Ok, the holidays are behind me. One friend called me to make sure I was ok, because I had made no blog entries in two weeks. Well, my Xmas letter took a little longer than usual this year; and there was the return to work. I think that trying to write about mathematics had affected my letter-writing style, and I had to work at recovering it. For the Xmas letter, one of my guiding principles is to not take myself too seriously.
For this blog, however, I worry about the line between seeming all-too-human and seeming off-the-wall. What comes across as human in a personal letter may seem flaky in mathematics. Maybe I’ll just have to be myself and leave it for you to judge.
Donal O’Shea’s “the Poincare’ conjecture” arrived from Amazon Wednesday. I read chapters 1 & 2 before getting caught up in routine, then read the rest of the book last night (Friday).
It was good. I want more. The author says, “I write for the curious individual who remembers a little high school geometry, but not much more – although I hope that those with substantial mathematical backgrounds will also enjoy the book.” I suspect he blew non-mathematics-majors out of the water more than once, but not often.
(I’m not sure I can be trusted at this any more than I can be trusted to assess whether “hot” Chinese food is “too hot” for a friend. I’ve had “too hot” exactly once in my life; it was homemade and the cook was going for “as hot as chemically possible”. In any restaurant I know, I automatically pour hot oil on whatever they serve me. It’s useless to ask me if something is too spicy. It may be equally useless to ask me if high school is sufficient for something about mathematics.)
Back to the Poincare’ conjecture. I recognized the author, Donal O’Shea, as one of the authors of “ideals, varieties, and algorithms”, so I knew I was getting a book by a mathematician about but not of mathematics.
I particularly liked a quotation from Perelman, the Russian mathematician who proved the conjecture (and declined the subsequent fields medal, and may decline the one million dollar millennium prize):
“I’m not good at talking linearly, so I intend to sacrifice clarity for liveliness.”
I was stunned by one piece of mathematics: any surface can be given a geometry under which it has constant curvature. I need to go track that down.
Some of you – if there are any of you – have just realized how ignorant I am of differential geometry – of which I do know something, but clearly way less than I thought.
I ordered “Ricci Flow and the Poincare’ Conjecture (Clay Mathematics Monographs)” by John Morgan. It may well be too advanced for me, but it was half the price of an undergraduate textbook, so why not?
I also ordered one of the easier “further readings”: “The Shape of Space (Pure and Applied Mathematics)” by Jeffrey R. Weeks.
And somewhere in Amazon I stumbled across another book by an author I like: “The Topology of Chaos: Alice in Stretch and Squeezeland” by Robert Gilmore. I have his “catastrophe theory for scientists & engineers”, “lie groups, lie algebras and some of their applications”, and “elementary quantum mechanics in one dimension”.
Of course I ordered his “topology of chaos”.
The Poincare’ conjecture strikes a familiar chord. The idea of a Ricci flow is to write a differential equation that smoothes out the curvature of a surface by letting it flow toward regions of less curvature, like multi-dimensional heat. What strikes me is that, once again, we see a problem in one field solved by moving it into another field; a problem in topology has been turned into a problem in partial differential equations. the challenge is that there is too much mathematics for any one person to do anymore.
I also looked at some old business. It seems that Eta Carinae looks ready to go supernova real soon now; of course, on a stellar time scale, that could be in another 100,000 years.
But don’t I have some books on supernovae? Yes, two; and one of them is only 10 years old, but it’s what they learned from a supernova in 2006 that set the deathwatch on Eta Carinae, so even that book of mine is too old to tell me why they think this star is ready to go. And I can’t say the book looks all that interesting, so I’ll not move it into the top 500 or so technical books I want to read. If Eta Carinae detonates in my lifetime, it should be as bright as the full moon, so it’s not like anyone will have tell me about it. (I do get my head out of my books often enough to notice a second moon.)
Now, let’s get back to principal component / factor analysis.