Archive for the ‘Mathematics’ Category

Topology and manifolds, part six

Holy crap, new jobs eat spare time and curtail blogging.

Moving on.

I’m hoping to finish this series with this post, because let’s face it I’ve let this drag on too long. We’ve established that “most” 3-manifolds are hyperbolic, for a suitably complicated and unstated definition of “most”, and we’ve established that if a 3-manifold is hyperbolic then the hyperbolic structures on that manifold are unique enough that we can define “the hyperbolic volume of this manifold” in a way that makes sense. Chronologically speaking, we’re now up to the late 1970’s or so in the history of this particular field. That’s when my academic grandfather, Bill Thurston (seen here promoting a line of women’s fashion in Paris…man, won’t that link screw up search indexing for this page?), together with another mathematician named Troels Jorgensen proved a remarkable and vexing theorem: that any collection of complete hyperbolic 3-manifolds with finite volume had to have a minimum-volume element.

Once again, I’m over-simplifying somewhat; the theorem in question proved a great deal more than that, and the statement above is just the most interesting consequence from my point of view. (I also haven’t defined what a “complete” manifold is; roughly speaking it just means “a manifold without holes”.) Compare the statement above to, say, positive numbers: there’s no such thing as “the smallest positive number”. Any time you think you’ve found the smallest positive number, just divide it by two: presto, you’ve found a smaller positive number! Jorgensen and Thurston said that didn’t happen for (complete) hyperbolic 3-manifolds. There is a smallest such manifold. More than that, there’s a smallest element in any non-empty collection of hyperbolic 3-manifolds you try to define. Pick a property that a hyperbolic 3-manifold might or might not have, and there will be a smallest manifold with that property and a smallest one without.

The vexing part? Jorgensen and Thurston didn’t tell you what a single one of those manifolds actually was.

The proof was entirely non-constructive: they only proved that all of these smallest manifolds existed, but gave you no idea what those manifolds might be, thus creating an infinite set of open questions. What’s the smallest hyperbolic 3-manifold? What’s the smallest orientable hyperbolic 3-manifolds? What’s the smallest cusped hyperbolic 3-manifold? (Pretend I’ve defined all of these terms.) For certain mathematicians, including myself, this is like red meat in front of a hungry dog. The fact that some of these questions have been open for thirty years…despite the fact that these are geometric objects, things that we could actually physically construct were it not for the trifling fact that the universe has the wrong curvature and not enough dimensions…has struck me at times as being both ridiculous and hilarious.

And this, finally, is where I enter the narrative. My modest claim to fame as a mathematician is that I, along with my academic father and my academic uncle, answered one of these questions, one of the big ones in my opinion. Thanks in part to me, mathematicians can now say “the smallest orientable hyperbolic 3-manifold is the Weeks Manifold.” Period. Full stop. The proof was long, and computer-assisted, and covered in three different papers the last of which is still griding through the gears of journal publication, but it’s done.

And that’s what I do for a living.

Or rather, that’s what I did for a living, because between the time I started this series of blog posts and today I left academia and am now working at Google in Canada as a software engineer, working on problems that have nothing, sweet jack-all to do with hyperbolic geometry and topology. How’s that for a commentary on my lack of blogging stamina?

Topology and manifolds, part five

Back to topology.

Last time, I was talking about how according to Thurston’s Geometrization Conjecture, any three-manifold can be broken into pieces in such a way that each piece has a geometric structure. Furthermore, out of the eight possible geometric structures, hyperbolic geometry is the only one for which the corresponding geometric three-manifolds haven’t been completely classified, and that’s one reason why hyperbolic three-manifolds are interesting.

Here’s a question: is the hyperbolic structure on such a three-manifold unique? Or, alternatively, can a manifold have two different hyperbolic structures?

This is a question with consequences, particularly if the answer is “hyperbolic structures are unique”. A manifold, in general, is a topological thing, not a geometric one. A geometric structure, at first glance, is something that we impose on the manifold: a choice that we make, of how to define “distance” and “angle” on that manifold. But if hyperbolic structures are unique, then that means we aren’t making a choice at all, but instead that “distance” and “angle” are determined by the manifold itself. Even though there’s nothing geometric in a manifold’s topological definition, somehow geometry springs forth from it anyway, unbidden.

Mathematicians love that kind of thing.

