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! |