Good Topics to Know For Seminar, Cont.
Well, here are a few more. My picks are more basic, I think - if you're looking for knowledge that will be useful often, best to start with the "easy" stuff.
Yeah, so this stuff isn't really easy, but I think these are the topics that seem to come up over and over again in GFT and the ASS. I'd be happy to talk about some of these when I get back.
In other news, the space that Pekka suggested turns out NOT to be even locally Ahlfors 2-regular, so it is not fit to be a counter-example. So it's back to the drawing board. I think we have so more ideas...
Hope all is well.
- Advanced Basic Complex Function Theory (advanced basic?! Here I mean things like the Koebe distortion theorem and other extremal problems; ie what conformal map minimizes the quantity _____?)
- Basic geometry of fractal spaces (properties of Cantor sets, Seipinski carpets, etc)
- Plane Topology, especially recognition theorems (i.e. give topological conditions for a subset of a metric space to be: a circle, an arc, a continuous image of [0,1], etc.)
- The most general change-of-variables theorem you can find
- Harmonic Measure and the Dirichlet Problem
- A very basic understanding of stochastic calculus
- A toolbox of dimensions (Topological, Hausdorff, Asymptotic, Box, Conformal, etc..)
Yeah, so this stuff isn't really easy, but I think these are the topics that seem to come up over and over again in GFT and the ASS. I'd be happy to talk about some of these when I get back.
In other news, the space that Pekka suggested turns out NOT to be even locally Ahlfors 2-regular, so it is not fit to be a counter-example. So it's back to the drawing board. I think we have so more ideas...
Hope all is well.
posted by Kevin at 4:29 AM
5 Comments:
My picks are more basic, I think - if you're looking for knowledge that will be useful often, best to start with the "easy" stuff.
.. Yeah, so this stuff isn't really easy, but I think these are the topics that seem to come up over and over again in GFT and the ASS.
I was wondering what you meant by "easy." Your ideas are good, and I'll say more about them in forthcoming comments.
Also, I often use the abbreviation "AnSS," myself, but I think in our crowd, it's a nonstandard term. q:
The most general change-of-variables theorem you can find
Do you mean in terms of pushforward measures T#dx for a transformation T, or something more like a CoArea formula?
Now that I think about it, both are fair game. (;
I was thinking, in particular, of the theorem we had in Mario's class last year that says: if $U,V$ are domains in $R^n$, and $f:U \to V$ is a homeo with property (N), then
$$\int_V g dm_n = \int_U (g \circ f) D\mu_f dm_n$$
for every measurable function $g:V \to R$, where $D\mu_f$ is the measure derivative of $f$.
ummm...I guess you'll need to run that through LaTeX to read it...sorry.
Anyway, it would be a cool seminar to talk about this, pushforward measures, and (co-)area formulas.
Anyway, it would be a cool seminar to talk about this, pushforward measures, and (co-)area formulas.
I agree. Now it's a matter of convincing someone to give this talk .. q:
I was thinking, in particular, of the theorem we had in Mario's class last year
Not to split hairs, but the notion of a measure derivative is a special case of a pushforward measure (where one uses a homeomorphism instead of a more general map).
[cf: notes from QC Maps class, 27 Sept '04]
It's a more tangible and useful special case, though: condition N permits you to use the Radon-Nikodym derivative and Lebesgue measure instead of this icky pushforward measure .. the principle being that measures "we know" are less icky than measures that "we don't know." q:
I'll ask around and see who I can convince to talk about this.
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