PT EN

Differentiable Manifolds

Program

Differentiable manifolds: local structure, submanifolds, Sard's Theorem, transversality, vector fields and flows. Fibre bundles, tangent and cotangent bundles of a manifold. Lie derivative of vector fields. Lie algebras. Lie groups (classical). Homogeneous spaces. Differential forms, exterior derivative. Symplectic forms. Integration on manifolds. Stokes' theorem. One or more of the following additional topics may also be covered:
- Riemannian manifolds. Curvature. Symmetric spaces. Classical examples.
- de Rham cohomology, singular cohomology and de Rham's theorem.
- Degree of a map. Index of a vector field. Applications.
- Distributions. Frobenius and Stefan-Sussmann theorems.

Research and Events

Events

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Defended Theses

  • Some aspects of descent theory and applications
      Rui Rodrigues de Abreu Fernandes Prezado (January 2024)
      Maria Manuel Clementino
      Fernando Lucatelli Nunes
  • Comparability between different systems: star-shaped and convex transform orders
      Beatriz Ferreira Santos (December 2023)
      Paulo Eduardo Oliveira
      Idir Arab
  • On Lax Idempotent Monads in Topology
      Carlos Miguel Alves Fitas (December 2023)
      Maria Manuel Clementino
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