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: JordanBrouwer separation theorem,
BursukUlam 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; PoincarĂ©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 
PoincarĂ© duality, Day I  
2/27 
PoincarĂ© 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 takehome! 