three (3) more lectures scanned ..

.. 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!

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.

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.

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 ..

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

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

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.