Category Archives: Math

The Three Phases of the Mathematical Life

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This autumn, I got the chance to ask a few questions of Ngô Bảu Châu.

If your jaw is not on the floor, it’s because (A) you’ve spent shockingly little time browsing the list of Fields Medal winners, and (B) you’re not Vietnamese.

A helpful Vietnamese journalist I met explained to be that Châu is “the biggest celebrity in Vietnam.” Châu won his Fields Medal in 2010 for proving—hands inside the vehicle, please, because this is a wild ride—a key relationship between “orbital integrals on a reductive group over a local field” and “stable orbital integrals on its endoscopic groups.”

In Vietnam, that relationship is apparently the one sizzling on tabloid covers.

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Châu is not your prototypical superstar. Even in Vietnam, apparently, he is a cryptic figure; not a chatty TV celebrity, but a silent legend. At the press conference where I met him, at the Heidelberg Laureate Forum, he gave some journalists terse one-sentence answers. Not because he was being standoffish, but because a mathematician like Châu never proves in ten lines what he can prove in just one.

I didn’t know what to ask him. I’m not a research algebraist and have never been mistaken for one. So I asked about his education, his youth in Vietnam, his mathematical coming of age.

How does Ngô Bảu Châu get to be Ngô Bảu Châu? Continue reading →

The Joy of Slightly Fishy Proofs

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I’ve got a proof stuck in my head.

It comes from a man as brilliant as he is grumpy. Last week in the staff room, one of my all-time favorite colleagues busted out a dozen-line proof of a world-famous theorem. Though I doubt it’s original to him, I shall dub it Deeley’s Ditty in his honor:

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This argument is sneaky like a thief. I admire but don’t wholly trust it, because it solves a differential equation by separation of variables, a technique I still regard as black magic.

What I can’t deny is that it’s about 17 million times shorter than the proof I usually show to students, which works via Taylor series:

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You might call this one Taylor’s Opus. It builds like a symphony, with distinct movements, powerful motifs, and a grand finale. It takes familiar objects (trig and exponential functions) and leaves them transformed. It’s rich, challenging, and complete.

It’s also as slow as an aircraft carrier making a three-point turn.

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Here we have two proofs. The first is a nifty shortcut, persuasive but not explanatory. Watching it unfold, I’m unsure whether I’m witnessing a scientific demo or a magic trick.

The second is an elaborate work of architecture. It explains but perhaps overwhelms. If an argument is a connector between one idea and another, then for some students, this will feel like building the George Washington Bridge to span a creek.

The question it prompts, to me, is: What do we want from a proof?

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Lemony Snicket’s “A Series of Unfortunate Series”

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Let me be very clear, as clear as the vials of tears that I keep on my desk: This story is a long and sad one. It converges to no happy ending, and perhaps does not converge at all, although as you read, you will find your own joy and sanity both converging swiftly to zero.

If you were to abandon this text and go read about something pleasant, like butterscotch pudding or statistical sampling, I would applaud your good judgment, and humbly beseech you to statistically sample a pudding on my behalf.

As for me, I am compelled to tell this tale to its sour end, because I am an analyst—a word which here means “someone who fusses over agonizing details, bringing grief to many and enjoyment to none.”

But if you insist on reading further, then you ought to meet the three poor students at the heart of this tale:

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Lines Beyond y = mx + b

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What are my professional goals? Many days, I’ve got precisely three: I want kids to feel curious, then frustrated, then ohhhhhhhh.

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Math is pretty great for this. It’s full of puzzles and mysteries. Why do the angles in a triangle always sum to the same thing? How many moons would fill the sun? Why is it so hard to roll double sixes? There’s plenty here to excite curiosity, to elicit frustration, and to satisfy the intellectual itch for ohhhhhhh.

But there’s a common problem: too often, kids can beat the puzzles without feeling any of those things.

I find this, for example, with lines in the coordinate plane.

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The Island of Democrats and Republicans

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On February 10th, the world lost Raymond Smullyan: logician, puzzlemaster, and blue-ribbon Gandalf lookalike.

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Even if you don’t know his name, you’ve probably wrestled with his logic puzzles. They share a whimsical sense of rigor: “You come to an island where there are two types of people: knights, who always tell the truth, and knaves, who always lie…”

They’re silly and frustrating and fun; everything mathematics should be. I love this origin story for how Smullyan first got into such puzzles:

On 1 April 1925, I was sick in bed… In the morning my brother Emile (ten years my senior) came into my bedroom and said: “Well, Raymond, today is April Fool’s Day, and I will fool you as you have never been fooled before!” I waited all day for him to fool me, but he didn’t.

Or did he?

Young Ray had spent all day expecting to be fooled. But the fooling had never come. Didn’t this constitute the greatest fooling of all?

I recall lying in bed long after the lights were turned out wondering whether or not I had really been fooled.

In Smullyan’s honor, I wanted to offer up my own amateur variant on his knights-and-knaves puzzles.

I call it: the island of Democrats and Republicans.

Now, Republicans and Democrats look identical to an outsider like you. But they always recognize one another immediately. And because of their mutual antipathy, they follow this strange custom:

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So, here comes your puzzle. Ten of them, really.

Continue reading →