Program of the First Italian School

Forino (AV), 17 July - 1 August 1998





0. General Introduction - A. Vinogradov



A comparison between Classical Physics and Mathematics:
  1. Observation mechanism in classical Physics
  2. Commutative algebrae
  3. States of a classical system
  4. Spectra of a commutative algebra
Spectral Theorem (manifolds as spectra of commutative algebrae).
Application: why it is wrong to regard set theory as the basis of mathematics.
Classical differential calculus: the ideas from mechanics and geometry which led to it.
Differential calculus on commutative algebrae. Theorem: classical differential calculus as a particular case of algebraic differential calculus.
Application: transforming algebraic geometry into differential geometry
Observability in Physics and Mathematics: paradoxes of actual paradigms.
What kind of mathematics is necessary to Quantum Observability? A first idea of Secondary Calculus.



1. First order differential calculus on manifolds

A. M. Vinogradov



  1. Tangent vectors and absolute and relative vector fields. D functor.
  2. Behaviour of tangent vectors and vector fields with respect to differentiable mappings of manifolds.
  3. The flow of a vector field. Lie derivative of vector fields. Commutators and Lie algebrae.
  4. Tangent covectors and differential forms. Tensors. Main operations on tensors. Algebra of differential forms.
  5. Behaviour of differential forms and covariant tensors with respect to differentiable mappings of manifolds. Lie derivatives of covariant tensors.
  6. Exterior differential and de Rham cohomology.
  7. Cartan's formula and homotopy property of de Rham cohomology.
  8. Integration theory as de Rham cohomology.




2. Introduction to differential calculus on commutative algebrae

I.S. Krasil'shchick



  1. Rinigs and commutative algebrae. Modules and bi-modules.
  2. Linear differential operatirs between modules
  3. Comparison with the analytical definition of differential operator.
  4. Examples of algebraic differential operators.
  5. Bi-module structure in the set of linear differential operators.
  6. Composition of differential operators
  7. Symbol of a linear differential operator. Algebra of symbols.
  8. Categories and functors, representative objects.
  9. Functors of the algebraic calculus and their representability.
  10. Derivations and multi-derivations.
  11. Differential forms and de Rham complex.
  12. Interior product and lie derivative. Comparison with the geometric approach.