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.
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.
posted by Kevin at 1:18 PM
3 Comments:
Also he speaks quite quietly.
That's an understandment, if I ever heard one. I think I bumped into him at a bookstore last year (just before he left) and when I said Hello, he didn't just nod, but said Hello back and asked me how I was doing!
I don't suppose he's much different in Finland?
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,
I'll add a corollary to that: I might be able to prove them on my own, but somehow, Mario's proofs make loads more "sense" than mine ever will: they are intuitive and clear and give no wasted effort.
For example, stereographic projection makes a lot more sense when Mario describes it. It even sounds cool, which it was before, I suppose. But now it is pretty cool. q:
I'm finally going to have to face the fact that I'm not really comfortable with modulus arguments.
I know exactly what you mean; modulus is still a foreign notion to me, too.
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