Quasiconformal Analysis on Fractals, Winter 2006

Question

2006-06-08T00:37:00.000-07:00

Hey Folks!

Thanks to Jasun for posting all the notes. They're great. I hope you all enjoyed the workshop too - I was really jealous.

So here's the question. I haven't done much research on this, so I don't know if it's impossible or totally trivial.

A theorem of Ahlfors and Beurling* states that every quasisymmetric map of the circle extends to a quasisymmetric map of the disk. That is, every quasicircle is the boundary of a quasidisk. Now, every bi-Lipschitz map is quasistymmetric, so every bi-Lipschitz image of the circle is the boundary of a quasidisk. But, does every bi-Lipschitz map of the circle extend to a bi-Lipschitz map of the disk?

The obvious thing to try would be to see if the Ahlfors-Beurling** extension for quasisymmetric maps produces a bi-Lipschitz map from bi-Lipschitz boundary data. However, I don't really remember how the Ahlfors-Beurling*** extension works, and since I'm on the road, I can't really look it up (a technical version is in Lehto's book, but Mario explained to me once the "right" way of thinking about it - damn my crappy memory!) Anyway, any insights would be appreciated.

Another way to think about this would be to use some Lipschitz extension theorems. However, the McShane extension wouldn't work because there is no way in that construction to ensure that the extension of a bi-lip homeo is a bi-lip homeo. (one only gets Lipschitz, not bi-Lipschitz, which is a result of the fact that one must apply the extension to the coordinate functions individualy). Maybe Kirzbaum is better? (again, I'm on the road so I can't look up the proof of kirzbaums theorem).

Ok, thanks!

*Thanks to Anonymus for correcting this reference.
**Ditto
***Ditto Ditto

last licks: a belated update.

2006-05-04T15:45:00.000-07:00

the last five lectures (32-36) are available at the usual place. the very last lectures have the flavor of a seminar talk, and for more depth or details, refer to the ICM preprints of M. Bonk and B. Kleiner.

sorry for the delay; after classes ended, i was hardly anywhere less than 10 meters from a coffee stand. q:

belated update.

2006-03-31T16:43:00.000-08:00

seven (7) new lectures available in PDF at the usual place. sorry for the delay.

to clarify matters, every monday or so is a double session of student talks for grade assessment. i've chosen not to take notes during these talks, but happily every speaker thus far has made LaTeX PDF handouts for the audience. if you would like some, then go bug these people:

a review of basic riemannian geometry (jason miller): in which we examine the riemann curvature tensor through connections, and compute the sectional curvature of hyperbolic n-space to be constant (-1).
on liouville's theorem (aaron magid): this is a proof of a rigidity theorem: given a domain in euclidean n-space (n ≥ 3) a thrice-differentiable conformal transformation is the restriction of a möbius transformation. the proof rests on differentiating to obtain an overdetermined system of pde, where our computations are motivated by appropriate curvature quantities from riemannian geometry.
the hyperboloid model for hyperbolic space (karl weintraub): this is a construction of an equivalent model for hyperbolic n-space, using the lorentz metric on a light cone, defining an appropriate projection, and demonstrating that it is isometric to the disc model.
on the measurable riemann mapping theorem (marie snipes): the outline: a quick introduction to sobolev spaces and quasiconformal mappings, then formulating the beltrami equation, and through means of clever singular integral operators (the cauchy and hilbert transforms), one proves that given two simply connected domains in the plane and a beltrami differential, there is a quasiconformal homeomorphism of the prescribed domains and with the prescribed dilatation.

FTP question

2006-03-23T00:40:00.000-08:00

Why is that ever since we migrated from math.lsa.umich.edu to itd.umich.edu, I have been unable to use sftp to get to my file space?

Before, from any terminal I could run sftp name@login.math.lsa.umich.edu and get to my space, but a similar command sftp name@login.itd.umich.edu yields an error "connection reset by peer". I get the same error using the JYU network as I do from my laptop.

Any ideas how I can fix this?

cool theorem

2006-03-20T04:18:00.000-08:00

Consider the following game:

Let epsilon > 0. Let U be a domain in the plane, and define "the ball" to be a point x in U. There are two players, A and B. A turn consists of

1) Player A picks a unit vector v.
2) Player B sets b= +/- 1.
3) The ball is moved from x to x+b*sqrt(2)*epsilon*v.

The goal of A is to move the ball outside of U. The goal of B is prevent A from doing so.

