Syllabus and Class Notes
Syllabus for the course
I will post my lecture notes for the course. Um ... so far I haven't done well at keeping a good record of lectures, but I vow I will improve.
- Manifolds and Tangent Spaces
- Immersions and Submersions (incomplete)
- Transversality (only the beginning)
- Sard's Theorem
- Manifolds with Boundary
- 1-Manifolds
- Winding Numbers
- Differential Forms
- Exterior Derivatives (missing the first-half)
- Does ker(curl)=Image(grad)?* and relevant scene from A Beautiful Mind
- Hodge Decomposition Theorem I*
- Hodge Decomposition Theorem II*
- Hodge Decomposition Theorem III*
- de Rham Cohomology
* denotes lectures drawn from a course I taught in 2010 on geometric topology (vector calculus, differential forms, homology, ...).
Class Projects
- Joe Paat, Partitions of unity, presentation
- Natalie Rich, Proof of Sard's Theorem
- David Robertson, Classification of surfaces, presentation
- Robert Sizemore, Classification of 1-manifolds, presentation
- Danielle Walsh, Partitions of Unity, presentation
Class Documents
Monkey saddle: has three directions of decrease (for the legs and tail). One equation is z = x^3 - 3xy^2. Here's a quick plot ... but
click here for a monkey on the saddle.
Alexander's horned sphere
From class on 4/1, we viewed
this youtube animation and saw this
this Wiki image. Here's Alexander's horned cat, from Bing's book:
