The second time I meet Cédric Villani is when I bump into him in the Eurostar terminal in Paris. But then how could I miss him? There are crowds of milling businessmen and weekending couples but there, amid the Sunday-night chaos of the Gare du Nord, is a figure who looks like he’s somehow slipped the space-time continuum: Lord Byron on a mini break. Or Baudelaire who has returned to Earth, only this time as a Parisian banker doing the Monday-Friday commute. He has a silk bow around his neck. His hair flows around his shoulders. And when I stop to say hello, he embraces me and starts telling me about his latest projects (dozens of them) and his trips (to everywhere, he’s always travelling) and the book he’s writing and then he rummages through his briefcase to give me a business card.

But instead of one business card, he hands me a stack of around 50, scattering another 50 or so over the floor. “Here! Take them!” he says. But, Cédric, I say, I only need one, but he presses them into my hands and as he does so drops a file of papers and a small case on the ground. The commuters and weekenders have, by now, started to take notice of the commotion in front of them, though he’s oblivious as he gesticulates and exclaims and then opens up the small case to show me what’s inside. “Spiders!” he says showing me a dozen different decorated hand-made spider brooches. “I always wear one,” he says and points to his jacket, where there is an ornate bejewelled spider crawling across his lapel.

As meetings with Cédric Villani go, this is not untypical. He looks like he’s just wandered in from the 19th century, speaks English with an accent that rivals that of Hercule Poirot, and has the physical dexterity of Manuel from Fawlty Towers, but he also happens to be one of the greatest mathematicians alive, a winner of the Fields medal, mathematics’ answer to the Nobel prize, handed out every four years.

I’ve met him three times by the time I read his new book, the one he was writing when I met him at the Eurostar terminal, Birth of a Theorem: A Mathematical Adventure. He’s spoken at two TEDxObserver events we’ve held and I’ve got to watch him close-up in action, both before and after the event. Both times, he delivered brilliant, arm-flailing, passionate expositions of the beauty of mathematics and why it matters. But what really marked him out of the norm wasn’t that, or the spiders and neckerchiefs: it was his intense engagement with the world around him. He was the only speaker who listened – intently – to almost half of the other talks. And afterwards, he talked – delightedly – to random well-wishers and stray journalists and the idly curious. To anyone and everyone. He takes on all comers, equally, with an openness and interest that is vanishingly rare in public life.

But it’s not until I read his book, an odd, fascinating account that attempts to render the mystery of the moment of mathematical revelation into a narrative a layperson can understand (you know, -ish, up to a point), that I realise that this isn’t despite his mathematical brilliance; it’s actually the cornerstone of it. The theorem – a new proof that explains Landau damping, a mysterious and complex property of plasma behaviour – was the result of a collaboration with a former student, Clément Mouhot, and he presents their emails to show how they bounced ideas back and forth. And that’s the least of it: there are conferences, seminars, encounters with random academics. Even his children, or “lambikins” as he refers to them at one point, play a part in his thinking and it’s not all that much of a surprise that when I try to catch up with him, I discover he’s in Senegal, and by the time I pin him down on Skype, he’s in Cameroon.

“It’s a project of the African Institute of Mathematical Sciences,” he says via a crackly line. “I go every year and teach. I have a link as well with Algeria because my parents were born there. But anyway, I also consider it part of life, to do such cooperative things. And Africa is such a warm and joyful place that I always come back to Europe full of energy.”

Collaboration is, he says, “maybe the single most important thing about the book. One of the greatest misconceptions about mathematics is that it’s a solitary activity in which you work with your pen, alone, in a room. But in fact, it’s a very social activity. You constantly seek inspiration in discussions and encounters and randomness and chance and so on.”

His life, and his work, owes so much to luck, to chance, he says. “Though the more interaction you have, the more luck appears.” And yet, he wasn’t born that way. For the first 18 years of his life, he says, he was painfully shy. “In class, the teacher would never hear the sound of my voice. I was like this for all my childhood. Now, there is hardly a day that passes without me giving a public lecture. Somehow, I went from one extreme to the other.

“And for a mathematician, I think you need both. You have to have an intense interior life and reflection to spend time thinking – really thinking – and then find the solution. But also to interact with other people and discuss and discuss.”

