Math 104

Winter 2018

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

Wednesday January 3

Chapter 1

Topological manifolds and their properties. Examples.

Friday January 5

Chapter 1

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

Monday January 8

Chapter 2

Smooth functions and smooth maps, diffeomorphisms

Tuesday January 9

x-hour instead of the class on January 15

Chapter 2

Partitions of Unity

Wednesday January 10

Chapter 2

Partitions of Unity Continuation

Friday January 12

Chapter 2

Tangent vectors and derivations

Monday January 15

MLK day classes moved to x-hour

 

Pushforwards and computation in coordinates

Tuesday January 16

Chapter 3

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

Wednesday January 17

Chapter 3

Tangent bundle, Vector fields on manifolds

Friday January 19

Chapter 3

Pushforwards of vector fields, Lie algebra of vector fields

Monday January 22

Chapter 4

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

Tuesday January 23

Chapter 4

Bundle maps and constructions with bundles

Wednesday January 24

Chapter 4

Covectors and tangent convectors on manifolds, cotangent bundle

Friday January 26

Chapter 5

Covectors and tangent convectors on manifolds, cotangent bundle

Monday January 29

Chapter 5

Differential of a function, pullbacks

Tuesday January 30

x-hour

Chapter 6

Maps of constant rank, Inverse function theorem

Wednesday January 31

Chapter 6

Proof of inverse function theorem

Friday February 2

Chapter 7

Rank Theorem, Implicit Function Theorem

Monday  February 5

Chapter 7

Rank Theorem, Implicit Function Theorem

Wednesday February 7

Chapter 7

Immersions, submersions and constant rank maps between manifolds

Friday February 9

Chapter 7

Embedded Submanifolds

Monday February 12

Middle of the term presentation and discussion Monday February 6-Friday February 16

Chapter 8

Immersed Submanifolds

Wednesday February 14

Chapter 8

Algebra of tensors and tensor fields on manifolds

Friday February 16

Chapter 11

Algebra of alternating tensors, differential forms

Monday February 19

Chapter 12

Wedge product

Wednesday February 21

Chapter 12

Exterior Derivative, cohomology

Friday February 23

Chapter 12

Orientation, orientation of the boundary of a manifold

Monday February 26

Chapter 13

Fubini Theorem without proof, Integration of differential forms on manifolds

Tuesday February 27

Chapter 14

Stokes Theorem

Wednesday February 28

Chapter 14

Stokes Theorem continuation

Friday March 2

Chapter 14

Vector calculus theorems and their relation to the Stokes Theorem.

Monday March 5

Chapter 14

Bordism groups. The pairing between cohomology and bordism groups given by the Stokes Theorem

Tuesday March 6

x-hour

Chapter 14

Bordism groups. The pairing between cohomology and bordism groups given by the Stokes Theorem

End of the term presentation and discussion Wednesday March 7 – Sunday March 11