Unfortunately, hyperbolic structures aren’t quite unique. It is possible for a manifold to have two different hyperbolic structures. Fortunately, those structures are going to be really, really similar. Mostow’s Rigidity Theorem implies that any two such structures are going to be isometric: that there’s going to be a map from the manifold to itself which converts distances and angles from the first hyperbolic structure to distances and angles from the second. That doesn’t imply that the two hyperbolic structures are the same (note that the map from the manifold to itself doesn’t have be the identity map), but it does imply that the two structures are going to have many of the same properties.

For example, we can use a hyperbolic geometric structure to define the hyperbolic volume of the entire manifold. The volume isn’t necessarily finite, but when it is Mostow’s Rigidity Theorem guarantees that the volume is well-defined: it doesn’t depend on which hyperbolic structure we choose to define. The hyperbolic volume is a property of the manifold itself, not the hyperbolic structure that we choose to impose upon it.

In other words, hyperbolic volume is an invariant of hyperbolic three-manifolds. Invariants are the mathematician’s version of measurements: they’re how mathematicians distinguish and classify mathematical objects, as if they were species of butterflies or types of stars.

Hyperbolic volume is and was my particular favourite mathematical playground. We’re getting closer, ever closer, to my answer to “So what do you do?” but we’re not quite there yet…

Topology and manifolds, part four

So in part two I described all of the compact two-dimensional manifolds, and in part three I discussed how each two-dimensional manifold has a geometric structure. Let’s finally start talking about three-dimensional manifolds.

Simply put, three-dimensional manifolds are much, much harder than their two-dimensional cousins. Most of the clever tricks that can be used to classify two-dimensional manifolds are next to worthless in higher dimensions. But fortunately there exceptions to that rule. Geometric structures, it turns out, are as important to three-dimensional manifolds as to two-dimensional ones.

It’s helpful to have at least one example in the back of your head, so let’s construct a three-dimensional analogue of a torus, called a three-torus. We constructed a torus before by taking a rectangle and gluing opposite edges of the rectangle together, left to right and top to bottom. To construct a three-torus, start with a three-dimensional cube, and glue opposite faces together: left to right, top to bottom, and front to back.

It’s impossible to do this physically, of course. In six dimensions it’d be easy, but as we’re stuck in three dimensions we  just have to pretend that the gluing has taken place without actually doing it, so that if we were standing in the cube and walked out through the left-hand face we would “wrap around” and reappear in the right-hand face, and similarly for front and back, and for top and bottom.

(Did you ever see that “Avengers” movie with Uma Thurman? Remember the scene where her character gets trapped in a room which seems to have two exits but every time she leaves through one she seems to reappear through the other? It’s like that. But I digress.)

If we can make that conceptual leap, then the result of all that gluing is a three-torus. There are annoying technical details to worry about concerning the vertices of the cube and what happens when we glue them together: while in the two-dimensional case the worst thing that could happen was that the vertices didn’t lay flat, in the three-dimensional case it’s possible that the vertices come together in such a way that we can’t even construct a chart around the resulting point, meaning that the resulting object isn’t a manifold at all!

Also, it’s the edges that might not lie flat in the three-dimensional case, not the vertices.

Did I mention that three-dimensional manifolds can be annoying?

Don’t worry about it. Instead, take my word for now that in the case of the three-torus the vertices behave, so we indeed have a three-manifold, and moreover the edges lay flat when we glue everything together, so the three-torus is a Euclidean manifold, i.e. it has a natural Euclidean geometric structure just like the two-dimensional torus does.

Recall that every two-dimensional manifold had some kind of geometric structure, either spherical, Euclidean, or hyperbolic. Is the same true in three dimensions? Sadly, no.

However, in 1982 William Thurston proposed Thurston’s Geometrization Conjecture, which says the next best thing. Oversimplifying hugely for a moment, Thurston’s Geometrization Conjecture says any three-manifold which doesn’t have a geometric structure can be cut into pieces in a canonical way, such that that the resulting pieces do have a geometric structure.

Thurston’s Geometrization Conjecture was proved quite recently by Grigori Perelman, who completed a program first proposed by Richard Hamilton to prove the conjecture. And since proving the Geometrization Conjecture also proves a long-standing open problem called the Poincaré Conjecture, which was one of the Clay Millennium Prize problems, Grigori Perelman became one of the first person to win one of those. William Thurston is also my academic grandfather: my Ph.D. advisor’s Ph.D. advisor. So you can see why I’m interested in this sort of thing.

