Math 620 Syllabus

This is a Qualifying Eligible (QE) course for the Math PhD with regular, graded HW and a comprehensive final exam.


Students 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.


  1. Basic constructions: smooth manifolds and maps, tangent and cotangent bundles, vector fields, differential forms, orientations, Lie bracket.
  2. Smooth maps: embeddings, immersions, submersions, Inverse Function Theorem, Rank Theorem, submanifolds.
  3. Lie groups and their Lie algebras, quotients of Lie group actions, Maurer-Cartan forms.
  4. Integration on manifolds, Stokes' Theorem.
  5. Flows of vector fields, foliations, Frobenius Theorem.
  6. Additional topics as time allows.


  • John M. Lee,  Introduction to Smooth Manifolds
  • William Boothby, Introduction to Differentiable Manifolds and Riemannian Geometry