Math 104
Winter 2015
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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Monday January 5 |
Chapter 1 |
Topological manifolds and their properties. Examples. |
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Wednesday January 7 |
Chapter 1 |
Smooth structures, atlases, Examples of smooth manifolds,
manifolds with boundary |
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Friday January 9 |
Chapter 2 |
Smooth functions and smooth maps, diffeomorphisms |
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Monday January 12 |
Chapter 2 |
Partitions of Unity |
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Wednesday January 14 |
Chapter 2 |
Partitions of Unity Continuation |
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Friday January 16 |
Chapter 3 |
Tangent vectors and derivations |
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Monday January 19 MLK day classes moved to x-hour |
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Tuesday January 20 x-hour instead of the class on January 19 |
Chapter 3 |
Pushforwards and
computation in coordinates |
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Wednesday January 21 |
Chapter 3 |
Tangent space to a manifold with boundary, tangent vectors
to curves, alternative definitions of tangent vectors |
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Friday January 23 |
Chapter 4 |
Tangent bundle, Vector fields on manifolds |
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Monday January 26 |
Chapter 4 |
Pushforwards of vector
fields, Lie algebra of vector fields |
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Wednesday January 28 |
Chapter 5 |
Vector bundles and examples, local and global sections of
vector bundles |
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Friday January 30 |
Chapter 5 |
Bundle maps and constructions with bundles |
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Monday February 2 |
Chapter 6 |
Covectors and
tangent convectors on manifolds, cotangent bundle |
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Tuesday February 3 x-hour instead of the class on February 6 |
Chapter 6 |
Differential of a function, pullbacks |
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Wednesday February 4 |
Chapter 7 |
Maps of constant rank, Inverse function theorem |
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Friday February 6 Carnival Holiday classes moved to x-periods |
Chapter 7 |
Proof of inverse function theorem |
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Monday February 9 Middle of the term presentation and discussion Monday
February 9-Friday February 13 |
Chapter 7 |
Rank Theorem, Implicit Function Theorem |
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Wednesday February 11 |
Chapter 7 |
Immersions, submersions and constant rank maps between
manifolds |
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Friday February 13 |
Chapter 8 |
Embedded Submanifolds |
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Monday February 16 |
Chapter 8 |
Immersed submanifolds |
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Wednesday February 18 |
Chapter 11 |
Algebra of tensors and tensor fields on manifolds |
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Friday February 20 |
Chapter 12 |
Algebra of alternating tensors, differential forms |
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Monday February 23 |
Chapter 12 |
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Wednesday February 25 |
Chapter 12 |
Exterior Derivative, cohomology |
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Friday February 27 |
Chapter 13 |
Orientation, orientation of the boundary of a manifold |
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Monday March 2 |
Chapter 14 |
Fubini Theorem without
proof, Integration of differential forms on manifolds |
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Wednesday March 4 |
Chapter 14 |
Stokes Theorem |
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Friday March 6 |
Chapter 14 |
Stokes Theorem continuation |
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Monday March 9 |
Chapter 14 |
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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End of the term presentation and discussion Thursday March
12 – Saturday March 14 |
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