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

# Advanced Topology -- Spring 2013

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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.
• homework + quizzes: 30%
• midterm: 30%
• final exam: 40%
• Letter Grades: A: 90-100, B: 80-89, C: 70-79, D: 60-69, F: <60

## Description

 Stokes theorem $$\int_ σ dω = \int_{∂σ} ω$$ Derivative vs boundary

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.

## Lecture notes

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

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

## Chapters

Updated daily. The arrow indicates the current chapter.

1. Introduction

2. Continuous differential forms

3. de Rham cohomology

4. Cubical differential forms

5. Cubical cohomology

6. From vector calculus to exterior calculus

7. Geometry

8. Integration of differential forms

Boundary operator with Excel:

## Tests

From a related course, 2011: