As of last week, I think we've settled this extension business in the smooth category. Now all that's left is to give a talk about it. You guessed it: there's always a catch.
Always. [sighs]
Oh well. I've got three weeks to pare it down into something manageable for two hours on a Thursday, and 'twill a friendly, forgiving crowd at that ..
Lately I don't feel like a newcomer anymore, but I still feel like a mathematical tyro. It's not an uncomfortable feeling, if only because of practice. I've now been accustomed to feeling foolish, unaware, and slow-witted about half of the moments of the day. It's a feeling like getting older, where you don't improve with age and instead, resign yourself to those faults and vices that will never go away.
I'm not certain what causes this sentiment. Perhaps it's the mental response to the end of this research investigation, and I'm assessing what, if anything, I''ve accomplished in my graduate career, or even during the brief years of my life.
Perhaps it's because we're nearing prospective student weekend, that time of year where we current grads meet and chat with our potential successors: our versions 2.0. I could swear that either the incoming grads are better prepared every year, or that I was woefully unprepared when first I came to Michigan.
Perhaps it is simply the timing of my third year, second semester at UM, and I realise that if all goes according to plan, then the game is half-over and the thesis nowhere near half-complete.
Most likely this is nothing new. I've been feeling old all my life, and maybe my body's biology has finally caught up.
Maybe I should say more positive things. It's too easy to resort to pessimism and disillusionment. So I guess I'll say a little about work.
It's more extension stuff, but I get to work once more with Sobolev spaces and revisit something slightly classic: the Dirichlet Problem, the Poisson extension operator, and its topological and regularity properties. In low dimensions, say 2 or 3, it might be a more viable tool than the Gehring-Schoenflies extension because of concreteness. Then again, harmonic homeomorphisms are a tricky hope: sometimes minimizing energy doesn't mean preserving mass, and in general, topological embeddings of space may be much to hope for.
Well, it's work. If I've learned anything as a grad student, it's that steady work is something to be appreciated and sought out, because good ideas are rare in their happenstance.
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