Math 225B:  Differentiable Manifolds

Class Schedule

The topics are tentative but the due dates are not.

Date  Tentative topic
 Homework                                                                          



1/6
 Sard's theorem
 HW1, due 1/13
1/8
 Transversality, Day I
1/10
 Transversality, Day II



1/13
 Morse functions HW2, due 1/22
1/15
 Whitney embedding theorem
1/17
 Orientations



1/20
 No class (Martin Luther King Day)
1/22
 Oriented intersection numbers
 HW3, due 1/27
1/24
 Degree



1/27
 Applications of degree: winding numbers
 HW4, due 2/3
1/29
 More applications of degree: Jordan-Brouwer separation theorem, Bursuk-Ulam theorem
1/31
 Class canceled



2/3
 The diagonal (more intersection theory; Euler characteristic) HW5, due 2/10
2/5
 Lefschetz fixed point theorem, Day I
2/7
 Class canceled



2/10 (Mon)
 Discussion
 HW6, due 2/19
2/12
 Lefschetz fixed point theorem, Day II; Poincaré-Hopf theorem
2/13 (Th)
 Framed cobordisms and the Pontryagin construction, Day I (lecture and discussion switched this week)
2/14
 Framed cobordisms and the Pontryagin construction, Day I



2/17
 No class (Presidents' Day)
2/19
 Applications of framed cobordisms including the Hopf degree theorem HW7, due 3/2
2/20 (Th)
 Classification of vector bundles (no discussion this week, lecture during discussion)
2/21
 Cobordisms and Thom's work

 
2/24
 Poincaré duality, Day I
2/26
 Poincaré duality, Day II 
2/28
 Thom isomorphism



3/2
 Consequences of Thom isomorphism HW8, due 3/13
3/4
 Hodge theory preliminaries
3/6
 Hodge theory, Day I



3/9
 Hodge theory, Day II
3/11
 Elliptic PDE's
3/13
 Elliptic operators are Fredholm




Final exam is take-home!


Last modified:  January 20, 2020.