I mentioned earlier this week that I’d been crafting a long post on the fabric of the Universe when I was sidetracked by relatively mundane political events. Now I’ve been sidetracked again by the entirely unexpected (to me) news of the death from melanoma, at age 65, of the Fields Medalist Bill Thurston, who devoted his life to understanding the shape of space.
One-dimensional topology is the study of curves and two-dimensional topology is the study of surfaces. Both subjects are quite well understood. Thurston was the king of three-dimensional topology, which gains additional interest from the fact that we perceive ourselves as living in a three-dimensional Universe. Three-dimensional topology attempts to classify all the possible shapes for that Universe.
One of course is also interested in four, five, six and many-dimensional topology, four dimensions being of particular interest because they can be used to model space together with time. But although three dimensions are more complicated than two and two are more complicated than one, it turns out that when you go much higher, a lot of things get simpler. Consider knots, for example. There are no knots inside a one or two dimensional space; a knot needs three dimensions in which to pass over and under itself. But in more than three dimensions, you can untie any knot just by pulling on its ends — roughly because the additional dimensions give it so much space in which to untangle itself. For those and related reasons, topology is often hardest in three and four dimensions — coincidentally (or maybe not) the very dimensions most relevant to the way we experience the world.
Thurston revolutionized three-dimensional topology in the 1980s with his geometrization conjecture, which says that any three-manifold (the three-dimensional analogue of a smooth curve or surface) can be cut up into pieces, each of which exhibits one of eight permissible geometries. The simplest of those geometries is the flat three-dimensional space you think you see around you, where you can draw three straight lines in mutually perpendicular directions and extend them forever. Another is the geometry of the three-dimensional sphere, which is an analogue of the two-dimensional surface of the earth, where any “straight” line eventually circles back to meet itself.
The geometrization conjecture was important, but what really mattered was the vast array of new techniques Thurston introduced for visualizing and understanding the structure of three-manifolds. When those techniques came on line in the early 1980s, he was widely acclaimed as the mathematician of the decade.
One thing that set Thurston apart was his insistence that mathematics is a human study, and that it’s the mathematician’s job to communicate not just theorems and proofs, but a unique way of thinking. Stories are often told of mid-twentieth century mathematicians (usually French) who, when asked a question about their work, would scribble a picture on the blackboard, deliberately stand in front of that picture to shield it from everyone else’s view, and then, having studied it a few minutes, erased the picture, turned around, and gave a purely formal explanation designed to obscure all of the motivation and insight. Nobody ever told a story like that about Bill Thurston. Here he is, talking about the mystery of three-manifolds; dip in at a random moment and chances are excellent you’ll hear him talking not about how he proved a theorem but about how he sees the world:
For more, have a look at Thurston’s partly autobiographical essay on Proof and Progress in Mathematics. Or check out his questions and answers on MathOverflow, where he (along with several other Fields Medalists and many others of a similar caliber) has been an active participant.
I’ve said before that in a just world, there would be some level of accomplishment that exempts you from mortality. In a just world, Bill Thurston would still be with us.
“Hey Bill, I heard your bachelor party had a Moebius stripper. How was she?”
“Pretty decent. But I never got to see her back side.”
Student: “I have to say, you’re the worst professor. You wouldn’t know you ass from a whole in the ground.”
Topologist: “Very true; but I could tell the difference between my ass and 2 holes in the ground.”
I was hoping that the NY Times would do a better job of describing his accomplishments. Instead we get this nonsense:
“very rarefied group in his field that thinks deep theoretical thoughts with no particular practical application”
“cosmologists have drawn on Dr. Thurston’s discoveries in their search for the shape of the universe”
“despite working in a realm of rather cold abstractions, [he] was personally very warm”
http://www.nytimes.com/2012/08/23/us/william-p-thurston-theoretical-mathematician-dies-at-65.html
My advisor once remarked about a paper published in a Computer Science journal words to the effect “You can tell Comp sci is not like mathematics yet, since they write to make it understandable.”
On the video lecture. Mathematics must be the only field where the leader in the field can give a lecture using and praising the PhD thesis of another guy, but that guy cannot get a job because his type of research is not respected.
