Math 104
Winter 2017
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 4 |
Chapter 1 |
Topological manifolds and their properties. Examples. |
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Friday January 6 |
Chapter 1 |
Smooth structures, atlases, Examples of smooth manifolds,
manifolds with boundary |
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Monday January 9 |
Chapter 2 |
Smooth functions and smooth maps, diffeomorphisms |
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Wednesday January 11 |
Chapter 2 |
Partitions of Unity |
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Friday January 13 |
Chapter 2 |
Partitions of Unity Continuation |
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Monday January 16 MLK day classes moved to x-hour |
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Tuesday January 17 x-hour instead of the class on January 16 |
Chapter 3 |
Tangent vectors and derivations |
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Wednesday January 18 |
Chapter 3 |
Pushforwards and
computation in coordinates |
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Friday January 20 |
Chapter 3 |
Tangent space to a manifold with boundary, tangent vectors
to curves, alternative definitions of tangent vectors |
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Monday January 23 |
Chapter 4 |
Tangent bundle, Vector fields on manifolds |
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Wednesday January 25 |
Chapter 4 |
Pushforwards of vector
fields, Lie algebra of vector fields |
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Friday January 27 |
Chapter 5 |
Vector bundles and examples, local and global sections of
vector bundles |
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Monday January 30 |
Chapter 5 |
Bundle maps and constructions with bundles |
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Tuesday January 31 x-hour |
Chapter 6 |
Covectors and
tangent convectors on manifolds, cotangent bundle |
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Wednesday February 1 |
Chapter 6 |
Differential of a function, pullbacks |
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Friday February 3 |
Chapter 7 |
Maps of constant rank, Inverse function theorem |
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Monday February 6 Middle of the term presentation and discussion Monday
February 6-Friday February 10 |
Chapter 7 |
Proof of inverse function theorem |
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Wednesday February 8 |
Chapter 7 |
Rank Theorem, Implicit Function Theorem |
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Friday February 10 |
Chapter 7 |
Immersions, submersions and constant rank maps between
manifolds |
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Monday February 13 |
Chapter 8 |
Embedded Submanifolds |
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Wednesday February 15 |
Chapter 8 |
Immersed Submanifolds |
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Friday February 17 |
Chapter 11 |
Algebra of tensors and tensor fields on manifolds |
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Monday February 20 |
Chapter 12 |
Algebra of alternating tensors, differential forms |
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Wednesday February 22 |
Chapter 12 |
Wedge product |
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Friday February 24 |
Chapter 12 |
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Monday February 27 |
Chapter 13 |
Orientation, orientation of the boundary of a manifold |
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Wednesday March 1 |
Chapter 14 |
Fubini Theorem
without proof, Integration of differential forms on manifolds |
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Friday March 3 |
Chapter 14 |
Stokes Theorem |
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Monday March 6 |
Chapter 14 |
Stokes Theorem continuation |
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Tuesday March 7 x-hour |
Chapter 14 |
Vector calculus theorems and their relation to the Stokes
Theorem. |
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Wednesday March 8 |
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 Thursday March
9 – Sunday March 12 |
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