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 me 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?

In his soft, low voice, he told a beautiful story. It cut against the romantic myth of the mathematician (lone genius, predestined, operating in a sphere beyond mortal reckoning). In fact, it resonated with the themes I encounter as a teacher: self-doubt, missteps, the need for helping hands.

Here, in three acts, is Châu’s tale.

“When I was in the sixth grade,” Châu confided, “I was not particularly interested in mathematics.” He lacked discipline. “I was in a very new-style school. We were free to do whatever we want.”

And what did he want? Well, not mathematics. Not yet.

When an opportunity came to join a gifted class, Châu leapt at it—but failed the entrance test. “I thought I was very good,” he remembers. “I was not.”

Rejection stung, and the sting drove him. He refocused his efforts. “There were so many problems I couldn’t do,” he says, but he drew inspiration from those who had walked this path before—and from the awe and reverence in which classmates held them.

“I heard stories about the older students who were in the Olympiad,” Châu says, referring to the international math competition. “Mathematicians had mythic status.”

As a teacher, I downplay the competitive side of mathematics. I preach patience and collaboration and encourage students to end proofs with “Kumbaya” as an alternative to “QED.” When they compare test scores, I cringe; I don’t want math to feel like a zero-sum tournament. But for many children (including, if I’m honest, my younger self) competition is part of the draw. Kids love what they feel good at. Measuring yourself against peers is how you know where you stand.

And soon, Châu stood atop the mountain. He won back-to-back golds in the Olympiad. No Vietnamese student had ever done that. It was only a few years from test-day failure to local legend.

“I do well with competition,” he says, smiling.

In a word, that’s the first phase of Châu’s mathematical journey: competition. Math was a ticket to an elite club—and now, Châu was in.

Châu earned a scholarship to study combinatorics in Hungary. Then, suddenly, the Berlin Wall fell. Borders were redrawn. Instead of combinatorics in Budapest, he found himself studying algebraic geometry in France.

The transition wasn’t easy. “I suffered quite a bit. This abstract algebra was very unfamiliar.”

I challenge you to find a group theory student who doesn’t identify with that.

Combinatorics is a concrete, problem-solving branch of math, concerned with the counting and rearrangement of objects. Abstract algebra is… abstract. Notoriously so. It studies nebulous collections of slippery objects defined not by their substance but by their properties and interactions.

“It was really hard for me,” Châu recalls. “I couldn’t get it. It was very painful.”

Says the Fields Medalist.

Châu hid his struggle below the surface. “My professors thought I was a fantastic student,” he remembers. “I could do all the exercises. I did well on exams.” Then he shakes his head, half-smiling. “But I didn’t understand anything.”

So it went throughout his undergraduate degree: a pantomime of understanding, and an anguish below. It’s an all-too-common tale for the mathematics student: you copy patterns blindly, not comprehending, until one day the patterns evade you, and your mathematical days are through.

Châu feared falling into that same trap.

It was his PhD that saved him. “I had one of the best advisors in the world,” Châu glows. That’s Gérard Laumon, with whom he later collaborated on his breakthrough work.

“I would come to his office every week. He would read with me one or two pages every time.” They went line by line, equation by equation, settling for nothing less than full comprehension.

Châu embraced this slow, deliberate pace as a gift. “It was very revealing for me.” For me, this strikes a resounding chord. Early in my teaching career, I saw my students as mired in meaningless symbols; I wanted them to step away from the notation and think about the ideas. More recently, I’ve come to believe the ability to read notation is vital: but you’ve got to read it, not just push it around the page. The work to invest mathematical symbols with meaning is slow and painstaking but utterly central to the project of becoming a mathematical thinker.

And that meticulous project of unpacking mathematical writing is how Châu spent his PhD. Week by week, he built his understanding, growing under Laumon’s mentorship from a competitor into a scholar.

Châu soon began work on the famous Langlands Program, which I think of as the transcontinental railroad of modern mathematics.

Almost fifty years ago, Robert Langlands laid out a sweeping vision for how to connect several distant branches of higher math. (Which branches? I’ll let you read up on it.) The project has drawn generations of ambitious mathematicians like Châu into its orbit.

Châu found himself attracted to a particularly vexing piece of the Langlands Program: proving the “Fundamental Lemma.”

(The name is something of an oxymoron; a “lemma” is typically an intermediate fact, proved on the path towards a grander, more exciting truth called a “theorem.” But Langlands named the “fundamental lemma” before its immense subtlety and difficulty became clear.)

By the time Châu arrived on the scene, the Fundamental Lemma stood as a crucial choke-point in the Langlands Program. It was the bottleneck stifling further progress.

So were rival mathematicians jockeying for primacy, racing to be the first to prove it? Was it the Olympiad all over again?

No, says Châu.

“I was helped a lot by people in my field,” he says. “Many people encouraged me to do so, in a very sincere way. I asked them for advice, and they would tell me what to learn. It was very open. I did not feel competition.”

And, within a few years, Châu managed to prove the Fundamental Lemma.

This feat earned him individual glory (fame, fortune, Fields) but did not come by individual labor alone. As NBA squads have learned, handing the ball to a superstar is no match for a fluid and cohesive offense. Individuals don’t win championships; teams do.

So not only did Châu build on past work, but he turned to his colleagues for guidance throughout the process, drawing on their expertise to patch holes, to strategize, and to overcome snags. In this, his work resembles pretty much every great human accomplishment I’ve ever witnessed, from my students’ best math to my colleagues’ best teaching to the kitchens at my favorite restaurants. People thrive thanks to other people.

Châu has thrived through competition, grown through mentorship, and advanced the field through collaboration. Now, he perhaps begins the fourth act of his mathematical tale: Leadership.

Though Chicago is his home now, Châu spends his summers back in Vietnam. He helps to run the Vietnam Institute for Advanced Study in Mathematics, and reaches out to young people to encourage them into mathematics. In the years after his upbringing (when Olympiad winners were revered as true Olympians) the cachet of mathematics had fallen in Vietnam. But after hitting a low point in the 1990s and 2000s, it is now on the rebound.

Thanks to Châu, mathematics is cool again.

Meanwhile, the Langlands program has entered a new phase. Of the earlier work, Châu says: “It was very structured. There were very clear task divisions. It all fit together.” It was as if Langlands had composed the table of contents—a crisp outline of ideas—and had left the other researchers to produce the text.

But with that phase complete, work today is different. “Now, there is no such division. [Langlands’] description is more like poetry. It is not programmatic.”

Châu pauses. “Now we are on our own.”

Only 44 years old, he has decades ahead of him in which to push mathematics forward. Younger researchers will be following his lead. And Ngô Bảu Châu, for one, knows what they can expect on the journey ahead.

© Heidelberg Laureate Forum Foundation / Flemming­ – 2016