Cédric Villani’s arrival, one afternoon last May, at a small café off the Champs-Élysées drew glances from a good fraction of the late-lunch crowd. His shoulder-length hair was parted almost symmetrically down the middle, and he wore his usual ensemble: a three-piece pin-striped black suit, a silver pocket watch and chain, a peacock-green cravat (purchased at a costume store for actors), an overstuffed backpack, and, pinned to his lapel like a biological specimen, a custom-made spider brooch. Having just wrapped up a national-radio segment at the station next door, he was stopping for a bite on his way back to the Institut Henri Poincaré, where he serves as director.

Villani has been called the Lady Gaga of French mathematicians. After winning the Fields Medal, math’s highest honor, in 2010, for what his award citation called “proofs of nonlinear Landau damping and convergence to equilibrium for the Boltzmann equation,” he embraced a role that many other medalists have dreaded—that of mathematical ambassador, hopscotching from event to event and continent to continent, evangelizing for the discipline. “We are the most hidden of all fields,” he told me. “We are the ones who typically interact the least with the outer world. We are also the field which is most emblematic of revulsion in school.” The French filmmaker Olivier Peyon, who first met Villani while shooting his 2013 documentary “Comment J’ai Détesté les Maths” (“How I Came to Hate Math”), says that the mathematician struck him immediately as a natural proselytizer. “He was funny, very—in French, we say pédagogique,” Peyon told me. “He knew how to speak about his art, about math.” (In the film, Villani delights in the absurd number of people who claim to have been last in their math classes. “How could so many be last?” he asks with mock surprise.)

As part of his outreach effort, Villani published a memoir, in 2012, when he was thirty-eight, about the professional journey that culminated in his worldwide recognition. The book has sold nearly a hundred thousand copies in France and has been translated into Farsi, German, Italian, Japanese, Korean, Portuguese, Romanian, and Serbian; the U.S. edition is out today from Farrar, Straus & Giroux. Written as a nonlinear and unapologetically jargon-laced medley of diary entries, e-mail exchanges, and complex physics equations (his French publisher categorized the text as a novel), “Birth of a Theorem: A Mathematical Adventure” is less about math than about mathematicians—how they live, how they work, and how they talk to one another. “I don’t want the reader to even try to understand what is being proved,” Villani said. “When closing the book, the reader should not know more about the math.” As if to illustrate his point, he explained the memoir’s titular theorem this way: “We established a mathematical proof of stability in plasmas without any collisions.” Clément Mouhot, Villani’s former doctoral student and close collaborator, who serves as a sidekick and foil in the book, put it more simply. They had solved a seventy-year-old mystery in theoretical physics, he said, answering “a question that was not answered before.”

Villani finished his lunch and found the nearest metro station. (He claims he hasn’t driven a car in five years.) He rode to Gare de Châtelet-Les Halles, one of Paris’s more cavernous transit hubs, and began looking for “Énergies,” a bronze sculpture by the artist Pierre-Yves Trémois. It bears the Schrödinger equation, which describes how the wave function of a quantum-mechanical system changes over time, and which relates to Villani’s work. He asked a gendarme for directions. The cop pointed the way, but warned him about thieves and pickpockets and told him to zip his backpack, whose contents—books, notes, laptop—were peeking out, freely accessible to the metro-riding public. “Oui, oui,” Villani said. He gets that all the time from the police.

“I was the most shy kid that you can imagine,” Villani told me recently. “Just the idea of speaking in class was making me sweaty and confused and all red with terror. My teachers were complaining that I was not speaking enough. And now I’ve been on TV more than any child in the whole school!” He described his first experience interacting with the media, around 2001, as a “real disaster.” But, he noted, things change with time and effort. He took a class in media relations a few years before he won the Fields, and now his engagements keep him busy. He told me about one day, last December, when he taught an online open course on differential equations for most of the morning, made an appearance with the French minister of education at two-thirty, recorded a television interview at four o’clock, a radio interview at five, another radio interview fifteen minutes later, and appeared twice on live TV—first on the popular nightly news show “Le Grand Journal,” at seven, and then on another program, at ten. Villani’s efforts in the past five years have helped an alliance of French and international math institutes amass more than twenty-four million dollars in donations. There’s no way he could have raised that kind of funding without the outreach, he said.

Given the chance, not many of Villani’s colleagues would choose fame over mathematics. “A mathematician would usually be very reluctant to say half-lies,” Mouhot said, or to omit or overstate something. Villani has taken flak for involving himself in politics (he campaigned for Paris’s mayor, Anne Hidalgo, last year) and for questioning the existence of dark matter, which many physicists believe, based on compelling indirect evidence, permeates the universe. But he considers this a part of his job—part of keeping the backpack open. In outreach, he said, “you should talk about things outside your range of expertise to give people more opportunities to relate to you.”

Many mathematicians are glad that Villani is willing to participate in public life, Mouhot said, so that they don’t have to. “You do it because you think it’s for the common cause,” he said. “For me, this would not be a strong enough motivation for spending my life in planes, sleeping very little,” and not having time to do research—“which is the life of Cédric in the last years.” Olivier Peyon, who has become a close friend of Villani’s, sees how he has struggled with this internal schism. “He knows that, for now, it’s important for him to fight for math in the large arena,” he said. “The price to pay is he has to stop doing math for himself.”

Recently, I asked Villani what he thought of a Times Op-Ed by the political scientist Andrew Hacker, from 2012, called “Is Algebra Necessary?” Why put kids through the torture of learning complex math, Hacker argued, when it’s about as useful as wisdom teeth? Why make it a requirement when it contributes to lowering high-school graduation rates? “I hate this,” Villani said. “If you only teach stuff that is useful, then what do we need to teach in school? Not much.” He was high on his soapbox now. “Languages were invented all around the world; technology was invented many times. Mathematics was developed once and collectively—your culture cannot be complete if you don’t have at least a glimpse of what is mathematical reasoning.”

After posing for a photograph in front of “Énergies,” Villani hopped a train from Châtelet-Les Halles toward the Institut Henri Poincaré, in the Fifth Arrondissement. He emerged near the Luxembourg Gardens, where two high-school girls recognized him. He stopped to chat and shake their hands. (This sort of thing, he told me later, happens several times a day.) Soon he arrived at the institute, a magnificent red-brick building on Rue Pierre et Marie Curie. Upstairs, displayed on Villani’s office door alongside the word “CHEF,” was an oversized photo of him leaping high into the air, his arms and legs outstretched, in front of a chalkboard. Now that he is a seasoned public speaker, Villani said, he feels comfortable telling kids who are shy, as he once was, to be proud of their diffidence. “Everybody is telling you to be daring and so forth. I don’t think there’s more merit one way or the other,” he said. He likened being shy or outgoing to Triumph and Disaster in Rudyard Kipling’s poem “If.” “Defeat and victory both are impostors,” he said. “It’s the same with celebrity.”