It turns out that there are eight different kinds of geometrical structure you have to worry about in three-dimensional geometry, but three of them are very familiar: the three-dimensional analogues of spherical, Euclidean, and hyperbolic geometries. It also turns out that geometric three-manifolds (i.e. manifolds which have geometric structures even before you cut them into pieces) have all been classified, except for hyperbolic three-manifolds. That, in my opinion, is the last great frontier when it comes to classifying three-dimensional manifolds. And it’s a huge frontier. Next I’ll try to describe my tiny piece of it.

Topology and manifolds, part three


So, what do spherical, Euclidean, and hyperbolic geometry have to do with manifolds? Well, it turns out that all two-dimensional manifolds have a natural geometric structure.

That statement will require some unpacking.

Let’s start with the torus, i.e. the surface of a donut, which is commonly described as follows: take a rectangle (imagine it to be made of some thin, flexible material) and then glue two opposite edges together to form a tube. Then bend the tube, and glue the ends of the tube together to get a torus. This is an example of a Euclidean geometric structure on a torus, which is a fancy way of saying that we’ve constructed a torus out of pieces of the Euclidean plane…namely, the rectangle…in such a way that all of the edges and corners glue together neatly.

This “gluing together neatly” part is important. Specifically, it’s very important that the corners come together neatly. In the case of the torus above, the rectangle had four corners, each of which was a right angle, that is, π/2 radians or 90 degrees. Those four corners all come together to form a single point after all of that gluing, and when they do those four angles add up to 2π radians or 360 degrees, i.e. a full circle. That’s exactly what we want to happen: the single “corner” of the resulting torus lies flat.

Let’s take another example. Instead of a single rectangle, let’s take six squares and glue them together to form the surface of a cube. Topologically, a cube is the same as a sphere, but geometrically we have a problem: the corners don’t lie flat. At each corner of the cube there are three right angles coming together, and that only adds up to 3π/2 radians, less than a full circle. Gluing six squares together in this way does not result in a Euclidean geometric structure on a sphere.

But not surprisingly, spherical geometry steps in here. In spherical geometry, the angles of every triangle, and indeed every polygon, add up to more than they would in Euclidean geometry. It’s possible to construct a square in spherical geometry whose angles are each 2π/3 radians, or 120 degrees…all four of them. If you take six of those squares and glue them together, you’ll get a spherical “cube” where the corners do lie flat: each corner is made up of three angles of 2π/3 radians, which adds up to a full circle. A sphere has, not surprisingly, a spherical geometric structure: we can construct it neatly out of pieces of the sphere.

One last example. This is the fun one, but you might want to draw some pictures as you go along.

Imagine two pairs of pants. Stitch all the holes together: waist to waist, leg to leg, and other leg to other leg. What you’ll end up with is a two-holed torus. Now, cut the two-holed torus back into two pairs of pants. Then, cut each pair of pants into two pieces with three cuts, as follows: from the left edge of the waist down the outside of the left leg to the bottom, from the right edge of the waist down the outside of the right leg to the bottom, and finally along the inside of both legs, from the left hem up to the crotch and down to the right hem. (Basically, cut the pants into the front half and the back half.) You’ve now cut your two-holed torus into four pieces in total. Each piece is a hexagon, made of three cuts and three half-hems (if you consider the waist to be a hem). When you stitch the hexagons back together to make the two-holed torus, the corners of the hexagons will come together in sets of four. If we want those corners to lie flat, the simplest way would be to ensure that each corner of each hexagon is a right angle; four of those will add up to a full circle, just like in the torus example at the beginning.

Hexagons with six right angles? Try drawing that. It’s just crazy talk…in Euclidean geometry. But we can construct right-angled hexagons in hyperbolic geometry. Thus, the two-holed torus has a hyperbolic geometric structure.

Whew! Let’s cut to the chase: every two-dimensional manifold has some kind of geometric structure, either spherical, Euclidean, or hyperbolic. Talking about the compact manifolds I described in part two, the sphere and the projective plane have spherical geometric structures, the torus and the Klein bottle have Euclidean geometric structures, and everything else has a hyperbolic geometric structure.

