The geometry of Manifold increases your understanding of the differentiable manifolds and vector fields and forms. From this [course_title] you will be able to have a clear conception on Banach Manifolds, Sard’s Theorem, Whitney’s Embedding Theorems, Freudenthal Suspension Theorem, the de Rham theorem as well as Riemannian Manifolds to help you solve various problems efficiently.
Assessment
This course does not involve any written exams. Students need to answer 5 assignment questions to complete the course, the answers will be in the form of written work in pdf or word. Students can write the answers in their own time. Each answer needs to be 200 words (1 Page). Once the answers are submitted, the tutor will check and assess the work.
Certification
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Course Credit: MIT
Course Curriculum
| Module: 01 | |||
| Manifolds: Definitions and Examples | 00:30:00 | ||
| Smooth Maps and the Notion of Equivalence | 00:45:00 | ||
| The Derivative of a Map between Vector Spaces | 00:30:00 | ||
| Inverse and Implicit Function Theorems | 00:45:00 | ||
| More Examples | 00:30:00 | ||
| Module: 02 | |||
| Vector Bundles and the Differential: New Vector Bundles from Old | 00:30:00 | ||
| Vector Bundles and the Differential: The Tangent Bundle | 00:30:00 | ||
| Connections | 01:00:00 | ||
| The Embedding Manifolds in RN | 00:30:00 | ||
| Sard’s Theorem | 01:00:00 | ||
| Module: 03 | |||
| Stratified Spaces | 00:45:00 | ||
| Fiber Bundles | 00:30:00 | ||
| Whitney’s Embedding Theorem, Medium Version | 00:30:00 | ||
| A Brief Introduction to Linear Analysis: Basic Definitions | 01:00:00 | ||
| A Brief Introduction to Linear Analysis: Fredholm Operators | 00:45:00 | ||
| Smale’s Sard Theorem | 00:45:00 | ||
| Module: 04 | |||
| Parametric Transversality | 00:30:00 | ||
| The Strong Whitney Embedding Theorem | 00:45:00 | ||
| Morse Theory | 01:00:00 | ||
| Canonical Forms: The Lie Derivative | 00:30:00 | ||
| Canonical Forms: The Frobenious Integrability Theorem | 01:00:00 | ||
| Module: 05 | |||
| Differential Forms and de Rham’s Theorem: The Exterior Algebra | 00:45:00 | ||
| Differential Forms and de Rham’s Theorem: The Poincaré Lemma and Homotopy Invariance of the de Rham Cohomology | 01:00:00 | ||
| Refinement The Acyclicity of the Sheaf of p-forms | 00:30:00 | ||
| The Poincaré Lemma Implies the Equality of Cech Cohomology and de Rham Cohomology | 00:45:00 | ||
| The Immersion Theorem of Smale | 00:30:00 | ||
| Assessment | |||
| Submit Your Assignment | 00:00:00 | ||
| Certification | 00:00:00 | ||
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