Introduction to differential forms, spring 2011

Lecturer

Pekka Pankka

Scope

10 cu.

Type

Advanced studies

Description

Differential forms give a coordinate free formalism for multivariable calculus and a point of view to modern analysis. This formalism can also be used
to understand basic concepts in algebraic topology as homology and cohomology. In this course we discuss how familiar methods from calculus give rise
to more abstract concepts in topology.

Background in algebraic topology or in differential geometry is not necessary.

Prerequisites

Real analysis and Topology II (or equivalent).

Lectures

Weeks 3-9 and 11-18, Tuesday 16-18 and Thursday 10-12 in room B321. (Note: Change in Tuesdays lecture time.)

Easter holiday 21.-27.4.

Methods of passing the course

The course can be passed by returning written solutions to weekly problem sets. Solutions are graded with scale 0-6. Problems discussed in the exercise sessions give extra credit. Course can be also passed by an exam. Contact the lecturer for details.

Topics (preliminary list)

• Multilinear algebra
• Vector fields and differential forms
• Simplicial complexes
• Integration
• Homology and de Rham cohomology
• Methods of calculation in cohomology
• Degree theory
• Classical applications of cohomology

Bibliography

• Tu: An Introduction to Manifolds
• Madsen - Tornehave: From Calculus to Cohomology
• Bott - Tu: Differential Forms in Algebraic Topology

Registration

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Lecture notes

Introduction and 1-forms
Multilinear algebra
From Orientation to Integration
Exterior derivative
Chains, Forms, and Integration
de Rham cohomology: Poincaré's lemma
de Rham cohomology: Exact sequences
de Rham cohomology: Homotopy methods
de Rham cohomology: Applications
Manifolds I
Integration on manifolds

Exercise sets

Problem set 1 Solutions to Problem set 1
Problem set 2 Solutions to Problem Set 2
Problem set 3 Solutions to Problem Set 3
Problem set 4 Solutions to Problem Set 4
Problem set 5 Solutions to Problem Set 5
Problem set 6 Solutions to Problem Set 6
Problem set 7 Solutions to Problem Set 7
Problem set 8 Solutions to Problem Set 8
Problem set 9 Solutions to Problem Set 9
Problem set 10 Solutions to Problem Set 10
Problem set 11
Problem set 12Solutions to Problem Set 12

Exercise groups