Math 124
Winter 2013
Current Problems 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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Monday January 7 |
Chapter 1 |
Topological manifolds and their properties. Examples. |
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Wednesday January 9 |
Chapter 1 |
Smooth structures, atlases, Examples of smooth manifolds, manifolds with boundary |
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Friday January 11 |
Chapter 2 |
Smooth functions and smooth maps, diffeomorphisms |
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Monday January 14 |
Chapter 2 |
Partitions of Unity |
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Wednesday January 16 |
Chapter 2 |
Partitions of Unity Continuation |
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Friday January 18 |
Chapter 3 |
Tangent vectors and derivations |
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Monday January 21 MLK day, no class |
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Tuesday January 22 x-hour instead of the class on January 21 |
Chapter 3 |
Pushforwards and computation in coordinates |
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Wednesday January 23 |
Chapter 3 |
Tangent space to a manifold with boundary, tangent vectors to curves, alternative definitions of tangent vectors |
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Friday January 25 |
Chapter 3 - Chapter 4 |
Tangent bundle, Maps of constant rank, Inverse function theorem |
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Monday January 28 |
Chapter 4 |
Proof of inverse function theorem |
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Tuesday January 29 x-hour possibly instead of one of the future lectures |
Chapter 4 |
Rank Theorem, Implicit Function Theorem |
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Wednesday January 30 |
Chapter 4 |
Immersions, submersions and constant rank maps between manifolds |
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Friday February 1 |
Chapter 5 |
Embedded Submanifolds |
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Monday February 4 Middle of the term presentation and discussion Monday February 4-Friday February 8 |
Chapter 5 |
Immersed submanifolds |
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Wednesday February 6 |
Chapter 8 |
Tangent bundle, Vector fields on manifolds |
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Friday February 8 Winter Carnival No class |
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Monday February 11 |
Chapter 8 |
Pushforwards of vector fields, Lie algebra of vector fields |
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Tuesday February 12 x-hour instead of the class on Friday February 8 |
Chapter 10 |
Vector bundles and examples, local and global sections of vector bundles |
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Wednesday February 13 |
Chapter 10 |
Bundle maps and constructions with bundles |
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Friday February 15 |
Chapter 11 |
Covectors and tangent convectors on manifolds, cotangent bundle |
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Monday February 13 |
Chapter 11 |
Differential of a function, pullbacks |
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Wednesday February 15 |
Chapter 12 |
Algebra of tensors and tensor fields on manifolds |
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Monday February 18 |
Chapter 14 |
Algebra of alternating tensors, differential forms |
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Wednesday February 20 |
Chapter 14 |
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Friday February 22 |
Chapter 14 |
Exterior Derivative, cohomology |
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Monday February 25 |
Chapter 15 |
Orientation, orientation of the boundary of a manifold |
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Wednesday February 27 |
Chapter 16 |
Fubini Theorem without proof, Integration of differential forms on manifolds |
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Friday March 1 |
Chapter 16 |
Stokes Theorem |
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Monday March 4 |
Chapter 16 |
Stokes Theorem continuation |
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Wednesday March 6 |
Chapter 16 |
Vector calculus theorems and their relation to the stokes Theorem. Bordism groups and the pairing between cohomology and bordism groups given by the Stokes Theorem. |
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Friday March 8 |
Wrap up |
Wrap up |
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End of the term presentation and discussion Sunday March 10 – Wednesday March 13 |
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