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touch of the galois

As you will no doubt already know, there’s been a lot of talk in the last few days about a potential proof of the abc conjecture. I just gave up my last professional non-fiction writing gig last week, which means that I no longer have any obligation to explain to you what that is, or write even vaguely short sentences about it.  but I still have the vestigial urge to find out, if only because of journalistic lure of an abstract mathematics page on Wikipedia being marked with the

Ambox currentevent.svg This article documents a current event. Information may change rapidly as the event progresses.

macro.

The thing is, the new proof is authored by Shinichi Mochizuki, who has been out doing his own deep explorations of mathspace on his own for so long, that everyone in the profession of math is having to race through his previous research to sufficiently understand his argument. Still, everyone can sense, Higgs Boson-like that this may be a big deal. When the rumor first began to emerge, the majority of professional mathematicians (as opposed to  you know,  the usual Diophantine analysis hangers-on) observed that via a reputational chain-of-trust calibration, whereby they were saying “well this isn’t my area, and it’s not this guy’s area either, but he’s closer to it than me, and I respect what he says in areas clone to mine, and  he says that it doesn’t look incoherent, and he wouldn’t say that without putting some of his reputation on the line, so I guess it might be legit. For now.”
I’m clearly about three links down the interpretative chain — I got the link about the abc conjecture from Hacker News, which was posted by somebody linking to a blog by one of these mathematicians saying that he couldn’t understand the proof, but golly. Dumbly, I immediately do want to understand the proof, even though the people who might be professionally qualified to understand this theory are themselves having to madly catapult themselves from newly-constructed research projects trebuchets to get near over the nearest conceptual ramparts.

I click on this link to mathoverflow, a Q&A site whose very existence I would not have conjectured until today. I mean, I don’t know molecular genetics, but sit me down with a copy of a Nature article and I can at least begin to get some dim silhouette of what’s going on. I can read something as “the noun verbs the other noun near this noun, prompting adverbal verbing over there in the bigger noun”, and at least begin to sketch out the correspondence.

I cannot even get a purchase on these explanations. This is mathematics, which mean that — to my mind at least — it is the study of the innate structure of correspondences themselves, which means I can’t even get a shape in my head. I read sentences like “I believe the Frobenioid associated to a number field is something close to the finite \’etale covers of Spec(OF) (equipped with some log structure) together with metrized line bundles on them, although it’s probably more complicated”, and I’m thinking: I won’t even be able to cut-and-paste that.  This is someone who knows his metrized line bundles, and they’re having to hand-wave.

Anyway, knowing it’s futile, I grab onto a word that seem relatively freight with meaning, but of which I have some dim recollection of. “Galois theory”. Okay, I’ve heard of Galois theory. Let’s call down Wikipedia on that, and see if it stirs any recollections and I can use it to hitch just a few inches higher up the chain.

Evariste Galois. Delineator of Galois theory, radical French republican, died in a duel. Oh, now I remember where I’ve heard of Galois theory.  I’m nineteen years old, and I’m in a maths class in college. This is pretty unusual in a British university unless you’re actually taking mathematics — usually you only take classes in the single topic you’re studying. I’m (partly) learning economics, though, so there’s some a little bit of catch-up in mathematical analysis to be done.

We’re being taught by what I now guess must have been a postgrad, and she’s the best explainer of maths-beyond-my-scope I’ve ever met. She’s also, she admits, incredibly hungover, and keeps getting sidetracked from the basic statistics she’s been sent to hitch us up to wander into her own topic of interest. Which, I guess now must be Galois theory, because  the bit that stuck in my mind was her elaboration on Evariste Galois. She had, she explained, a huge math crush on Evariste, and who wouldn’t? Flunked two colleges, fought to restore the Republic, imprisoned in the Bastille, and managed to scribble down the thoughts that would lead to several major fields of mathematics, before dying in a duel — either romantic or political — at the age of twenty.

Well, I’m nineteen at the time, so as a nineteen year old I’m thinking “I still have a year to pull that off”. But listening to this in cloisters of St Hilda’s,  I observe the  same reputational chain effect. Here is clearly the coolest person I’ve yet met at Oxford, and she is clearly in awe of someone else who is, I guess, her to the power of some unknown value of fascinating. I don’t understand Galois theory, but my tutor has already dedicated her life to it. There’s no way that either of us is ever going to live up to Evariste, but maybe just by lining up him as a goal, and pushing off in that general direction, perhaps we’ll get somewhere interesting.

Do we have to understand completely to be pulled along in its wake? Is it foolish to even queue up behind those who are so far behind the front lines? Isn’t this how we feel our way ahead, tied together by emotions, but walking together toward the truth?

7 Responses to “touch of the galois”

  1. Dave Green Says:

    >”I just gave up my last professional non-fiction writing gig last week…”

    - OK, I’ll bite: where can I find your fiction writing..?

  2. Danny O'Brien Says:

    Haha, no, I was clumsily trying to be explicit about writing tech journalism, but didn’t want to say either journalism or tech, because it’s been columns for the last few years, and the rest of the post was about math not tech.

  3. Jouni K. Seppänen Says:

    Here’s some wonderful writing about the ABC conjecture: http://www.math.harvard.edu/~mazur/papers/scanQuest.pdf

  4. John Baez Says:

    Great post! The romance of mathematics is a wonderful thing!

    “I read sentences like “I believe the Frobenioid associated to a number field is something close to the finite etale covers of Spec(OF) (equipped with some log structure) together with metrized line bundles on them, although it’s probably more complicated”.”

    And the scary part is, that’s someone trying to reduce Shinichi Mochizuki’s work to the usual old stuff! I mean, everyone in that business knows you should look at the finite etale covers of Spec(O_F), and so on. What’s weird about Mochizuki’s work is that he claims to be using ideas from the foundations of mathematics – set theory – to create new ‘universes’ in which to do math. And then, he’s claiming to take mathematical structures from one universe and transport them to another.

    If you look at his homepage, you’ll see a sign saying ‘Inter-universal geometer’ next to a picture of him staring bravely off into space. That gets the idea across as well as anything.

    But here’s a passage from his 4th paper:

    “In particular, one must continue to extend the universe, i.e., to
    modify the model of set theory, relative to which one works. Here, we recall in passing that such “extensions of universe” are possible on account of an existence axiom concerning universes, which is apparently attributed to the “Grothendieck school” and, moreover, cannot, apparently, be obtained as a consequence of the conventional ZFC axioms of axiomatic set theory [cf. the discussion at the beginning of §3 for more details]. On the other hand, ultimately in the present series of papers [cf. the discussion of [IUTchIII], Introduction], we wish to obtain algorithms for constructing various objects that arise in the context of the new schemes/universes
    discussed above — i.e., at distant Θ^±\ell NF-Hodge theaters of the log-theta-lattice — that make sense from the point of view the original schemes/universes that occurred at the outset of the discussion.”

  5. Danny O'Brien Says:

    That’s a great description-of-the-shape of it! Thank you John!

  6. Waider Says:

    One of Greg Bear’s books runs into this, a bit. Essentially he has this idea, extrapolated somewhat from present reality, that a pathfinder like Mochizuki effectively alters reality through his discovery in a way that makes his work graspable by others. I seem to recall him using this idea in both Distress and a short story.

  7. alwayslurking Says:

    Greg Egan, I think…

                                                                                                                                                                                                                                                                                                           

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