But I said in the interlude that all of this geometry had something to do with three-dimensional manifolds, didn’t I? Yes, yes I did…

Interlude about geometry

Before I try to explain what’s up with three-dimensional manifolds, I need to talk a bit about geometry, specifically spherical vs. Euclidean vs. hyperbolic geometry.

Euclidean geometry is “high-school” geometry, the geometry of the plane put into axiomatic form by Euclid of Alexandria. By which I mean, Euclid replaced the study of actual physical points and lines with the study of formal mathematical objects called “points” and “lines”. Euclid’s “points” and “lines” were inspired by real-world points and lines, and have properties based on the properties of real-world points and lines. For example, there’s a single, unique “line” through any two distinct “points”. But Euclid’s “points” and “lines” are just a formalism. They have the same relationship to real-world points and lines that poems have to roses.

And like poems, Euclid’s formalism permits analogies.

What if, instead of a plane, we consider the surface of the sphere? And instead of straight lines in the plane, what if we define a “line” to be a great circle on the surface of the sphere, such as the equator? Then we get spherical geometry instead of Euclidean geometry. The “lines” in spherical geometry don’t look like the “lines” in Euclidean geometry, but they act like lines in lots of important ways: they intersect in points, they subtend angles with one another, their segments have well-defined lengths, and so on. The formalisms of Euclid’s geometry are just a language, and by applying that same language to spherical geometry we can highlight the similarities between the two.

At the same time, we discover the differences. Triangles, for example: in Euclidean geometry, the angles of a triangle always add up to 180 degrees, but in spherical geometry they always add up to more than 180 degrees. Spherical geometry is finite: no shape can have an area greater than the area of the entire sphere. There’s no such thing as similar triangles in spherical geometry. Also, most famously (if that’s the word) there’s no such thing as parallel lines.

Hyperbolic geometry, like spherical geometry, uses the language of “points” and “lines” to describe a completely different space from the Euclidean geometry. But hyperbolic geometry is much stranger and weirder than spherical geometry. It’s hard to visualize: it doesn’t live on any surface in Euclidean space the way that spherical geometry lives on the sphere. And it turns the properties of spherical geometry on their heads. The angles of a triangle in hyperbolic geometry always add up to less than 180 degrees. Hyperbolic geometry is infinite…but bizarrely, no triangle can have area greater than pi. (Circles, on the other hand, grow as large as they want, exponentially with respect to their radius.) There are no similar triangles in hyperbolic geometry, but there area infinitely many lines “parallel” to any given line. And so on.

It’s just bizarre. It’s beautiful and it’s frustrating and it’s bizarre, and I’ve been studying it for so long that I’m more comfortable with it than I am with Euclidean geometry most of the time. I live in a Euclidean world, but I tend to think in a hyperbolic one.

Next time I’ll try to explain what this has to do with manifolds.

Topology and manifolds, part two

Continuing my detailed answer to the question, “So what do you do?”…

When a mathematician sees the definition of a mathematical object, they want to know what all the possible examples of that object are. It’s a pretty hard-wired reflex. So what are all the possible examples of manifolds?

Well, that’s a rather huge question. One-dimensional manifolds are easy: the circle and the real line. That’s it. Two-dimensional manifolds, i.e. surfaces, are harder, but the question has largely been answered. If we restrict ourselves to compact two-dimensional manifolds, then the question was answered at the beginning of the twentieth century, although the i’s weren’t dotted and the t’s weren’t crossed until the 1920’s.

I haven’t explained what compact means yet. Compact is, loosely speaking, a geometrical notion of finiteness. Of the two one-dimensional manifolds, the circle is compact, but the real line is not, because of the way it extends to infinity. Compact manifolds are an important subclass of manifolds, if only because they’re easier to study.

The list of compact two-dimensional manifolds starts with the sphere, e.g. the surface of the earth. Next is the torus, e.g. the surface of a doughnut. After that, picture the surface of a doughnut with two holes. Mathematicians usually just call this a “two-holed torus” or a “genus-two surface” if they’re getting technical. (I once misread an old topology book which used the term “quoit” and thought the word referred to something with the shape of a genus-two surface, but it turns out a “quoit” is just a ring. Pity.) Then there’s a three-holed torus, a four-holed torus, and so on.

