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?

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