The first call for applications for the new edition of the PhD Program in Mathematics UC|UP (starting September 2018) will be open during January 02, 2018 - February 02, 2018
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.