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
Lectures |
Sections in Text |
Brief Description |
Monday January 7 |
Chapter 1 |
Topological manifolds and their properties. Examples. |
Wednesday January 9 |
Chapter 1 |
Smooth structures, atlases, Examples of smooth manifolds, manifolds with boundary |
Friday January 11 |
Chapter 2 |
Smooth functions and smooth maps, diffeomorphisms |
Monday January 14 |
Chapter 2 |
Partitions of Unity |
Wednesday January 16 |
Chapter 2 |
Partitions of Unity Continuation |
Friday January 18 |
Chapter 3 |
Tangent vectors and derivations |
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 |
Wednesday January 23 |
Chapter 3 |
Tangent space to a manifold with boundary, tangent vectors to curves, alternative definitions of tangent vectors |
Friday January 25 |
Chapter 3 - Chapter 4 |
Tangent bundle, Maps of constant rank, Inverse function theorem |
Monday January 28 |
Chapter 4 |
Proof of inverse function theorem |
Tuesday January 29 x-hour possibly instead of one of the future lectures |
Chapter 4 |
Rank Theorem, Implicit Function Theorem |
Wednesday January 30 |
Chapter 4 |
Immersions, submersions and constant rank maps between manifolds |
Friday February 1 |
Chapter 5 |
Embedded Submanifolds |
Monday February 4 Middle of the term presentation and discussion Monday February 4-Friday February 8 |
Chapter 5 |
Immersed submanifolds |
Wednesday February 6 |
Chapter 8 |
Tangent bundle, Vector fields on manifolds |
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 |
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 |
Wednesday February 13 |
Chapter 10 |
Bundle maps and constructions with bundles |
Friday February 15 |
Chapter 11 |
Covectors and tangent convectors on manifolds, cotangent bundle |
Monday February 13 |
Chapter 11 |
Differential of a function, pullbacks |
Wednesday February 15 |
Chapter 12 |
Algebra of tensors and tensor fields on manifolds |
Monday February 18 |
Chapter 14 |
Algebra of alternating tensors, differential forms |
Wednesday February 20 |
Chapter 14 |
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Friday February 22 |
Chapter 14 |
Exterior Derivative, cohomology |
Monday February 25 |
Chapter 15 |
Orientation, orientation of the boundary of a manifold |
Wednesday February 27 |
Chapter 16 |
Fubini Theorem without proof, Integration of differential forms on manifolds |
Friday March 1 |
Chapter 16 |
Stokes Theorem |
Monday March 4 |
Chapter 16 |
Stokes Theorem continuation |
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. |
Friday March 8 |
Wrap up |
Wrap up |
End of the term presentation and discussion Sunday March 10 – Wednesday March 13 |
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