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.

O’Shea’s “the Poincare conjecture” is, by and large, a pleasant mix of biography and history, and it mentions some fascinating mathematics, but – quite appropriately – it barely scratches the surface. I want more. I hope, at the very least, that some of its readers end up saying things like:

“He makes (Euclid’s) geometry sound a lot more interesting than it was in high school.”

“He talked about the non-Euclidean mathematicians rather than non-Euclidean mathematics. Where do I find the mathematics?”

“What do you mean, there are an infinite number of ways to do calculus in 4D space?”

There are 3 things that stand out for me in “the Poincare conjecture”. 

The easiest to mention is its reference to V. I. Arnold’s “On Teaching Mathematics”, which can be found in HTML and PDF at

http://pauli.uni-muenster.de/~munsteg/arnold.html

and

http://www.math.boun.edu.tr/instructors/ozturk/arnold.pdf

respectively. It’s a tirade – by an outstanding mathematician – about abstract and formal mathematics divorced from its roots. My favorite story from it is of the French schoolchild who is asked, “what is 2+3?” He answers, “it’s 3+2, because addition is commutative.”

True, but some of us were hoping to hear “5”. Some of us think “5” is the essential answer.

He also says he taught group theory to schoolchildren in Russia, and the notes of the class are available in English. That book just arrived this afternoon. I haven’t had a chance to look at it, but i’m eager to see what he did.

The next easiest to mention is O’Shea’s “further reading”. Here is where I found Weeks’ “the shape of space”, which tries to do what Arnold says: play with surfaces, play with ways to understand 3D manifolds, do not just go for abstraction. Weeks was a student of Bill Thurston, who apparently not only had an awesome geometric imagination, but also encouraged intuition in others. Some notes and a book of Thurston’s are listed, along with a single-volume exposition of Perelman’s proof of the Poincare conjecture, “ricci flow and the Poincare conjecture” by Morgan & Tian. 

The Thurston book is on its way to me. The “Ricci flow” has already arrived. 

Yes, I bought it too. No, I can’t read it. But until I can, I’ll know that there’s more for me to understand in my intermediate books.

The most outstanding thing in “the Poincare conjecture” was his statement that “… Poincare & Klein’s work implied that any surface could be given a geometry in which it had a constant curvature….”

That blew me away. I’m embarrassed, but not as much as I might be: none of my books ever phrased it that way. I have books about surfaces of constant curvature, and none of them said that they included, in some sense, all surfaces.

It took me a while to find out what O’Shea was talking about. I looked through old familiar – obviously not that familiar – texts, but Weeks gave it to me. He phrased it as “every surface can be given some homogeneous geometry.” And he went on to explain how.

The embarrassing part is that it’s not only a named theorem, but it’s one of those “the major theorem of this book” theorems. 

Its name is Gauss-Bonnet.

What it says is that an apparently geometric quantity (the total curvature, i.e. the integral of the curvature at each point) is equal to a topological invariant (the Euler characteristic or Euler number, vis. the number of vertices plus faces minus edges).

Ok. But that doesn’t say anything about changing the geometry.

But here’s what O’Shea and Weeks mean. Take a donut. More precisely, take the surface of a donut. Or take an inner tube. These are 2D surfaces if we imagine them of zero thickness – like all the lines we drew with straight-edge or compass in high school, that really had width but we imaged they didn’t. 

The donut sits in 3-space, and it acquires an induced geometry from that 3-space: we can compute the lengths of paths on this donut, as we can compute lengths of paths on the surface of the earth, using Euclidean geometry but staying on the surface. We’re not allowed to tunnel from here to china; must go by land and sea. Ok, by plane, too.

More to the point, we can compute the curvature at every point on the surface of the donut; and then compute the total curvature by integrating. 

We get zero. The curvature isn’t zero everywhere: it’s positive on the outer parts, and negative on the inner parts, but the total is zero.

Now we do something topological. We cut the donut and unbend it so that it becomes a cylinder. Then we cut the cylinder the long way and – voila’ – we have a flat piece of paper.

The total curvature of the cylinder is zero, like the donut, but more importantly, the curvature of the cylinder or of the piece of paper is zero everywhere.

It was that unbending, stretching the inner part of the donut and compressing the outer, that changed the curvature to zero everywhere.

We have given the donut a homogeneous geometry, one in which it has constant curvature. The key restriction is that we have preserved the Euler number, so each of the total curvatures remains zero. in particular the constant curvature is zero. Topological transformations won’t let us change that.

Want to learn more about this? For intuition and understanding, read Weeks. For the actual mathematics, I’d go with O’Neill’s “elementary differential geometry”, 2nd ed. If you want to see even more about it, Bloch’s “a first course in geometric topology and differential geometry” will talk about topological, polyhedral, and smooth structures on surfaces. (All of these are undergraduate texts, and the Weeks book is probably accessible to high school readers.)

Yes, the three structures coincide for surfaces. But it was only a few years ago that I learned “topologists study three types of manifolds – topological or continuous… piecewise linear… differentiable….” it was news to me. (That’s the opening sentence of the introduction to another one of those books I can’t read yet, freed & uhlenbeck’s “instantons and four-manifolds”.)

I bought the Bloch book because it would show me all three structures between two covers. I have just started actually reading it. If you’ll pardon my saying so, “Yummy.”

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. 

Bloch, Ethan D.; A First Course in Geometric Topology and Differential Geometry.

