Program of the Second Italian
School
Forino, 17 July - 1 August 1998
1. Basic constructions in the first order algebraic differential calculus. de Rham and Spencer complexes on differential manifolds. Natural operations on differential forms and on jet forms. Differential homotopy. The 'covering pass' method in differential geometry. Classification of first degree forms (Darboux lemma) and of second degree closed forms. Symplectic and contact geometry. Hamiltonian mechanics. Morse lemma.
2. de Rham cohomology with values in a flat connection. Multiplicative structures on de Rham type cohomologies. Exterior and interior subcomplexes and related cohomological techniques. Computation rules for de Rham type cohomologies. de Rham theorem. de Rham cohomology of manifolds with singularities and of simplicial complexes.
3. Cohomological integration theory. Newton and Leibnitz formulae as boundary operators. Suspention isomorphism. n-dimensional cohomology of n-dimensional manifolds. Cohomological version of generalised Stokes formula. Numerical marking of n-dimensional cohomology classes. Cohomological nature of integral calculus.
4. Applications of cohomological integral calculus and de Rham cohomology. Degree of mappings between manifolds of the same dimension. Existence of symplectic structures on compact manifolds. First elements of characteristic classes.