Perhaps a stupid question but you say “One-dimensional topology is the study of curves and two-dimensional topology is the study of surfaces.” I thougth a curve existed in two-dimensions (x,y) and a surface in three (x,y,z)?
Andy: A curve is a one-dimensional object. Some (but not all) curves can be imbedded in the plane (with coordinates x,y). (Google for “twisted cubic” to find a curve that can’t be imbedded in the plane.) A surface is a two-dimensional object. Some (but not all) curves can be imbedded in euclidean three-space (with coordinates x,y,z). (Google for “Klein bottle” to find a surface that can’t be imbedded in three-space.)
What’s usually interesting to the mathematician is the dimension of the object itself, not the dimensions of the ambient spaces in which the object can be imbedded.
Re 5 and 6:
Not a stupid question at all. Consider a soap bubble, maybe one stretched on a metal frame of some shape. That bubble forms a surface. It is 2 dimensional, even though it curves through 3-D space. There are various ways to define ‘dimension’, some more general than others, but for most purposes your direct untuition is godd enough. In a small area, how many independent directions can you head off in.
I see, a curve is itself one dimensional because there is really no difference between say y=x^2 and y=x if you are sort of travelling along the curve because you can only travel to the right or to the left, or along the x-axis. With a surface you can only travel along the x and y axis so it’s two dimensional but lives in x,y,z.
… sort of.
@Andy. Yes exactly. The technical way of saying what you just did in 8 is this. In a very small area of what we are talking about, curve, surface, etc, can we smoothly bend it until we get a normal line or normal plane, or normal 3-space. Or normal N dimensional box. But it is important to notice the “in a very small space” part.
A standard example is a bobius strip. It is 2-d but has only one side. That is not like a normal surface!
Ken B, In slightly less technical language and omitting the bending part, smooth or otherwise, am I correct that your standard example of a bobius strip is a 2-dimensional curve which may be embedded in more than one 3-dimensional or n-dimensional box? Definitely not a normal surface.
@Bill B:
Oops. Mobius, not bobius. My mad :)
Yeah, a mobius strip in a small area looks pretty normal. But in its totality it is not, it is non-orientable. That is there is no up or down.
A mobius can be embedded easily in 3d. Take a long strip of paper, twist it once ie holding the lwft end fixed, turn the right end upside down. Now glue the ends together. You get a loop with a twist int. If you trace your finger on it you will find 1 side, 1 edge. There will certainly be dems on youtube too.
“I thougth a curve existed in two-dimensions (x,y) and a surface in three (x,y,z)?”
There are multiple ways to describe such things and they all have their uses. As Professor Landsburg noted, there are two main ways to describe curves and surfaces, first via local (one doesn’t need a coordinate system to cover the entire object, just a piece for every part of the object) internal coordinate systems which are respectively one and two dimensional, and via external coordinate systems, which can be much higher in dimension. These are the “intrinsic” and “extrinsic” descriptions of the object.
Knot theory, for example, is the study of the ways that a loop can be described in three dimensions where the loop is allowed to move as long as the loop never passes through itself. As such, it is a study of extrinsic descriptions of a loop. But all of these loops have the same intrinsic description, they’re just a circle.
As Professor Landsburg noted, the intrinsic description is considered more useful because it is completely independent of how the object is placed in another space. Even such things as the shape of the object in that bigger space or how it intercepts a vector field (think a fluid flowing through the larger object) can be encoded as intrinsic data.
Even with this focus, extrinsic descriptions turn out to be very useful. For example, one can often derive information about a higher dimensional object from classifying how loops and surfaces of spheres or tori can embed in that object. These can be used as a means for finding holes, for example.
@Ken B:
Ok, an uninteresting typo was always a possibility but when I googled bobius . . . .
Well, to say the least, I gained a new perspective on your comment and the descriptive power of mathematics
@Bill B: HA! I thought I had a big vocabulary and a twisted sense of humour. Clearly I have more to learn ….
“Hey Betty, I heard your bachelorette party had a bobius stripper. How was he?”
@AMTBuff: Twisted!
:)