So I met with my advisor yesterday for the first time in .. wait, six weeks?
I'm still inclined towards disbelief, but between my advisor's one month holiday with his family and the travelling-planning issues for a week before and after, I suppose it must be right. Wow.
At any rate, we met and I showed sufficient progress, but apparently I was working on the wrong metric space. Instead of Euclidean space Rn, the results were meant for its 1-point compactification, or if one wishes concreteness, on the n-sphere Sn in Rn+1.
Thus for more than a week, I was worried about a issue that is, in fact, a non-issue: the (local) Sobolev regularity of the inversion map x → |x|-2 x is problematic near the origin in Rn, but the equivalent mapping is a diffeomorphism on Sn, so I was worried over nothing ..
.. well, sort of.
Something else is now suspect.
For those mathematically-inclined, I'm trying to prove an extension theorem for a particular type of homeomorphism on particular types of domains. In my efforts to resolve the (non)issue, I began a systematic examination in how "large" a sub-domain can the old and new homeomorphisms agree in value.
This in turn led to the study of the functions' behaviors for points approaching infinity, and rather interesting limits, at that! Apparently these same limits might cause trouble for another function in a different part of the proof. It would have been fine were we in Rn, but it may go wrong in Sn.
I suppose some difficulties never go away, do they? (;
On a less technical note, I gave my advisor a preprint of an article of my co-author's and mine, and he didn't seem too displeased by it. Then again, he didn't get much time to read it during our meeting, so I suppose we'll wait and see ..
We also talked a little about the future, and my previous guess now seems true: after this term I'll have gotten all my course requirements completed (both cognate and distributional), and tentatively my oral Prelim Exam is set at the end of the Fall 2005 Term.
If I pass, it would mean that I'll be a Ph.D. candidate by January. Zounds! "Candidate" has such a nice ring to it, too .. q:
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