Math 9071: Topology of fiber bundles (Differential Topology) (syllabus, also here). (Detailed syllabus not yet posted)
Lectures: Wachman 527, TR 12:30-1:50
Office Hours: Tuesday and Thursday, 11:00-12:00. Office hours by appointment are also possible, arranged one day in advance. Or just come to my office: if I have time, I'll see you
The course will begin with introductory lectures on general facts about fiber bundles, but will focus primarily on topological aspects of vector bundles. Topics include Chern classes (and obstructions to triviality), connections, an introduction to Lie groups as needed for the study of principal G-bundles, groups of bundle isomorphism of a given vector bundle, special structures of the tangent bundle (involutive structures such as complex and CR structures). Time permitting, there will be an introduction to K-theory and the Atiyah-Singer index theorem.
Notes by the instructor based on a book in progress.
Kobayashi: Differential Geometry of Complex Vector Bundles.
Kobayashi: Transformation Groups in Differential Geometry.
Milnor & Stasheff: Characteristic Classes.
Husemoller: Fibre Bundles.
Gilkey: Invariance Theory, The Heat Equation, and the Atiyah-Singer Index Theorem
MATH 8061-8062. Differential Geometry and Topology I, II.