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math214:home [2020/04/22 12:35]
pzhou [Lectures] |
math214:home [2020/12/18 21:23] (current)
pzhou |
| * typo in the note: one page 2, right column. I wrote incorrectly that, "if p∈K∩gK then p,gp∈K". The correct statement is that "if g∈GK, then there exist p∈K such that g⋅p∈K". | * typo in the note: one page 2, right column. I wrote incorrectly that, "if p∈K∩gK then p,gp∈K". The correct statement is that "if g∈GK, then there exist p∈K such that g⋅p∈K". |
| * 03-20: Ch 19, Distribution. covered up to Frobenius Thm (statement and sketch of proof) | * 03-20: Ch 19, Distribution. covered up to Frobenius Thm (statement and sketch of proof) |
| * HW9: Ch 21: 1,5,9,16. Ch 19: 1. | * HW9: Ch 21: 1,5,9,16. Ch 19: 1. [[hw9-sol]] |
| * Week 10. From Wednesday on, we will start to use the [[https://www3.nd.edu/~lnicolae/Lectures.pdf| lecture note of Nicolascu ]] for topics about connections, denoted as [Ni] | * Week 10. From Wednesday on, we will start to use the [[https://www3.nd.edu/~lnicolae/Lectures.pdf| lecture note of Nicolascu ]] for topics about connections, denoted as [Ni] |
| * 03-30: Riemannian metric: an introduction with examples and no proofs.[Lee] | * 03-30: Riemannian metric: an introduction with examples and no proofs.[Lee] |
| * Week 11: | * Week 11: |
| * [[04-06]]: {{ :math214:note_6_apr_2020.pdf |ipad note}}Holonomy, interpretation of Curvature[Ni] 3.3.4 | * [[04-06]]: {{ :math214:note_6_apr_2020.pdf |ipad note}}Holonomy, interpretation of Curvature[Ni] 3.3.4 |
| * Errata of ipad note: page 1, right column. It should be: if $A = \omega \ot (e_\alpha \ot \delta^\beta)$, for ω∈Ω1(U) and eα the local frame of E previously chosen, and δβ the dual frame of E∗, then $dA = d(\omega) \ot (e_\alpha \ot \delta^\beta).Thisisbecaused e_\alpha =0, d \delta^\beta=0bythedefinitionoflocalconnectiondonE|_U$. | * Errata of ipad note: page 1, right column. It should be: if $A = \omega \otimes (e_\alpha \ot \delta^\beta)$, for ω∈Ω1(U) and eα the local frame of E previously chosen, and δβ the dual frame of E∗, then $dA = d(\omega) \otimes (e_\alpha \ot \delta^\beta).Thisisbecaused e_\alpha =0, d \delta^\beta=0bythedefinitionoflocalconnectiondonE|_U$. |
| * 04-08: {{ :math214:note_8_apr_2020.pdf |ipad note}} Connection on Tangent Space. Levi-Cevita connection. | * 04-08: {{ :math214:note_8_apr_2020.pdf |ipad note}} Connection on Tangent Space. Levi-Cevita connection. |
| * 04-10: {{ :math214:note_10_apr_2020_3_.pdf | ipad note}}. Levi-Cevita connection for bi-invariant metric on Lie group. Begin Geodesic equation. | * 04-10: {{ :math214:note_10_apr_2020_3_.pdf | ipad note}}. Levi-Cevita connection for bi-invariant metric on Lie group. Begin Geodesic equation. |
| * 04-20: {{ :math214:note_20_apr_2020_2_.pdf |ipad note}} (typo about the critical point for f(x,y,z)=xyz, it should be {yz=0,xy=0,zx=0} which is a union of three coordinate axixes. | * 04-20: {{ :math214:note_20_apr_2020_2_.pdf |ipad note}} (typo about the critical point for f(x,y,z)=xyz, it should be {yz=0,xy=0,zx=0} which is a union of three coordinate axixes. |
| * 04-22: {{ :math214:note_22_apr_2020_2_.pdf |ipad note}}. This follows [Ni]'s lecture note, section 5.2. | * 04-22: {{ :math214:note_22_apr_2020_2_.pdf |ipad note}}. This follows [Ni]'s lecture note, section 5.2. |
| | * 04-24: {{ :math214:note_24_apr_2020_3_.pdf |ipad note}} |
| | * [[hw13]] |
| | * Week 14: de Rham cohomology |
| | * 04-27: {{ :math214:note_27_apr_2020_2_.pdf | ipad note}} |
| | * 04-29: {{ :math214:note_29_apr_2020_2_.pdf | note}} Poincare duality. {{ :math214:mv-sequence.pdf |Excerpt from Bott-Tu}}, p23-24, on MV sequence. |
| | * 05-01: {{ :math214:note_1_may_2020_2_.pdf | note}} Singular, Cech, Morse cohomology. |
| | |
| | ** [[hwsol | Students Homework Solutions]] ** |
| |
| | |
| | ===== Final ===== |
| | [[final]] and [[final-solution]] |
