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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:
  • Class Web-Page:
  • 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.


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.


Updated daily. The arrow indicates the current chapter.

1. Introduction

  1. Topology in real life
  2. Topology via Calculus
  3. Why do we need differential forms?

2. Continuous differential forms

  1. Differentials
  2. Examples of differential forms
  3. Algebra of differential forms
  4. Wedge product of continuous forms
  5. Exterior derivative
  6. Properties of the exterior derivative

3. de Rham cohomology

  1. Calculus and algebra vs topology
  2. Closed and exact forms
  3. Closedness and exactness of 1-forms
  4. Quotients of vector spaces
  5. Homotopy
  6. Simple connectedness
  7. de Rham cohomology
  8. Change of variables for differential forms

4. Cubical differential forms

  1. Cell decomposition of images
  2. Boundary operator of cubical complex
  3. Cubical complexes
  4. Discrete differential forms
  5. Chains vs cochains
  6. Algebra of discrete differential forms
  7. Calculus of discrete differential forms
  8. Dual space
  9. Discrete exterior derivative
  10. Continuous vs discrete differential forms

5. Cubical cohomology

  1. Cochain complexes and cohomology
  2. Cohomology of figure 8

6. From vector calculus to exterior calculus

  1. Forms vs vector fields and functions
  2. Identities of vector calculus
  3. Hodge duality of differential forms
  4. Hodge duality of cubical forms
  5. Discrete Hodge star operator
  6. Second derivative and the Laplacian
  7. Diffusion

7. Geometry

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

8. Integration of differential forms

  1. Orientation
  2. Integration of differential forms of degree 0 and 1
  3. Orientation of manifolds
  4. Integral theorems of vector calculus
  5. Integration of differential forms of degree 2
  6. Properties of integrals of differential forms
  7. General Stokes Theorem

Boundary operator with Excel:


From a related course, 2011: