Math 225B: Differentiable Manifolds
|
| Date | Tentative topic |
Homework
|
| 1/9 |
Sard's theorem |
HW1, due 1/18 |
| 1/11 |
Transversality, Day I |
|
| 1/13 |
Transversality, Day II |
|
| 1/16 |
No class (Martin Luther King Day) |
|
| 1/18 |
Morse functions |
HW2, due 1/23 |
| 1/20 |
Whitney embedding theorem |
|
| 1/23 |
Orientations |
HW3, due 1/30 |
| 1/25 |
Oriented intersection numbers |
|
| 1/27 |
Degree |
|
| 1/30 |
Applications of degree: winding numbers |
HW4, due 2/6 |
| 2/1 |
More applications
of degree: Jordan-Brouwer separation theorem,
Bursuk-Ulam theorem |
|
| 2/3 |
The diagonal (more intersection theory; Euler
characteristic) |
|
| 2/6 |
Lefschetz fixed point theorem, Day I | HW5, due 2/13 |
| 2/8 |
Lefschetz fixed point theorem, Day II; Poincaré-Hopf theorem | |
| 2/10 |
Framed cobordisms and the Pontryagin construction, Day I | |
| 2/13 |
Framed cobordisms and the Pontryagin construction, Day II | HW6, due 2/22 |
| 2/15 |
Applications of
framed cobordisms including the Hopf degree
theorem |
|
| 2/17 |
Classification of vector bundles |
|
| 2/20 |
No class
(Presidents' Day) |
|
| 2/22 |
Cobordisms and Thom's work |
HW7, due 3/6 |
| 2/24 |
Poincaré duality, Day I | |
| 2/27 |
Poincaré duality, Day II | |
| 3/1 |
Thom isomorphism |
|
| 3/3 |
Consequences of
Thom isomorphism |
|
| 3/6 |
Hodge theory
preliminaries |
HW8, due 3/13 |
| 3/8 |
Hodge theory, Day I |
|
| 3/10 |
Hodge theory, Day II |
|
| 3/13 |
Sobolev spaces in a nutshell |
|
| 3/15 |
The basic estimate |
|
| 3/17 |
Sketch of proofs of Hodge theorem |
|
| Final exam
is take-home! |