Kostroma, Russia,
February 1 - February 12, 2007
Last updated March 13, 2007

This school is organized in cooperation with and under the scientific direction of Prof. A. M. Vinogradov (Università di Salerno, Italy).


  1. Secondary Calculus
  2. Courses
  3. Prerequisites for beginers
  4. List of participants
  5. Seminars and Diplomas
  6. Organizing committee
  7. School photos. (last updated April, 2007.)
  8. 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.

First Order Calculus on Manifolds (by G. Moreno)
  1. 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.
  2. 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.
  3. 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.
  4. 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.
  5. 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.
  6. 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.

Theory of Distributions and Contact Geometry (by M. Bächtold)
  1. Distributions, their Symmetries and Characteristics
  2. Theorem of Frobenius
  3. Lemma of Darboux closed for 2-forms
  4. Morse Lemma for isolated singularities of functions
  5. Lemma of Darboux for 1-forms
  6. Local classification of distributions of co-dimension 1
  7. Contact manifolds and the Jacobi bracket
  8. Symplectic manifolds and the Poisson bracket

Introduction to Geometry of Finite Jet Spaces (by L. Vitagliano) --
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.

List of participants:

Also among participants:

Seminars and Diplomas

Diplomas of participation in the School were handed to all participants. Moreover, there were examinations in all courses, which were organized as follows. For each course, students received a list of examination problems. To pass an examination, one had to solve a reasonably large number of problems. Students having passed an examination received diplomas certifying this fact.

The following students have passed examinations in Introduction to Differential Calculus over Commutative Algebras: The following students have passed examinations in First Order Calculus on Manifolds: The following students have passed examinations in Theory of Distributions and Contact Geometry: The following students have passed examinations in Introduction to Geometry of Finite Jet Spaces: The following student have passed examinations in Classical Mechanics and Symplectic and Poisson Geometry:

Organizing committee:

M. Bächtold, C. Di Pietro, G. Moreno, V. Kalnitsky, R. Piscopo, M. M. Vinogradov, L. Vitagliano.

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