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

 

Lectures

Sections in Text

Brief Description

Monday January 5

Chapter 1

Topological manifolds and their properties. Examples.

Wednesday January 7

Chapter 1

Smooth structures, atlases, Examples of smooth manifolds, manifolds with boundary

Friday January 9

Chapter 2

Smooth functions and smooth maps, diffeomorphisms

Monday January 12

Chapter 2

Partitions of Unity

Wednesday January 14

Chapter 2

Partitions of Unity Continuation

Friday January 16

Chapter 3

Tangent vectors and derivations

Monday January 19

MLK day classes moved to x-hour

 

 

Tuesday January 20

x-hour instead of the class on January 19

Chapter 3

Pushforwards and computation in coordinates

Wednesday January 21

Chapter 3

Tangent space to a manifold with boundary, tangent vectors to curves, alternative definitions of tangent vectors

Friday January 23

Chapter 4

Tangent bundle, Vector fields on manifolds

Monday January 26

Chapter 4

Pushforwards of vector fields, Lie algebra of vector fields

Wednesday January 28

Chapter 5

Vector bundles and examples, local and global sections of vector bundles

Friday January 30

Chapter 5

Bundle maps and constructions with bundles

Monday February 2

Chapter 6

Covectors and tangent convectors on manifolds, cotangent bundle

Tuesday February 3

x-hour instead of the class on February 6

Chapter 6

Differential of a function, pullbacks

Wednesday February 4

Chapter 7

Maps of constant rank, Inverse function theorem

Friday February 6

Carnival Holiday classes moved to x-periods

Chapter 7

Proof of inverse function theorem

Monday February 9

Middle of the term presentation and discussion Monday February 9-Friday February 13

Chapter 7

Rank Theorem, Implicit Function Theorem

Wednesday February 11

Chapter 7

Immersions, submersions and constant rank maps between manifolds

Friday February 13

Chapter 8

Embedded Submanifolds

Monday February 16

Chapter 8

Immersed submanifolds

Wednesday February 18

Chapter 11

Algebra of tensors and tensor fields on manifolds

Friday February 20

Chapter 12

Algebra of alternating tensors, differential forms

Monday February 23

Chapter 12

Wedge product

Wednesday February 25

Chapter 12

Exterior Derivative, cohomology

Friday February 27

Chapter 13

Orientation, orientation of the boundary of a manifold

Monday March 2

Chapter 14

Fubini Theorem without proof, Integration of differential forms on manifolds

Wednesday March 4

Chapter 14

Stokes Theorem

Friday March 6

Chapter 14

Stokes Theorem continuation

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.

End of the term presentation and discussion Thursday March 12 Saturday March 14