Birkhauser 1997; ISBN 0 8176 3840 7.

[topology, geometry;4 feb 2008]

upper division mathematics.

it is about 3 kinds of structure on surfaces: topological, simplicial, and differential. the restriction to surfaces makes it a quite readable introduction. if you’re like me, you’ve seen each of these structures, but never together.

Freed, Daniel S. and Uhlenbeck, Karen K.; Instantons and Four-Manifolds.

Springer, 1984. ISBN 0 387 96036 8.

[topology, geometry;4 feb 2008]

from the first paragraph of the preface: “This book is the outcome of a seminar…. to go through a proof of Simon Donaldson’s Theorem…. the nonsmoothability of certain topological four-manifolds…. by studying the solution space of … the Yang-Mills equations….”

in other words, this is a highly advanced book. but who could resist the title? certainly not me.

O’Neill, Barrett; Elementary Differential Geometry.

Academic Press, 1997 (2nd Ed); ISBN 0 12 526745 2.

[differential geometry;4 feb 2008]

the first edition revolutionized the teaching of the subject, by introducing differential forms to undergraduates. the prerequisites are multivariate calculus and linear algebra. the 2nd ed. has more material on intrinsic geometry, e.g. more material on the gauss-bonnet theorem. 

this is one of “the books”. it’s also one of the very small set of which i have deliberately bought a later edition primarily to thank the author for the first edition.

O’Shea, Donal; The Poincare’ Conjecture;

Walker Publishing Co., 2007.  OSBN 0 8027 1532 X

[history of math, topology, geometry;4 feb 2008]

“This book is about a single problem. Formulated by a brilliant French mathematician, Henri Poincare, over one hundred years ago….” a pleasant mix of biography and history, and it mentions some fascinating mathematics, but – quite appropriately – it barely scratches the surface. i want more, but this is a good introduction.

Morgan, John and Tian, Gang; Ricci Flow and the Poincare Conjecture.

American Mathematical Society, 2007; ISBN 0 8218 4328 4

[topology;4 feb 2008]

this is the proof of the poincare conjecture. i may never understand it, but i couldn’t resist it. 

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.

 

Beyer, Jinny; Designing Tesselations.

Contemporary Books 1999; ISBN 0 8092 2866 1.

[frieze & wallpaper groups; 3 feb 2008]

this is a rare find. it is written by a woman who designs quilts; she made it her business to learn about symmetry, and to get a couple of mathematicians to help her out. this is a visually magnificent and mathematically accurate presentation of all the wallpaper groups (and the frieze groups). 

Greenberg, Marvin Jay; Euclidean and Non-Euclidean Geometries.

W.H.Freeman & Co., (2nd Ed) 1980. ISBN 0 7167 1103 6.

[euclidean & non-euclidean geometry; 3 feb 2008]

this would be my first choice if decide to work through the subject. it has a comfortable writing style despite being a math book.  this one i have only browsed, but it reads easier than martin’s “foundations”. oh, it’s intended as lower-division mathmatics. hey, i’m not too proud to read it – once, but not any more.

Hartshorne, Robin; Geometry: Euclid and Beyond.

Springer, 2000; ISBN 0 387 98650 2.

[euclidean & noneuclidean geometry; 3 feb 2008]

and more. a fairly wide-ranging text, upper-division mathematics. i wouldn’t pick it up first, but i’m glad to have it. i would probably go through this after greenberg and after martin, for all the auxiliary material (e.g. field extensions).

Heath, Sir Thomas L.; Euclid: the Thirteen Books of the Elements.

Dover, 1956 (2nd Ed.); ISBN 0 486 60088 2, 60089 0, and 60090 4.

[geometry; 3 feb 2008]

i daresay this is almost the definition of geometry. what makes these books extaordinarily informative is the commentary by Heath. you should have this handy whenever you read a modern book about euclidean geometry. in fact, if you have this book by your side when you’re taking high school geometry, your geometry teacher will either love you or hate you.

Martin, George .; The Foundations of Geometry and the Non-Euclidean Plane.

Springer  1975 (2nd printing 1986); ISBN 0 387 90636 3.

[euclidean & non-euclidean geometry; 3 feb 2008]

upper-division. what i love about this book is its list of 26 equivalents to euclid’s parallel postulate. nevertheless, a fairly dry book. i would probably go through it after i had done greenberg. (i’ve more than browsed this book, but not by much. i read this in preference to greenberg because martin looked more like a math book.)

Martin, George .; Transformation Geometry.

Springer  1982. ISBN 0 387 90636 3.

[geometry, symmetry, frieze & wallpaper groups; 3 feb 2008]

introductory, requiring no college math. the mathematics of frieze and wallpaper groups. i’ve played with them, but i want to work through this book.

Pedoe, Dan; Geometry and the Visual Arts.

Dover, 1983; ISBN 0 486 24458 X

[geometry; 3 feb 2008]

an introductory book. the first three of its nine chapters are devoted to vitruvius, durer, and da Vinci. not your usual geometry book. not a text but a chance to play with geometry as found in art. i think you better have had high school geometry; at least know what a straightedge-and-compass construction is. from the preface: “This book can be taken by the general reader as a diversion into the by-ways of history, with glimpses of the enormous importance of geometry to such people as….”

 

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).

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.

http://www.universetoday.com/2007/06/21/come-on-eta-carinae-explode-already/

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.