This is a Qualifying Eligible (QE) course for the Math PhD with regular, graded HW and a comprehensive final exam.
PrerequisitesStudents taking Math 620 are required to have taken real analysis (at the level of Math 531) and are encouraged to have studied point-set topology (at the level of Math 411). They are expected to be familiar with the following concepts: differentiation and integration in R^n, topological spaces, continuous maps, subspace topology, compactness.
- Basic constructions: smooth manifolds and maps, tangent and cotangent bundles, vector fields, differential forms, orientations, Lie bracket.
- Smooth maps: embeddings, immersions, submersions, Inverse Function Theorem, Rank Theorem, submanifolds.
- Lie groups and their Lie algebras, quotients of Lie group actions, Maurer-Cartan forms.
- Integration on manifolds, Stokes' Theorem.
- Flows of vector fields, foliations, Frobenius Theorem.
- Additional topics as time allows.
- John M. Lee, Introduction to Smooth Manifolds
- William Boothby, Introduction to Differentiable Manifolds and Riemannian Geometry