Isomorphismes

William Thurston, geometrizer of manifolds

Sometimes I like to spend an hour looking at something I barely understand. The inside of this guy’s mind has got to be so interesting, but it’s been shaped by geometry rather than words, so it’s very hard for him to express it. The geometry shaping it is also quite less limited than the square space we hit baseballs in, so it’s hard to draw as well.

I can offer some help on grokking what he’s saying, but there’s simply no way to absorb this stuff quickly. That said, I wouldn’t mind being able to imagine the platonic forms inside Bill Thurston’s head.

GLOSSARY-LIKE DISCUSSION

1. Topology. You want to understand why identifying the left and right side of a wide rectangle (declaring left = right, so that when you leave the left side of the Mario screen you appear again on the right) is the same as cutting a long strip of paper and taping the two ends together.
   
   (There’s a slight variation on that game that results in a famously weird space—the one-sided, single-edged Möbius strip).
   
   
   
   
2. Quotient spaces. You want to understand what it means to quotient a space. I can give a few examples. ℝ/ℤ would be the unit real interval [0,1) — kind of a microcosm of the real numbers themselves. The Western chromatic musical scale quotients 13 notes into an octave. It’s not that A4 is “equal” to A8, but it shares the same structural relationship as do E4 and E8.
   
   
   
   An orbifold is a manifold that’s been quotiented. Like if you took the plane and made an equivalence class of the vertical [0,1)’s with all the [1,2)’s and [2,3)’s and etc., you would be looking at an infinitely wide strip with all the verticality “wrapped up” in [0,1) — not gone, just wrapped up into one microcosm.
   
   You could also think about Groundhog Day (the analogy doesn’t work precisely). He’s living through the same span of time over and over because it’s been quotiented along the time dimension (the result of the division is a length of one day)
   
   Oh … equivalence classes are another thing you have to know about. I haven’t written about them yet. WVO Quine came up with a sensible definition of “what is 2” using the concept. And as Terence Tao wrote, when one uses a “noise-tolerant” definition — like if a lot of different ways of saying something can be taken to mean the same thing — that’s another example of an equivalence class.
   
   Back to the music theory for a second—there are multiple ways you could set up equivalence classes.
   
   • octaves — it’s not as if all “G♯” notes sound the same — but when we talk about octaves it’s usually with reference to the same-sounding-ness of twice-as-fast frequencies
   • inversion — I can do a major A triad as AEG, GAE, EGA … it’s a combinatorics thing; 3! ways. No, they don’t all sound the same, but when I use the word “triad” I am equivalence-classing over the kind of sameness that they do have.
   • enharmonics — Sure, D♯♯ and F♭ sound the same — but conceptually they’re very different, and the notes around D♯♯ will be different than the notes around F♭.
   • slight errors — players of the cello or the voice know that pitch is a continuous variable—however we might reasonably call 398 Hz = 400 Hz = A4.
   • transposition — Certainly composers choose the key of D (or, if they’re Stephen Sondheim, F♯) for a reason — but if a song isn’t within your vocal range you can always subtract or add a certain fixed pitch (in notes-space, not in Hz-space!) from every note and the piece will sound “the same” — not exactly the same, but it will recognisably be a pub song — I mean, the US national anthem
   
   
   If I say “Hand me that glass”, I don’t mean to reference the glass at a particular orientation, rotation, or place in the room—I mean to equivalence-class ∀ such configurations of the glass—they all mean “that glass”. And if I say “Hand me a glass” — “Which glass? This glass?” — “Any glass!” then I’m equivalence-classing ∀ glasses within a certain distance from you.
   
    
3. Hyperbolic geometry.  In square space, four right angles ∟ add up to the whole shebang 360°. But in the logical abstract it needn’t be that way. What if “space” consisted of 3 right angles ∟, or 12? Something to think about.
   
   Oh — and what if it took one number of azimuthal ∢ right-angles to make the whole pie round, and took a different number of planar right-angles to make that whole pie round? Yeah, that would be weird too.
   
   
   
   
4.  Watch the 20-minute movie Not Knot, where they explain that links—knots made of several (closed/looped/circular) ropes rather than just one rope—biject uniquely to the complement of some hyperbolic geometrical space.
   
   Since hyperbolic geometric spaces had already been explored a bit before the 1980’s, now everyone had a fun tool to unite concepts and ad-lib toward new ones. The new bijection opened up the gates to some easy logical shortcuts. I drew a picture of the way this kind of logic goes in talking about a clever way someone thought of to generate random normals with little computation.
   
