For basic information about the course, see the course syllabus.
Since students will have to write a paper for the course, here's a list of suggested topics.
March 28: Smooth manifolds, vectors on manifolds. Optional reading: Bump, Chapter 6
March 30: Derivations, vector fields and flows. Optional reading: Bump, Chapter 6
April 1: Lie brackets of vector fields, commutators of flows. Lie algebras of Lie groups. Optional reading: Bump, Chapter 7,8
April 4: Lie algebras and Lie groups, examples, the Campbell-Baker-Hausdorff formula. Optional reading: Bump, Chapter 5
April 6: Uniqueness of Lie groups for Lie algebras, universal covers. Optional reading: Bump, Chapter 13
April 8: Fundamental groups of classical groups. Optional reading: Bump, Chapter 13
April 11: Fundamental groups and weight lattices.
April 13: The Weyl group and normalizers of tori.
April 15: Uniqueness of Cartans and Borels.
April 18: Compact groups, Peter-Weyl. Optional reading: Bump, Chapters 1-4
April 20: Compact Lie groups and Cartan involutions.
April 22: Uniqueness of compact forms, split forms.
May 9: Borel-Weil theorem.
May 11: Borel presentation of cohomology of G/B.