LEC # | TOPICS |
---|---|
1 | Manifolds: Definitions and Examples |
2 |
Smooth Maps and the Notion of Equivalence
Standard Pathologies |
3 | The Derivative of a Map between Vector Spaces |
4 | Inverse and Implicit Function Theorems |
5 | More Examples |
6 | Vector Bundles and the Differential: New Vector Bundles from Old |
7 | Vector Bundles and the Differential: The Tangent Bundle |
8 |
Connections
Partitions of Unity The Grassmanian is Universal |
9 | The Embedding Manifolds in R^{N} |
10-11 | Sard’s Theorem |
12 | Stratified Spaces |
13 | Fiber Bundles |
14 | Whitney’s Embedding Theorem, Medium Version |
15 |
A Brief Introduction to Linear Analysis: Basic Definitions
A Brief Introduction to Linear Analysis: Compact Operators |
16-17 | A Brief Introduction to Linear Analysis: Fredholm Operators |
18-19 | Smale’s Sard Theorem |
20 | Parametric Transversality |
21-22 | The Strong Whitney Embedding Theorem |
23-28 | Morse Theory |
29 | Canonical Forms: The Lie Derivative |
30 |
Canonical Forms: The Frobenious Integrability Theorem
Canonical Forms: Foliations Characterizing a Codimension One Foliation in Terms of its Normal Vector The Holonomy of Closed Loop in a Leaf Reeb’s Stability Theorem |
31 | Differential Forms and de Rham’s Theorem: The Exterior Algebra |
32 |
Differential Forms and de Rham’s Theorem: The Poincaré Lemma and Homotopy Invariance of the de Rham Cohomology
Cech Cohomology |
33 |
Refinement
The Acyclicity of the Sheaf of p-forms |
34 | The Poincaré Lemma Implies the Equality of Cech Cohomology and de Rham Cohomology |
35 | The Immersion Theorem of Smale |
Calendar
Course Info
Learning Resource Types
notes
Lecture Notes
assignment
Problem Sets