Differential Calculus over Commutative Algebras and
Smooth Manifolds
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)
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
Questions and suggestions should go to
Jet NESTRUEV, jet @ diffiety.ac.ru.