| Date |
Tentative topic
|
Homework
|
|
|
|
1/6
|
Cotangent bundles and 1-forms |
HW1,
due 1/15 |
1/8
|
Class canceled (due to wildfires) |
|
1/10
|
Differential forms |
|
|
|
|
1/13
|
More on differential forms |
|
1/15
|
Mayer-Vietoris sequence; some homological algebra |
HW2,
due 1/22 |
1/17
|
Integration |
|
|
|
|
1/20
|
No class (Martin Luther King Day) |
|
1/22
|
Stokes' theorem |
HW3,
due 1/29
|
1/24
|
Applications of Stokes' theorem |
|
|
|
|
1/27
|
Evaluating cohomology classes, degree |
|
1/29
|
Lie derivatives |
HW4,
due 2/5 |
1/31
|
Homotopy properties of de Rham cohomology |
|
|
|
|
2/3
|
Relationship between d and [,] |
|
2/5
|
Transversality, Day I |
HW5,
due 2/12 |
2/7
|
Transversality, Day II |
|
|
|
|
2/10
|
Morse functions |
|
2/12
|
Whitney embedding theorem,
orientations |
HW6,
due 2/19 |
2/14
|
Orientations |
|
|
|
|
2/17
|
No class (Presidents' Day)
|
|
2/19
|
Oriented intersection numbers, degree |
HW7,
due 2/26
|
2/21
|
Applications of degree: winding numbers |
|
|
|
|
2/24
|
More applications of
degree: Jordan-Brouwer separation theorem, Borsuk-Ulam
theorem; the diagonal |
|
2/26
|
Lefschetz fixed point theory, Day I |
HW8,
due 3/5 |
2/28
|
Lefschetz fixed point
theory, Day II; Poincaré-Hopf theorem |
|
|
|
|
3/2
|
Statement of Lefschetz
fixed point theorem; Hopf degree theorem |
|
3/4
|
de Rham cohomology with compact supports |
HW9,
due 3/16
|
3/6
|
Poincaré lemma; Poincaré duality |
|
|
|
|
3/9
|
Completion of proof of Poincaré duality; Thom
isomorphism |
|
3/11
|
Consequences of Thom
isomorphism; Poincaré duals |
|
3/13
|
Kunneth formula and Lefschetz fixed point
theorem |
|
|
|
|
|
Final exam is
take-home!
|
|