Math 104
Winter 2018
Topics in Topology
Lecture Plan
This lecture plan is tentative and will be updated irregularly. The homework page will be updated on the regular basis
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Lectures |
Sections in Text |
Brief Description |
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Wednesday January 3 |
Chapter 1 |
Topological manifolds and their properties. Examples. |
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Friday January 5 |
Chapter 1 |
Smooth structures, atlases, Examples of smooth manifolds, manifolds with boundary |
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Monday January 8 |
Chapter 2 |
Smooth functions and smooth maps, diffeomorphisms |
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Tuesday January 9 x-hour instead of the class on January 15 |
Chapter 2 |
Partitions of Unity |
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Wednesday January 10 |
Chapter 2 |
Partitions of Unity Continuation |
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Friday January 12 |
Chapter 2 |
Tangent vectors and derivations |
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Monday January 15 MLK day classes moved to x-hour |
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Pushforwards and computation in coordinates |
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Tuesday January 16 |
Chapter 3 |
Tangent space to a manifold with boundary, tangent vectors to curves, alternative definitions of tangent vectors |
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Wednesday January 17 |
Chapter 3 |
Tangent bundle, Vector fields on manifolds |
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Friday January 19 |
Chapter 3 |
Pushforwards of vector fields, Lie algebra of vector fields |
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Monday January 22 |
Chapter 4 |
Vector bundles and examples, local and global sections of vector bundles |
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Tuesday January 23 |
Chapter 4 |
Bundle maps and constructions with bundles |
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Wednesday January 24 |
Chapter 4 |
Covectors and tangent convectors on manifolds, cotangent bundle |
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Friday January 26 |
Chapter 5 |
Covectors and tangent convectors on manifolds, cotangent bundle |
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Monday January 29 |
Chapter 5 |
Differential of a function, pullbacks |
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Tuesday January 30 x-hour |
Chapter 6 |
Maps of constant rank, Inverse function theorem |
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Wednesday January 31 |
Chapter 6 |
Proof of inverse function theorem |
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Friday February 2 |
Chapter 7 |
Rank Theorem, Implicit Function Theorem |
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Monday February 5 |
Chapter 7 |
Rank Theorem, Implicit Function Theorem |
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Wednesday February 7 |
Chapter 7 |
Immersions, submersions and constant rank maps between manifolds |
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Friday February 9 |
Chapter 7 |
Embedded Submanifolds |
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Monday February 12 Middle of the term presentation and discussion Monday February 6-Friday February 16 |
Chapter 8 |
Immersed Submanifolds |
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Wednesday February 14 |
Chapter 8 |
Algebra of tensors and tensor fields on manifolds |
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Friday February 16 |
Chapter 11 |
Algebra of alternating tensors, differential forms |
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Monday February 19 |
Chapter 12 |
Wedge product |
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Wednesday February 21 |
Chapter 12 |
Exterior Derivative, cohomology |
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Friday February 23 |
Chapter 12 |
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Monday February 26 |
Chapter 13 |
Fubini Theorem without proof, Integration of differential forms on manifolds |
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Tuesday February 27 |
Chapter 14 |
Stokes Theorem |
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Wednesday February 28 |
Chapter 14 |
Stokes Theorem continuation |
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Friday March 2 |
Chapter 14 |
Vector calculus theorems and their relation to the Stokes Theorem. |
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Monday March 5 |
Chapter 14 |
Bordism groups. The pairing between cohomology and bordism groups given by the Stokes Theorem |
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Tuesday March 6 x-hour |
Chapter 14 |
Bordism groups. The pairing between cohomology and bordism groups given by the Stokes Theorem |
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End of the term presentation and discussion Wednesday March 7 – Sunday March 11 |
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