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

 

 

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

 

 

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

Wedge product

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