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Differential Calculus over Commutative Algebras and Smooth Manifolds
Alexandre Vinogradov
A program of the course at the 5-th Italian Diffiety School,
(San Stefano del Sole, July 19-31, 2002)

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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)

1.2. First order differential calculus on manifolds

• 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

1.3. Introduction to differential calculus on commutative algebrae

• 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

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The exam has been passed by the following students:

1. Valentina Golovko (University of Moscow)
2. Leonid Stunzhas (University of Moscow)
3. Angela Cardone (University of Naples)
4. Antonio De Nicola (University of Naples)
5. Daniele Signori (University of Milano)
6. Ida Del Prete (University of Naples)
7. Jan Tuitman (University of Groningen)
8. Nicola Carchia (University of Naples)
9. J.H. Sikkema (University of Groningen)
10. Silvana Martucci (University of Salerno)
11. Giovanni Moreno (University of Naples)

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