed to introduce to symplectic and Poisson Geometry as geometrical foundations of classical mechanics.

- Andrey Ardentov (Pereslavl, Russia),
- Svetlana Azarina (Voronezh, Russia),
- Irina Bobkova (Saint-Petersburg, Russia),
- Andrey Borozdin (Saint-Petersburg, Russia),
- Maxim Brodovskii (Ivanovo, Russia),
- Ilya Kachkovskiy (Saint-Petersburg, Russia),
- Victor Kasatkin (Saint-Petersburg, Russia),
- Vladimir Kotov (Saint-Petersburg, Russia),
- Nastya Lashova (Saint-Petersburg, Russia),
- Vitaliy Levtchenko (Saint-Petersburg, Russia),
- Eugenia Lysenko (Saint-Petersburg, Russia),
- Alexey Mashtakov (Pereslavl, Russia),
- Yurii Naumov (Ivanovo, Russia),
- Dmitri Pavlov (Saint-Petersburg, Russia),
- Artem Pimachev (Saint-Petersburg, Russia),
- Svyatoslav Pimenov (Saint-Petersburg, Russia),
- Darya Romanova (Saint-Petersburg, Russia),
- Olga Sergeeva (Saint-Petersburg, Russia),
- Andrei Shevliakov (Moscow, Russia),
- Alexandre Smirnov (Saint-Petersburg, Russia),
- Leonid Sudov (Saint-Petersburg, Russia),
- Maxim Tkachuk (Kiev, Ukraine).

**Also among participants:**

- Prof. Olga Kunakovskaya (Voronezh, Russia),
- Prof. Andrey Obukhovski (Voronezh, Russia),
- Prof. Yurii Zelinskii (Kiev, Ukraine),
- Prof. Mikhail Zvagelsky (Saint-Petersburg, Russia).

The following students have passed examinations in

- Andrey Ardentov (Pereslavl, Russia),
- Andrey Borozdin (Saint-Petersburg, Russia),
- Ilya Kachkovskiy (Saint-Petersburg, Russia),
- Victor Kasatkin (Saint-Petersburg, Russia),
- Vitaliy Levtchenko (Saint-Petersburg, Russia),
- Evgenija Lysenko (Saint-Petersburg, Russia),
- Artem Pimachev (Saint-Petersburg, Russia),
- Darya Romanova (Saint-Petersburg, Russia),
- Olga Sergeeva (Saint-Petersburg, Russia).

- Andrey Borozdin (Saint-Petersburg, Russia),
- Ilya Kachkovskiy (Saint-Petersburg, Russia),
- Victor Kasatkin (Saint-Petersburg, Russia),
- Vladimir Kotov (Saint-Petersburg, Russia),
- Vitaliy Levtchenko (Saint-Petersburg, Russia),
- Darya Romanova (Saint-Petersburg, Russia),
- Olga Sergeeva (Saint-Petersburg, Russia).

- Ilya Kachkovskiy (Saint-Petersburg, Russia),
- Victor Kasatkin (Saint-Petersburg, Russia),
- Vladimir Kotov (Saint-Petersburg, Russia).

- Irina Bobkova (Saint-Petersburg, Russia),
- Andrey Borozdin (Saint-Petersburg, Russia),
- Ilya Kachkovskiy (Saint-Petersburg, Russia),
- Victor Kasatkin (Saint-Petersburg, Russia),
- Vladimir Kotov (Saint-Petersburg, Russia),
- Darya Romanova (Saint-Petersburg, Russia),
- Olga Sergeeva (Saint-Petersburg, Russia).

- Andrey Borozdin (Saint-Petersburg, Russia).

Questions and suggestions should go to