Sumários
Mayer-Vietoris sequence
9 novembro 2016, 14:00 • Sílvia Nogueira da Rocha Ravasco dos Anjos
Short exact sequence of complexes and the associated long exact sequence in cohomology. The Mayer-Vietoris Sequence. Example: computation of the cohomology of the sphere of dimension d.
Homotopy invariance
7 novembro 2016, 13:00 • Sílvia Nogueira da Rocha Ravasco dos Anjos
Computation of the cohomology of a circle. Homotopy invariance. Manifolds with the same homotopy type and contractible manifolds. Examples.
Integration of Differential forms. De Rham cohomology.
2 novembro 2016, 14:00 • Sílvia Nogueira da Rocha Ravasco dos Anjos
Integration of differential forms on manifolds and Stokes Theorem. The de Rham complex and its cohomology, an important invariant of a differentiable manifold.
Differential forms
31 outubro 2016, 13:00 • Sílvia Nogueira da Rocha Ravasco dos Anjos
Definition of differential forms and more generally, of tensor fields. The elementary operations with differential forms: exterior product, inner product and pull-back. The differential and the Lie derivative, which give rise to the Cartan calculus. Cartan's magic formula.
Transformation groups (cont.)
26 outubro 2016, 14:00 • Sílvia Nogueira da Rocha Ravasco dos Anjos
Smooth structures on homogeneous spaces. Examples. An infinitesimal action of a Lie algebra on a manifold is an anti-homomorphism of Lie algebras.