This site is devoted to mathematics and its applications. Created and run by Peter Saveliev.

# Advanced Topology -- Spring 2013

### From Mathematics Is A Science

**MTH 632 Advanced Topology.** Differential forms and cohomology, and applications. PR: MTH 331. 3 hours.

- Time and Place: 12:30 pm - 1:45 pm TR Smith Hall 516
- Instructor: Peter Saveliev (call me Peter)
- Office: 325 Smith Hall
- Office Hours: 2:30 - 4:30 TR, or by appointment
- Office Phone: x4639
- E-mail: saveliev@marshall.edu
- Class Web-Page: math02.com
- Prerequisites: Math 331 Linear algebra, the ability to understand and write proofs.
- Text: Applied Topology and Geometry (draft), specific chapters listed below
- Goals: understanding continuous and discrete differential forms and their relation to the topology of the space.
- Evaluation: midterm, final exam, weekly homework, quizzes.
- Grade Breakdown:
- homework + quizzes: 30%
- midterm: 30%
- final exam: 40%

- Letter Grades: A: 90-100, B: 80-89, C: 70-79, D: 60-69, F: <60

See also Course policy.

## Contents

## 1 Description

Differential forms provide a modern view of calculus. They also give you a start with algebraic topology in the sense that one can extract topological information about a manifold from its space of differential forms. It's called cohomology. Prerequisites: just linear algebra, in the sense of theory of vector spaces. Frequently, this material is only seen in more advanced linear algebra courses, or group theory.

## 2 Lecture notes

They will appear exactly as you see them in class and, as the course progresses, will be updated daily.

- Chapter 1a
- Chapter 1b
- Chapter 2a
- Chapter 2b
- Chapter 2c
- Chapter 3a
- Chapter 4a
- Chapter 4b
- Chapter 4c
- Chapter 5a
- Chapter 6a
- Chapter 6b

You may have to "hard" refresh to get the updated version of the text: "Ctrl+R", or "Ctrl+F5", or "Shift +R", etc.

## 3 Chapters

Updated daily. The arrow indicates the current chapter.

1. Introduction

2. Continuous differential forms

- Differentials
- Examples of differential forms
- Algebra of differential forms
- Wedge product of continuous forms
- Exterior derivative
- Properties of the exterior derivative

3. de Rham cohomology

- Calculus and algebra vs topology
- Closed and exact forms
- Closedness and exactness of 1-forms
- Quotients of vector spaces
- Homotopy
- Simple connectedness
- de Rham cohomology
- Change of variables for differential forms

4. Cubical differential forms

- Cell decomposition of images
- Boundary operator of cubical complex
- Cubical complexes
- Discrete differential forms
- Chains vs cochains
- Algebra of discrete differential forms
- Calculus of discrete differential forms
- Dual space
- Discrete exterior derivative
- Continuous vs discrete differential forms

5. Cubical cohomology

6. From vector calculus to exterior calculus

- Forms vs vector fields and functions
- Identities of vector calculus
- Hodge duality of differential forms
- Hodge duality of cubical forms
- Discrete Hodge star operator
- Second derivative and the Laplacian
- Diffusion

7. Geometry

- Geometry in calculus
- Metric tensor in dimensions 1 and 2 <--(last)
- Geometric Hodge duality
- Lengths of digital curves

8. Integration of differential forms

- Orientation
- Integration of differential forms of degree 0 and 1
- Orientation of manifolds
- Integral theorems of vector calculus
- Integration of differential forms of degree 2
- Properties of integrals of differential forms
- General Stokes Theorem

Boundary operator with Excel:

## 4 Tests

- Final exam: Tuesday May 7, 12:45 - 2:45.
- Advanced Topology: midterm
- Advanced Topology: exercises
- Advanced Topology -- Spring 2013 -- final exam

From a related course, 2011: