Conservation of Paranoia / Planning Ahead.

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? (;

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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:

a month passes ..

.. and my advisor has returned from .. where he was, earlier.

  (I never asked where he went, and he never mentioned it)

Again I am accountable for my mathematical deeds. Now all I need to do is find him and allow myself to be held accountable.

I guess it hasn't been too unproductive of a month .. but there are few results to speak of. Earlier in June, co-author and I ran into some technical troubles in the final stage of our proof ..

  (for those keeping score at home, this is the second project, not the first)

.. so now the project has been put back on indefinite hold. It only adds to my deep-rooted suspicion that as much as I like geometric problems, I'm likely little good at them and should stick to more analytical topics, such as PDE.

I did have an idea to circumvent that technical difficulty, the other day, but I suppose it has to wait until the weekend. Thesis work comes first.