Calculus on Manifolds

## Calculus on Manifolds

#### Math 4B03/6B03, Winter 2012 Dr. Ben Mares

$$\int_M d\omega = \int_{\partial M} \omega.$$

(Credit: Abstruse Goose)

### Announcements

• Doodle link for extra optional lecture, location TBA
• Some of you may be interested in the undergraduate summer school at Notre Dame. See the link at the bottom of the page.
• I always appreciate constructive feedback via my anonymous survey.
• If you get stuck, please feel free to come to office hours and/or discuss with your classmates!
• Office hours: Tu Th 1-2pm, or by appointment.

### Final evaluation

This is suggested reading for the upcoming lectures. Be warned that I sometimes change my mind, but I will do my best to keep this up-to-date.

Tuesday, Apr 3

Topics: Cohomology of $S^n$ using Mayer-Vietoris, Outline of Mayer-Vietoris, closing perspectives

References: M&T Chapters 4&5

### The big picture

Where have we been?
• Intro (M&T Chapter 1)
• Tensor algebra and tensor calculus (Hitchin Chapter 2, Section 5)
• Alternating algebra (M&T Chapter 2, excluding characteristic polynomials (2.16))
• de Rham cohomology and Poincaré lemma (M&T Chapter 3)
• Homotopy (M&T Chapter 6, only 6.7, 6.8, 6.10)
• Smooth manifolds (M&T Chapter 8)
• Differential forms on manifolds (M&T Chapter 9, excluding 9.16 and after).
• Integration on manifolds (M&T Chapter 10)
• Chain complexes and cohomology (M&T Chapter 4)
• Mayer-Vietoris sequence (M&T Chapter 5)
Where are we going?
• On to do exciting new things!