It is not to hard to show that A can always win. In fact, if u_ep(x) = (epsilon)^2 k where k is the minimum number of turns in the optimal winning strategy for A and x is the starting point of the ball, then u_ep is bounded independently of x depending only the diameter of U. What is really interesting is that the function u defined by

u(x)=lim_(epsilon ->0)u_ep(x)

satisfies the pde

|grad(u)|div((grad u)/|grad u|) = -1 on U
u=0 on bdry U.

This is the same pde that governs the mean curvature flow. Creepy.

after long last ..

2006-03-10T14:56:00.000-08:00

there are five (5) new lectures available here. i apologize for the delay; after spring break i forgot to stop by shapiro library and scan the lot of them.

one lecture has no theorems in bold ink; a classmate's pen ran out of ink, and he borrowed mine.

Question

2006-02-27T03:54:00.000-08:00

Hi Folks!

Sorry I've been out of touch. You know, I've been trying to integrate myself (was area man, now volume man, plus constant). Anyway, I haven't had enough coffee, and I seem to proving contradictions in mathematics. Can anyone verify or debunk the following seemingly obvious statement? I have hopelessly confused myself.

Let C be the 1/3 Cantor set. Let s=log2/log3. Then Hausdorff s-measure(C)=1=Hausdorff s-content(C).

Right? Philosophy: For the Cantor set, the obvious cover is the most efficient. Of coursing proving this is a pain - see notes from Juha's course last semester.

Am I crazy?

lectures 17 through 19 ..

2006-02-20T13:31:00.000-08:00

.. ready for download at the usual place.

a word of caution: the last few lectures are somewhat technical in nature, so keep your eyes open wider for errors.

now we're up to sixteen.

2006-02-13T12:47:00.000-08:00

.. lectures, that is. Lectures 14, 15, and 16 from last week and today are up for download at the usual place. Be warned:

In case the resolution looks a little strange, the original paper notes were on blue paper (it was the only 3-hole punch paper I could find without paying for it) and I saved these notes as greyscale JPEG graphics and then PDF files.
I oscillated with what the 'k' means in the definition of a CAT(k) space .. that is, if it is a nonpositive or a nonnegative parameter. The meaning, I hope, should be understood.
The latest lecture may be error prone. Mario himself refers to it as "the hardest theorem in elementary mathematics," and I might agree with him. It's a tricky manipulation of symbols, and now I know how my Calculus II students from yesteryear must have felt! q:

Wow

2006-02-07T10:38:00.000-08:00

Check out this beautiful survey article for ICM by Mario.

you know the drill.

2006-02-06T13:15:00.000-08:00

Lectures 11 through 13 can be downloaded here in PDF format. I felt the urge to kill the snotty nosed-brat who was hogging the library scanner, but now I'm calm and he gets to live.

Anyways, enjoy .. and for the record, the disparate dry humor in Lecture 13 is M. Bonk's, and not mine. I'm hardly that clever.

this week's SAnS title/abstract

2006-02-02T08:58:00.000-08:00

hi Kevin, and all other interested parties;

I've submitted an abstract for UM Student Analysis Seminar this week, and the format will be a little different from previous times and traditional seminars.

This is a call for open problems: they don't need to be high-powered and result in a publication, but an opening to stimulate thought. The goal is to raise interest in varying subfields in analysis, and possibly lead to collaborations amongst us students.

So, anyone know a good problem? q:

Tues, 7 Feb 2006 (5-6 pm) Title: Open Problem Session Speaker: none Abstract: Interested parties will bring unsolved problems (related to mathematical analysis; possibly research level) and we will spend the hour discussing them, and hopefully make progress in solving them. When choosing problems, we would prefer that the background not be too lengthy; problems of a general background (e.g. complex analysis, measure theory) are encouraged.

three (3) more lectures scanned ..

2006-01-30T12:41:00.000-08:00

.. and the link is here.

The latest lecture (today) went to reasonably general matters, and there was a handout of possible talk topics for non-candidate students (omitted here).

For the record, the guy sitting at the computer next to me brought a very fragrant cup of coffee, and being hungry and thirsty I nearly killed him and stole the aromatic, steaming cup .. q:

Met with Kai and Pekka!

2006-01-26T13:18:00.000-08:00

Thanks for the notes Jasun! It seems we're still in introductory material, but it's quite nice to see the proofs spelled out so clearly. Mario's courses are great for just that reason - some day I'm going to have to use these facts, and I could probably prove them on my own, but it'll take me forever if I don't really work through the proofs now!

