Difference between revisions of "User talk:Harold"
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| + | == A collection of thoughts on Morse theory == | ||
| + | |||
| + | I'm currently trying to figure out why in Morse homology, the degree of the attaching map of a certain <m>n</m>-cell is somehow equivalent to the number of gradient flow lines. | ||
| + | The setup is as following. Let <m>(M, g)</m> be a closed (i.e., compact and connected) smooth Riemannian manifold (without boundary), and suppose <m>f: M \to \mathbb{R}</m> is a smooth map satisfying the following properties: | ||
| + | # for each <m>x \in \mathrm{Crit}(f) := { p \in M : df_p = 0 }</m>, the Hessian <m>\mathrm{Hess}(f): T_pM \times T_pM \to \mathbb{R}</m> is non-degenerate. '''TODO Define the Hessian.''' | ||
| + | # <m>f|_{\mathrm{Crit}(f)}: \mathrm{Crit(f)} \to \mathbb{R}</m> is injective. | ||
| + | |||
== Caveat with xymatrix == | == Caveat with xymatrix == | ||
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* [[/Xymatrix caveat]] | * [[/Xymatrix caveat]] | ||
See how you can scroll right? [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 22:08, 14 February 2017 (UTC) | See how you can scroll right? [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 22:08, 14 February 2017 (UTC) | ||
| + | |||
| + | == Some copy-and-paste-help == | ||
| + | |||
| + | It's good to render diagrams in tables, if only because they look a bit sparse with the white background (unless they're huge), try these: | ||
| + | {| class="wikitable" border="1" style="overflow:hidden;max-width:35em;" | ||
| + | |- | ||
| + | | <center><span style="font-size:1.2em;"><mm>\text{YOUR MATH HERE}</mm></span></center> | ||
| + | |- | ||
| + | ! Comment | ||
| + | |} | ||
| + | To float to the right: | ||
| + | <div style="float:right;margin:0px;margin-left:0.2em;"> | ||
| + | {| class="wikitable" border="1" style="overflow:hidden;max-width:35em;margin:0px;" | ||
| + | |- | ||
| + | | <center><span style="font-size:1.2em;"><mm>\xymatrix{A \ar[r]^f & B}</mm></span></center> | ||
| + | |- | ||
| + | ! Comment | ||
| + | |} | ||
| + | </div>Now you can write here and reference the diagram on the right | ||
| + | * Lists and everything | ||
| + | ** Baby | ||
| + | Hope it helps [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 22:15, 14 February 2017 (UTC) | ||
Latest revision as of 20:49, 15 February 2017
A collection of thoughts on Morse theory
I'm currently trying to figure out why in Morse homology, the degree of the attaching map of a certain [ilmath]n[/ilmath]-cell is somehow equivalent to the number of gradient flow lines. The setup is as following. Let [ilmath](M, g)[/ilmath] be a closed (i.e., compact and connected) smooth Riemannian manifold (without boundary), and suppose [ilmath]f: M \to \mathbb{R}[/ilmath] is a smooth map satisfying the following properties:
- for each [ilmath]x \in \mathrm{Crit}(f) := { p \in M : df_p = 0 }[/ilmath], the Hessian [ilmath]\mathrm{Hess}(f): T_pM \times T_pM \to \mathbb{R}[/ilmath] is non-degenerate. TODO Define the Hessian.
- [ilmath]f|_{\mathrm{Crit}(f)}: \mathrm{Crit(f)} \to \mathbb{R}[/ilmath] is injective.
Caveat with xymatrix
Hey, try this page:
See how you can scroll right? Alec (talk) 22:08, 14 February 2017 (UTC)
Some copy-and-paste-help
It's good to render diagrams in tables, if only because they look a bit sparse with the white background (unless they're huge), try these:
To float to the right:
- Lists and everything
- Baby
