more of “The Poincare Conjecture”

 

It seems a little unfair that I listed Weeks’ “The Shape of Space” in my bibliography some time before I listed O’Shea’s “the Poincare conjecture”. Oh, “the shape of space” is a fine book, and I’m very glad I got it, but it was O’Shea who got me started – restarted – on all this.

I noticed this omission when I was editing the bibliography. I omitted it just because I wanted to say more about the O’Shea book but hadn’t figured out exactly what to say.

books for the Poincare Conjecture

 The following books have been added to the bibliography.

First I read O’Shea’s “Poincare”; it’s way past time it showed up in the bibliography. Then I bought Weeks and “Ricci” based on O’Shea’s “further reading”. One is high school – barely – and the other is high research. I’ve skimmed Weeks once and have settled down to read it again. “Ricci” is beyond me, as I expected.

While trying to find out how a surface could be given a geometry in which it had constant curvature, I ended up at the back of O’Neill; he’s an old favorite. While trying to sort out topological and differential structures, I took another look at “Instantons” – right up there with “Ricci Flow” – and then discovered why I had bought Bloch in the first place: I could read him. 

books on euclidean & non-euclidean geometry

 

Since I hope that some readers of “the Poincare Conjecture” might be interested in Euclidean & non-Euclidean geometry, I have taken a look through my geometry bookshelf and constructed the following entries.

Until several years ago I had never gone back to look at high school geometry; there was this huge gap in my education, between high school geometry and Riemannian geometry (the geometry of general relativity).

When I perceived that gap, I acted on a couple of recommendations, acquiring Martin’s “Foundations” and Greenberg’s “Non-Euclidean”. The Hartshorne was a later acquisition: he’s known for a challenging book on algebraic geometry, and I just had to see what he did with Euclid.

I also picked up a few ancient used high school geometry books, but I’m not going to include them! And for now, I’ll omit Hilbert’s “Foundations” because we’re getting his work second-hand in the three non-Euclidean geometry books.

I wish I’d known how readable Heath’s “Euclid” was; i’d have bought it early on.

One of the things that fascinates me most in plane geometry is the frieze and wallpaper groups. (Sorry, go look them up! For me, the marvel was that we could say there are only 7 frieze and 17 wallpaper groups.) Martin’s “transformation” and Beyer’s “tessellations” book are devoted to them.

Pedoe I recognized as a co-author of a classic: Hodge & Pedoe “methods of algebraic geometry”; Hodge is he of the “Hodge decomposition” and the “Hodge star operator”. I bought Pedoe’s book as much for his name as for its title.

the poincare’ conjecture & other stuff

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).