Alexandre Vinogradov

A program of the course at the 5-th Italian Diffiety School,

(San Stefano del Sole, July 19-31, 2002)

1.1.General Introduction

- Observation mechanism in classical physics
- Commutative algebras
- States of a classical system
- Spectra of a commutative algebra
- Spectral theorem (manifolds as spectra of commutative algebras)

- Tangent vectors and absolute and relative vector fields; D functor
- Behaviour of tangent vectors and vector fields with respect to smooth mappings of manifolds
- The flow of a vector field; Lie derivative of vector fields; commutators and Lie algebras
- Tangent covectors and differential forms; tensors; main operations on tensors; algebra of differential forms
- Behaviour of differential forms and covariant tensors with respect to differentiable mappings of manifolds; Lie derivatives of covariant tensors
- Exterior differential and de Rham cohomology
- Cartan's formula and homotopy property of de Rham cohomology
- Cohomological theory of integration

- Rings and commutative algebrae; modules and bi-modules
- Linear differential operators between modules
- Comparison with the analytical definition of differential operator
- Examples of algebraic differential operators
- Bi-module structure in the set of linear differential operators
- Composition of differential operators
- Symbol of a linear differential operator; algebra of symbols
- Categories and functors; representative objects
- Functors of the algebraic calculus and their representability
- Derivations and multi-derivations
- Differential forms and the de Rham complex
- Interior product and Lie derivative. Comparison with the geometric approach

- Valentina Golovko (University of Moscow)
- Leonid Stunzhas (University of Moscow)
- Angela Cardone (University of Naples)
- Antonio De Nicola (University of Naples)
- Daniele Signori (University of Milano)
- Ida Del Prete (University of Naples)
- Jan Tuitman (University of Groningen)
- Nicola Carchia (University of Naples)
- J.H. Sikkema (University of Groningen)
- Silvana Martucci (University of Salerno)
- Giovanni Moreno (University of Naples)

Questions and suggestions should go to school @ diffiety.ac.ru.