|
週次 |
日期 |
單元主題 |
|
Week 1 |
2/25 |
Levi-Civita connection. Reference: [CE, §1.0] and [dC, §2.3] |
|
Week 6 |
4/01,4/03 |
Spring Break |
|
Week 2-1 |
3/04 |
the method of moving frame, Riemann curvature tensor. Reference: [CE, §1.4] and [dC, §4.2] |
|
Week 2-2 |
3/06 |
basic properties of Riemann curvature tensor, first variation formula, Jacobi field. Reference: [CE, §1.1 and 1.4] and [dC, §4.2 and 5.1] |
|
Week 3-1 |
3/11 |
the coefficients of the Gaussian coordinate. Reference: [CE, §1.4] and [dC, §4.2] |
|
Week 3-2 |
3/13 |
notions of curvatures (sectional, Ricci, scalar), second variational formula, index lemma. Reference: [CE, §1.6 and 1.8] and [dC, §9.2 and 10.2] |
|
Week 4-1 |
3/18 |
index lemma and its application. Reference: [CE, §1.8] and [dC, §10.2] |
|
Week 4-2 |
3/20 |
Bonnet--Myers theorem, Rauch comparison theorem, crash course on covering spaces and fundamental groups. Reference: [CE, §1.9 and 1.10] and [dC, §9.3 and 10.2],
see Bredon: Topology and Geometry, ch.III . MR and Vick: Homology theory, ch.4 . MR for the covering spaces and fundamental groups |
|
Week 5-1 |
3/25 |
covering spaces and fundamental groups (continued), application of Bonnet--Myers theorem. Reference: [CE, §1.9]
see note0325 for a brief introduction to tensor calculus |
|
Week 5-2 |
3/27 |
Cartan--Hadamard theorem, Cartan--Ambrose--Hicks theorem. Reference: [CE, §1.11 and 1.12] and [dC, §7.3 and 8.2] |
|
Week 7-1 |
4/08 |
Hodge star and Laplace--Beltrami operator. Reference: [W, §4.1] |
|
Week 7-2 |
4/10 |
Hodge theorem (assuming regularity and compactness). Reference: [W, §4.2] |
|
Week 8-1 |
4/15 |
Sobolev space and some basic properties, Sobolev embedding. Reference: [W, §4.3] |
|
Week 8-2 |
4/17 |
properties of Sobolev space, Rellich lemma, elliptic operator and elliptic estimate (on the torus). Reference: [W, §4.3 and 4.4] |
|
Week 9-1 |
4/22 |
difference quotient, elliptic regularity (on the torus). Reference: [W, §4.3 and 4.4] |
|
Week 9-2 |
4/24 |
regularity and compactness of Laplace (on any Riemannian manifold). Reference: [W, §4.5] |
|
Week 10-1 |
4/29 |
introduction to the local index theorem and the heat kernel approach. |
|
Week 10-2 |
5/01 |
midterm |
|
Week 11-1 |
5/06 |
intersection pairing and signature |
|
Week 11-2 |
5/08 |
the invariance of lambda, the criterion for homeomorphic spheres, the examples of Milnor. |
|
Week 12-1 |
5/13 |
calculation of lambda of the examples of Milnor. |
|
Week 12-2 |
5/15 |
TIMS 2015 Mini-Course on Topology and Geometry of Ricci Solitons |
|
Week 13-1 |
5/20 |
Bochner Laplacian |
|
Week 13-2 |
5/22 |
Bochner formula and its applitcations. |
|
Week 14-1 |
5/27 |
Killing vector field on negatively curved manifold. |
|
Week 14-2 |
5/29 |
Eigenvalue estimate, Laplacian and second fundamental form. |
|
Week 15-1 |
6/03 |
Reilly's theorem. |
|
Week 15-2 |
6/05 |
Reilly's proof of Aleksandrov theorem, introduction to harmonic maps. |
|
Week 16-1 |
6/10 |
harmonic maps on surfaces, Hopf differential. |
|
Week 16-2 |
6/12 |
Final presentation |
|
Week 17-1 |
6/17 |
Final presentation |
|
Week 17-2 |
|
Final presentation |