But then, it’s not the only misconception that people have about mathematics, he says. “People think it’s about numbers. It starts with numbers, for sure, but in a mathematical proof or in a mathematician’s life, there are hardly any numbers. It’s all about concepts and logical reasoning. It’s more like being a lawyer or a detective. If there are several problems, you have to find who is the guilty party by using very good reasoning. That’s the key thing. Do you know Columbo? He always used to find the bad guy using intuition and then used logic to prove it. And these are the same two key steps that a mathematician uses. He uses intuition to guess the right problem and the right solution and then logic to prove it.”

What Birth of a Theorem makes clear is how many steps there are to that process. Even for the gifted, cutting-edge, world-leading mathematics is a prolonged intellectual battle against stacked odds. Villani recounts the endlessly frustrating steps and mis-steps, as well as the pressure. He knows he only has one shot at the Fields medal, though; he can’t even bring himself to say the name out loud. It’s only awarded to mathematicians under 40 and when the book begins he’s 35. There are compensations, though, and a subplot of the book is an insightful romp through elite academia. Midway through, he accepts a six-month fellowship at the Institute for Advanced Study in New Jersey, one of the most prestigious establishments in the world, dedicated to theoretical research, where, Villani writes, all earthly cares and whims are catered for: “An army of secretaries stands ready to answer any question and resolve any difficulty. An apartment will be waiting for you only a few minutes from your office. The excellent dining hall will save you the trouble of looking for a restaurant.”

But then it’s the mixture of the banal and the sublime that makes the book so unusual. Everything goes in: as well as the excitable, but largely unintelligible email exchanges with Clément, there are pages of equations, paeans to French bread and cheese that he misses so much, and even his dreams. “The unconscious brain plays such an important role,” he explains. “And I wanted that in there in some way.”

And when the breakthrough finally comes, there is no mistaking it. In the book, he writes of “the miracle” when “everything seemed to fit together as if by magic” and quotes André Weil on it, who calls it “lucid exaltation”, a climax that “unlike sexual pleasure” may last “for hours at a time, even days”.

The memory of it is still vivid, says Villani. “It’s all very vivid. That period which is described in the book. The whole period is like a moment of grace.”

Some mathematicians, he says, believe they receive inspiration directly from God, “that he was whispering to them”. And, others, as he describes in the book, suffer some sort of mental disorder, though Villani points out it’s not just mathematicians but those involved in many fields of creative arts. But even the saner ones seem like they’d be somewhat trying domestic partners. He recounts his wife, Claire, shouting: “This is getting really weird!” at him at one point when he is in full flow.

Because what Villani explains so well is how mathematics lies somewhere between an art and science. “There is a famous quote about mathematics being the poetry of sciences,” he says. “And there really is something in it. First the enormous role of intuition. Inspiration is the key to mathematical work. You need both inspiration and rigour… and mathematicians and poets are people who believe in the power of words, of concepts and giving names to concepts.”

And maths is all around us, he points out. A new theorem is a breakthrough in human thought. A new way of understanding the world. “And you will never know how it will be used in technology. Fourier analysis was invented at the beginning of the 19th century and it was just a theory. Just a theory about the contradictions of heat and where that goes. And now Fourier analysis is everywhere. It’s in this Skype call. It’s in any communication, any cellphone, everything. Dozens and dozens of theorems have been built on Fourier and the work of hundreds of thousands of engineers… In science, you are always part of history.”

It’s why he’s so passionate about communicating what it is that mathematics is, and what mathematicians do, and why research for the sake of research is the engine of all human progress. And, why, he says, “my book is almost entirely emotions. I wanted to make it known how emotional the life of a mathematician is. It’s a world of progress and adventure and emotion.”

And openness. Which brings us on to politics. Because what is happening now in Europe is a “tragedy”, he says.

“Closing borders and being afraid and suspicious of immigrants, this is a tragedy for Europe. We’re heading for a political catastrophe at a very big level.” He was in China when the attack on Charlie Hebdo took place, and, “When I returned I had the impression that it was a different country.” But, he’s unlikely to be leaving any time soon. The life of the mind is one thing, but he is, when all is said and done, a Frenchman, and the country may have produced more Fields medal winners than almost anywhere else, but it’s also pretty good at other things too. He’s been offered all sorts of posts in America. “And my salary would be four times my current one, and my schedule would be much less work, but it would be less fun, less interesting, there’d be less interaction with people… and, you know, the bread…” he says and shakes his head sadly. “You just can’t get good bread.”

Birth of a Theorem by Cedric Villani is published by Random House (£14.99). Click here to order a copy for £11.99