The Three Phases of the Mathematical Life

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.


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

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:


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:


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.


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”


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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The Professor with a Billion Students


This September in Germany, between talks at the Heidelberg Laureate Forum, I managed to catch a few minutes with Cornell professor John Hopcroft.

He’s a guy with bigger things on his mind.

“I’m at a stage in my life,” he says, “where I’d like to do something which makes the world better for a large number of people.”

Skimming Hopcroft’s C.V., you start to wonder: Um… hasn’t he done that already?


Born to a janitor and a bookkeeper, he grew up to become a foundational figure in computer science. Exhibit A: His textbooks on automata, algorithms, and discrete math have been adopted across the world. (His most recent one—on data science—is free online.) Exhibit B: He has a distinguished research record, highlighted in 1986 with a Turing Award— the closest thing to a Nobel for computer science. And finally, Exhibit C: During a decorated teaching career, he was twice named Cornell’s “most inspiring” professor.

With all this, you’ve got to figure he’s done at least a little good for a few people, right?

Well, Hopcroft has a larger number in mind: 1.3 billion.


Hopcroft has become an advisor to Li Keqiang, the Premier of China. He describes this as “the opportunity of a lifetime”: to transform Chinese education for the better.

“They have one quarter of the world’s talent,” Hopcroft says, “but their university educational system is really very poor.”

What makes Hopcroft—working-class Seattle-ite turned Ivy League professor—think he can leave his mark on a country as vast, distant, and internally diverse as China? Isn’t this like a swimmer trying to steer an aircraft carrier?


“A couple of things are going in my favor,” he says. First, he is apolitical. “I don’t have any special agenda to push in China,” Hopcroft explains. “I’m pushing education.”

The second is subtler, and carries echoes of Hopcroft’s engineering background.

“I understand the scale of the problem,” Hopcroft says.

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It’s Obvious Why Students Cheat; We Just Can’t Agree on the Reason

If you love cringes – and hey, who doesn’t? – then walk into a school and try to start a conversation about cheating.

Depending on the school, I suspect you’ll find a superficial consensus (cheating is terrible! and, thankfully, our students do it very rarely!) masking deep rifts. Is the problem with cheating that it undercuts your own learning? That it steals glory from classmates in the zero-sum competition for grades? That it betrays the teacher’s trust? Are all acts of cheating equally terrible, and if not, what does that mean for “zero tolerance” policies?

We all know cheating is bad. But we seem unable to talk honestly about why.

So, I offer up these dialogue-starting cartoons, a few badly drawn meditations on the most basic question: Why do students cheat?


Is cheating a crime of character, or of opportunity? Continue reading