Math 225A: Differentiable Manifolds

Date  Tentative topic 
Homework

9/23 
Review of topology and linear algebra 
HW1,
due 9/30 
9/26 
Review of differentiation 

9/28 
Manifolds; examples 

9/30 
Smooth functions and smooth maps 
HW2,
due 10/7 
10/3 
The inverse function theorem 

10/5 
Submersions 

10/7 
Immersions and embeddings  HW3,
due 10/14 
10/10 
Tangent spaces, Day I  
10/12 
Tangent spaces, Day II  
10/14 
The tangent bundle  HW4,
due 10/21 
10/17 
Cotangent bundles and 1forms  
10/19 
Lie groups  
10/21 
Vector bundles, Day I  HW5,
due 10/28 
10/24 
Vector bundles, Day II  
10/26 
Tensor products 

10/28 
Tensor and exterior algebra 
HW6,
due 11/4 
10/31 
Differential
kforms 

11/2 
De Rham cohomology 

11/4 
MayerVietoris sequence; some homological
algebra 
HW7,
due 11/14 
11/7 
Integration 

11/9 
Stokes' theorem  
11/11 
No class (Veterans Day) 
HW8,
due 11/18 
11/14 
Applications of
Stokes' theorem 

11/16 
Evaluating cohomology classes, degree 

11/18 
Lie derivatives 
HW9, due 12/2 
11/21 
Homotopy
properties 

11/23 
Vector fields 

11/25 
No class (Thanksgiving break) 

11/28 
Vector fields and Lie derivatives 

11/30 
Relationship
between d and [,] 

12/2 
Frobenius theorem 

Final exam
is takehome! 