That’s not the complete list. Have you ever seen a Klein bottle? Wonderful stuff. A Klein bottle is an example of a non-orientable surface, a class of two-dimensional manifolds that can’t actually be constructed in three-dimensional space. (The examples at the link cheat a bit by letting the surface pass through itself.) There’s a list of non-orientable two-dimensional manifolds that parallels the list of many-holed tori, of which the Klein bottle is the second on the list. The first is the projective plane, which is a Moebius band glued to a disk in a wonderfully mind-bending way which I won’t go into here.

So: sphere, torus, two-holed torus, and so on, plus projective plane, Klein bottle, and so on. Two simple, if infinite, lists containing every possible compact two-dimensional manifold. What’s more, if you present a mathematician with some arbitrary compact two-dimensional manifold, it takes only a few very simple calculations before the mathematician can tell you where in the two lists the manifold has to appear. It’s the best possible outcome to this kind of mathematical problem.

What about three-dimensional manifolds?

Oy. You have no idea.

Topology and manifolds, part one

One additional caveat: this is not going to be as detailed as it could be. If you’re looking for a detailed technical online resource I figure Wikipedia, Wolfram MathWorld, and half a dozen other websites already have that covered. This is just me, writing my detailed answer to “So what do you do?” Onward.

What is a manifold? It’s something on which we can do a generalized form of geometry1.

“Geometry” is a good starting point. It’s something most people can visualize neatly: triangles, circles, and angles are familiar concepts. For many people it’s also their first and only exposure to proofs, the specialized, stilted form of language that is mathematicians’ bread and butter. Euclid of Alexandria is so famous for these kinds of formal arguments that the high school geometry of triangles, circles, and angles is properly known as Euclidean geometry.

The problem is, Euclidean geometry is a bit too abstract. It takes place on an infinite, flat plane, an ideal object bearing no resemblance to that thing we build skyscrapers, tennis courts, and particle accelerators on: the surface of the Earth, which is neither infinite nor flat. The surface of the Earth looks infinite…okay, really big…and it looks kinda-sorta flat depending on where you are, but that’s just because we’re really tiny and close to the ground. In reality, it’s a surface not at all like the ideal Euclidean plane. For that and other reasons, it’s necessary to generalize the mathematics of the Euclidean plane to other, more interesting shapes.

That’s where manifolds come in. A manifold is something which, like the surface of the Earth, looks like the Euclidean plane if you’re very small and standing right up next to it.

The technical definition of a manifold involves charts and atlases: a manifold is something which has an atlas made of charts, where each chart looks likes a portion of the Euclidean plane. The charts necessarily overlap, and together they map the entire manifold. On any one chart we can attempt to apply the mathematics of the Euclidean plane, but we can’t necessarily do that to the entire manifold, much like how we can lay flat any chart in an atlas of the Earth but can’t lay flat an entire globe.

The technical definition also has a very great deal to say about how those charts overlap. The short version is, how smoothly the charts overlap determines how much and what kind of mathematics you can do on the manifold. But let’s not go there.

I also should point out that what I’ve talking about so far are two-dimensional manifolds. We can define, say, three-dimensional manifolds by using “charts” which look like portions of Euclidean three-dimensional space. How such charts can come together to form an entire atlas is already much harder to visualize than in the two-dimensional case. It’s also a lot more interesting, in many ways. More on that later.

1And calculus. In fact calculus on manifolds is probably more important than geometry on manifolds. But the word “calculus” tends to frighten people.

Topology and manifolds, introduction

Be a pure mathematician for any length of time and you will come to dread the question “So what do you do?” Okay, perhaps “dread” is too strong a word. You’ll at least learn to limit your answer, because the questioner is almost certainly not asking “Would you tell me in detail about your job?” More likely, what the questioner is really asking is “Would you please summarize what you do in the hope that it will provide a hook for further small talk?” (It won’t.)

The correct answer is “I’m a mathematician.” Occasionally people will ask for more detail, at which point you trot out a brief summary of your field of study that you’ve refined over and over again with family members. Very, very rarely, you’ll be asked to explain in detail. Cherish these moments.

Because those moments come up far too infrequently, I thought I might write a detailed explanation here. More to come in future posts.

(Yeah, this post is just to shame me into writing the other posts later.)