| LEC # | TOPICS | LECTURE NOTES |
|---|---|---|
| 1 | Manifolds: Definitions and Examples | (PDF) |
| 2 | Smooth Maps and the Notion of Equivalence Standard Pathologies | (PDF) |
| 3 | The Derivative of a Map between Vector Spaces | (PDF) |
| 4 | Inverse and Implicit Function Theorems | (PDF) |
| 5 | More Examples | (PDF) |
| 6 | Vector Bundles and the Differential: New Vector Bundles from Old | (PDF) |
| 7 | Vector Bundles and the Differential: The Tangent Bundle | (PDF) |
| 8 | Connections Partitions of Unity The Grassmanian is Universal | (PDF) |
| 9 | The Embedding Manifolds in RN | (PDF) |
| 10-11 | Sard's Theorem | (PDF) |
| 12 | Stratified Spaces | (PDF) |
| 13 | Fiber Bundles | (PDF) |
| 14 | Whitney's Embedding Theorem, Medium Version | (PDF) |
| 15 | A Brief Introduction to Linear Analysis: Basic Definitions A Brief Introduction to Linear Analysis: Compact Operators | (PDF) |
| 16-17 | A Brief Introduction to Linear Analysis: Fredholm Operators | (PDF) |
| 18-19 | Smale's Sard Theorem | (PDF) |
| 20 | Parametric Transversality | (PDF) |
| 21-22 | The Strong Whitney Embedding Theorem | (PDF) |
| 23-28 | Morse Theory | (PDF) |
| 29 | Canonical Forms: The Lie Derivative | (PDF) |
| 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 | (PDF) |
| 31 | Differential Forms and de Rham's Theorem: The Exterior Algebra | (PDF) |
| 32 | Differential Forms and de Rham's Theorem: The Poincaré Lemma and Homotopy Invariance of the de Rham Cohomology Cech Cohomology | (PDF) |
| 33 | Refinement The Acyclicity of the Sheaf of p-forms | (PDF) |
| 34 | The Poincaré Lemma Implies the Equality of Cech Cohomology and de Rham Cohomology | (PDF) |
| 35 | The Immersion Theorem of Smale | (PDF) |