   But this is in general how mathematicians solve impossible-sounding problems. I use a little bit of logic in domain X, as long as it’s easy there. Then I use this equivalence that somebody figured out to port the stuff into domain Y. Then I do that’s easy in domain Y. Then I either go back to my original domain or maybe I use some more equivalences to do easy stuff in domain ℤ, ℚ, Linear, and so on—always only using “obvious” logic in the particular domain, and letting the equivalences keep me right as I convert the problem across domains. The “link”-to-hyperbolic-complement-space was one such. Other examples include Fourier-to-regular domain, polynomials-to-sequences, equivalences-across-NP-complete-problems, graphs-to-matrices, matrices-to-characters, Lie-groups-to-matrices, …..
   
   Oops — just used another common maths word without defining it. Bijections are one-to-one mappings from the source domain onto the entire whole of the target domain. For example a strictly monotonic function from ℝ→ℝ uniquely assigns members ∈ℝ to other members ∈ℝ — in such a way that no value is reused and every value is used.
   
   A strictly monotone function injects the source into && surjects the source onto the target—which means it can be inverted. (By contrast, a non-monotonic, up-and-down-looking function, re-uses values, so going in reverse you couldn’t tell which usage the 3 had come from.)
   
   If ∃ a bijection between X and Y, then ∃ a correspondence between X and Y. When mathematicians are trying to speak casually, they will often say something like “You can’t comb a hedgehog” or “You can turn any 3-manifold into a 3-sphere”. “You can do” is their way of saying ∃ a bijecting function that relates the two: ƒ(X)=Y. If ∄ a bijection, then it’s impossible to put X and Y into correspondence — there’s no earthly or heavenly way in which these two things could be made to look alike. For example, maps must fail to correctly show the globe because ∄ a bijection between a globe and a plane. (They also fail because of distortions; that would be asking for a conformal, area-preserving bijection instead of merely a bijection.)
   
   They also show how spaces-with-stuff-removed can biject to completely unexpected things. A punctured plane is equivalent to the surface of a cylinder, for instance. (?!?!) The punctured surface of a ball is equivalent to a (not-punctured) plane, for instance. (‽‽) Hey, I don’t make this stuff up, I’m just reporting the facts.
   
   I guess in this talk he is showing different pictures of the associated geometry of various links. 
   
    
5. Look up Hopf fibrations, one-point compactification, nilgeometry, solvegeometry, Lie groups (they’re groups, but continuous rather than discrete), Hopf circles,  …. on Wikipedia. Be forewarned: this may turn into a months-long reading project.
   
   
   
   
   
6.  Complements. Not Knot talks confusingly on this topic (“it’s not empty space, it’s space that’s not even there” … I think that way of talking only makes sense to mathematicians).
   
   As I said in (2), spaces-with-stuff-removed can be homeomorphic to something completely unexpected. If you remove a point from the plane you introduce cylindricity around that point. Kind of unexpected that poking a hole in a square space makes a circular space, but that’s logic for you—always pointing out that illogical-sounding things are in fact inescapably true. 
   
   The symbology for complements looks too similar to the symbology for quotients. Sorry, not my decision. ℝ\ℚ = the irrationals ℚ∁. ℂ\ℚ∁ = the curliness of √−1 without the ridiculously, insanely thick thickness of the continuum. A manageable space in which not all sequences converge. ℝ\Transcendentals = Algebraics. Another eminently reasonable number system that does everything you’d want without the messy insanity.
   
   ℚ\0 = all fractions, minus zero. This is a punctured thing. ℝ²\0 = the punctured plane. ℝ³\0 = the cubic solid we seem to live in (Newton’s rigid rods) minus a point in the center of the universe. I don’t know if ℝ³\0 bijects to a looped thing like ℝ²\0.
   
   

   

   

   
   The world\Snoopy. Logically it’s equivalent to the punctured cubic thing I just described. Kind of boring, I thought removing Snoopy would be more devastating.
   
   BTW, you can also adjoin things, like ℚ adjoin i = ℂ\ℚ∁ mentioned above. I like this one if you can’t tell. ℝ adjoin ∞ is the one-point compactification of the line (as long as ∞ is defined to be ± ∞ so you can get there from the left or right)
   
   
7. Symmetries.  The peace sign has a 3-way symmetry. Mirror images are 2-way symmetries. You could draw a flower with a 5-fold symmetry or a 12-fold symmetry and so on. The concept itself isn’t confusing, but the way Thurston and Not Knot talk fluidly, assuming without making explicit the implications of identification, quotienting by symmetry, topological gluing, point/line removal, and complementation together, is overwhelming.