In other news, I talked with Kai for quite some time today. He's *really* sharp. Also he speaks quite quietly. He mentioned that a result Mario and I had proven could also be used in to show more. If you remember, last semester I gave a seminar talk outlining the proof that a 2-reg, LLC metric plane is quasiconvex. The proof used a co-area formula, and the general idea appears a few places in the literature - in vague form in Semmes paper (where the result was first shown, by showing the much stronger result that a Poincaré inequality holds), and explicitly in one of Mario's papers. Anyway, Mario and I worked out the details over the summer. Last week Kai gave a talk about Fubini's theorem, and mentioned the Co-Area formula. Afterwards, I told him about the quasiconvexity result, and he mentioned today that the same method may be able to be used to show directly that a 2-reg LLC metric plane is a Loewner space. This would be REALLY cool because it approaches the strength of Semmes result - to see why, read Theorem 9.10 in Juha's LAMS.

After that I met with Pekka. We discussed that the space he had considered as a counter-example to his theorem didn't really suit the theorem - it isn't proper, and isn't locally 2-regular in a satisfactory sense. However, he believes (and I'm inclined to trust him) that this indicates that in fact a stronger theorem is true, and that his space is a counter-example to that the stronger theorem is sharp. So, we now have three steps: 1) Formulate the stronger theorem (check), 2)Prove that the space is indeed a counter-example to show the stronger theorem is sharp, and 3) prove the stronger theorem. Unfortunately, I am more comfortable with 3) than 2) - I'm finally going to have to face the fact that I'm not really comfortable with modulus arguments. I have no excuse for this! I feel terrible about it and I'm working hard to really "get" what the idea behind some common modulus and capacity results are. Yeesh.

Ok, that's all for now.

updated: more lectures scanned.

2006-01-23T13:48:00.000-08:00

We're up to the present day, now: my longhand notes for Mario's class are available in PDF format at the following URL:

http://www.math.lsa.umich.edu/~jgong/qfractals/

As usual, the good ideas are his and the errors are mine.

At the moment, there are seven (7) lectures, with plenty more to come as the Winter Term progresses.

Good Topics to Know For Seminar, Cont.

2006-01-18T04:29:00.000-08:00

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.

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.

a request for topics ..

2006-01-17T21:37:00.000-08:00

I'm not sure if this blog is meant solely for QC analysis, so this might be a bit tangential.

We've begun a theme at Student Analysis Seminar, titled "A Toolbox of Ideas: Useful Concepts in Analysis" or some permutation of those words. The idea is to empower us students, and in particular the younger students, so that the faculty-run analysis seminars are easier to break into.

Earlier we had a brainstorming session and thought of the following ideas (generously recorded by Marie):

~~Sobolev Spaces & Weak Derivatives~~ (I spoke on this today, in fact)
Type and Cotype
Quasiconformal/Quasisymmetric mappings
Distributions & Currents
Metric Measure Spaces & common constructions
Capacity
K-Convexity
Several Complex Variables Intro Topics

At any rate, Kevin: I was wondering if you had any suggestions about what would be useful to know, when walking into the analysis seminars.

First Project

2006-01-12T13:58:00.000-08:00

So I've met with Pekka a little bit and we've discussed a problem to begin work on. Namely, we'll be trying to show that a result he proved recently with Zoltan Balogh and Sari Rogovin is sharp. The result extends a some classic theorems of Gehring (who else) to locally Ahlfors regular metric spaces. See here for a more detailed discussion.

Jasun has also sent me the first 3 sets of lecture notes. If we can get Mario's permission, I'll post them here as well. I haven't read them yet, but when I do I'll leave some comments here. Does anyone know if there are going to be homework problems. Now that I don't really have to do them, I kind of hope there are some!

If you'd like to be a member of this blog, and make posts, just send me an email.

Welcome

2006-01-10T02:01:00.000-08:00

Hi Folks! This is the not-so-long awaited math blog. The idea here is two-fold.

1) We can use this blog to talk about Mario's analysis class here, even though I'm in Finland. Hopefully Jasun will be scanning notes - I'll post them here (with his permission), and then we can discuss. I'll also try to do the homework - we could discuss that here too. I'll try to post the solutions I come up with.

2) I'll talk here about what research I'm doing with Pekka in Finland this semester. I think many of you will find it interesting.

Anyway, please feel free to leave comments and/or questions. I miss you guys and I look forward to working with you, however remotely, this semester.