Math 215B Course Schedule
Below is the tentative schedule, which may be adjusted as necessary.
Week 1: (Jan 10-12)
- differentiable manifolds and tangent bundles; fiber bundles and vector bundles; manifolds with
boundary;
(Please review topological manifolds)
Week 2: (Jan 17-19)
-
Embeddings, immersions and submersions; Lie groups and principal bundles;
Week 3: (Jan 24-26)
-
Regular values and transversality; Sard Theorem and Ehresmann's fibration theorem;
Week 4: (Jan 31-Feb 2)
-
Embeddings and immersions in Euclidean space: existence (Whitney embedding theorems), obstructions, "turning a sphere inside out" ala Smale; tubular neighborhood theorem;
Week 5: (Feb 7- 9)
-
Vector fields and integral curves; flows; differential forms; connections and curvature
Week 6: (Feb 14-16)
-
deRham theory, Stokes theorem and Poincare lemma; orientation
Week 7: (Feb 21-23)
-
Classification of bundles; universal bundles and classifying spaces.
Week 8: (Feb 28-Mar 2)
-
Poincare duality, Thom isomorphism and intersection theory; degrees and linking numbers
Week 9: (Mar 7- 9)
-
Euler characteristic and self intersections; Lefschetz fixed-point theorem and Poincare-Hopf theorem;
Week 10: (Mar 14-16)
-
Classical Morse Theory and applications;