Friday, December 24, 2004

A Fine Paper (not mine!)

I was having printer troubles a few days ago in Ann Arbor. As I scowled at the page of feebly printed text (the toner cartridge was low), a friend and fellow second-year looked over and shook his head.
"You might want to try taking out the cartridge and shaking it around before putting it back. Sometimes that lets me print for another 30 pages or so."

Squinting at the lightly printed words, he gave me a quizzical look.

"You know that our break is only ten days long, right?" he asked.

I nodded cautiously.

"Do you really want to spend it by reading math papers and ~possibly~ getting confused and/or irritated by said papers?"

"Oh, no. I know one of the authors of this paper," I replied, waving the printout in the air. "Very good mind, his! It should make for good reading."

"Your time off, your decision." he said, giving up.

"Well, it can't be any more painful than my relatives asking me: 'when are you going to get a real job?'" I added.

He grins. "Good point."

It is a fine paper, and you can read it here: [link].

If anything, it demonstrates clarity of thought. It discusses very well the motivation and connection between the 'classical' theory of QR (quasiregular) mappings and this more substantial subclass, called QRG (quasiregular gradient) mappings. Their example of a radial stretch is also rather illuminating.

Now that I think about it, this QRG is a very nice and natural object to study, as well as a startling intersection of many ideas. Within it, there is:
  • this aforementioned Theory of QR/QC Mappings;

  • a nod to Differential Topology, because these maps are complex-valued gradients (to be understood in the weak sense), and hence an analogue to exact 0-forms;

  • an undercurrent of Elliptic PDE, which arrives purely from the characterization of the primitives of these gradients;

  • even a little Fourier Analysis, although it was introduced in the examination of homogeneous QRG mappings and the testing of a conjectured optimal Holder exponent.
Perhaps my only disparaging remark about this paper is that the authors make a conjecture in Section 4, but in their Concluding Remarks they admit that the conjecture has been solved .. by one of the authors, no less!

I suppose that this is not so much a technical fault but a personal idiosyncracy of one of the authors. After all, who can fault someone for answering a question?

At any rate, this paper is good mathematics. I only wish I can write so well .. or rather, come up with such good results!

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