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math214:home [2020/04/21 17:12]
pzhou [Lectures] 		math214:home [2026/02/21 14:41] (current)
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     * typo in the note: one page 2, right column. I wrote incorrectly that, "if $p \in K \cap gK$ then $p, gp \in K$". The correct statement is that  "if $g \in G_K$, then there exist $p \in K$ such that $g \cdot p \in K$".      * typo in the note: one page 2, right column. I wrote incorrectly that, "if $p \in K \cap gK$ then $p, gp \in K$". The correct statement is that  "if $g \in G_K$, then there exist $p \in K$ such that $g \cdot p \in K$". 
   * 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] 
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   * 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 $\omega \in \Omega^1(U)$ and $e_\alpha$ the local frame of $E$ previously chosen, and $\delta^\beta$ the dual frame of $E^*$, then $dA = d(\omega) \ot (e_\alpha \ot \delta^\beta)$. This is because $d e_\alpha =0, d \delta^\beta=0$ by the definition of local connection $d$ on $E|_U$. +      * Errata of ipad note: page 1, right column. It should be: if $A = \omega \otimes (e_\alpha \ot \delta^\beta)$, for $\omega \in \Omega^1(U)$ and $e_\alpha$ the local frame of $E$ previously chosen, and $\delta^\beta$ the dual frame of $E^*$, then $dA = d(\omega) \otimes (e_\alpha \ot \delta^\beta)$. This is because $d e_\alpha =0, d \delta^\beta=0$ by the definition of local connection $d$ on $E|_U$. 
     * 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.
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     * Calculus of Variation. Jacobi Field. Reference: Milnor Morse Theory (available online). Lee, Riemannian manifold, Ch 10 (available through library online, or {{ :math214:lee_ch_10_rm_mfd.pdf |here}}), Nicolescu's lecture note (following Milnor).      * Calculus of Variation. Jacobi Field. Reference: Milnor Morse Theory (available online). Lee, Riemannian manifold, Ch 10 (available through library online, or {{ :math214:lee_ch_10_rm_mfd.pdf |here}}), Nicolescu's lecture note (following Milnor). 
     * 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. 
 +    * 04-24:  {{ :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]]
math214/home.1587489179.txt.gz · Last modified: 2026/02/21 14:44 (external edit)

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