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!
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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.
posted by Kevin at 1:58 PM
5 Comments:
Does anyone know if there are going to be homework problems.
I think Mario adopted Juha's philosophy that: "candidates attend, and pre-candidates give talks."
Mind you, I only paid attention to the first part of that .. q:
Namely, we'll be trying to show that a result he proved recently with Zoltan Balogh and Sari Rogovin is sharp.
Is this the same work as in this preprint on the Scuola Normale Superiore server?
As long as we're on the subject: how exactly do you define the Sobolev space W^{1,p}(X, Y) for when X and Y are metric spaces? Do you need to embed the spaces into some large-dimensional euclidean space?
so no homework... :(
The preprint you found is indeed the one. It's pretty interesting - they prove a new covering theorem which is a twist on the 5B and use it to produce their result. As for the Sobolev space, they use upper gradients. f is in W^{1,p}(X,Y) if f has an upper gradient g in L^p(Y) and there is an there is an x_0 in X with u(x)=d_Y(f(x),f(x_0)) in L^p(Y). If X is proper, then each such f is absolutely continuous on p-a.e. rectifiable curve. The paper says that one should "notice" this fact, but I think it is not clear. Even in the case X=Y=R^2, it is not so easy to show - in Mario's class we had to use a technique involving rectangles, and in Juha's class we had to us e Fuglede's Lemma and some results about p-quasicontinuous representatives. One thing I hope to do here is understand this fact...
I see now that Jasun has already posted the first three lectures on his webspace, so there is no need to pot, so there is no need to post them on the blog. Maybe we can continue this arrangement and I'll just post links to the same address?
Maybe we can continue this arrangement and I'll just post links to the same address?
Sounds good to me. I wouldn't mind adding a disclaimer of the following type:
"Jasun has been known to be incapable of being able to think and write at the same time. Caveat emptor." q:
By the way, I'll upload more scanned notes by end of this week. Promise.
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