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The Truth Behind the Math: The Surprising Path of a Theorem [Q&A]

Mathematician Cédric Villani shares the actual conversations, emails and even dreams that led to his Fields Medal-winning proof

Apr 14, 2015 |By Clara Moskowitz

Cédric Villani’s new book describes the work that went into a mathematical proof of nonlinear Landau damping. If that sounds like Greek to you, have no worries. The subject of the book is not the math—which has to do with the stability of waves in a hot gas—but rather the world of mathematicians. Like few have before, Villani—a mathematician at the University of Lyon in France and director of the Henri Poincaré institute—imparts a sense of what it is like to search for mathematical truths.

He tells the story of the thrill of discovering the nut of a new problem, the exhilaration of making a breakthrough, as well as the devastation of realizing he has made a mistake and the frustration of reaching an impasse. He shares many emails between himself and his main collaborator, as well as conversations with colleagues, his own thought processes and calculations and even dreams that offered insight into his problem. Villani takes the unusual approach of not trying to translate all of the mathematical language in discussions or letters, but rather sharing them verbatim. Although the technical subtleties will be over many readers’ heads, seeing some of the original words and equations that led to the theorem better communicates the way mathematics truly works. We see the types of difficulties that arise and the techniques the researchers employ to try to deal with them, the way new ideas strike and the paths that turn out to be dead ends. We even feel the disappointment when Villani gets a letter from a journal rejection the first version of a paper describing the theorem, and later the elation when he learns that he has won a Fields Medal—the mathematics equivalent of the Nobel Prize—for his work on the proof.

Scientific American spoke to Villani about the book, the prize and the life of a mathematician.

[An edited transcript of the conversation follows.]

The book takes a unique approach to writing about a complicated subject. Instead of trying to translate the mathematics you do into lay language, you share the original emails and conversations that developed the theorem, letting readers get a flavor of what the research process was like, even if they don’t understand all the technical details.  Why did you choose to do it this way?
There are two ways to explain a technical job. One is to describe your job in very simple words and that is the usual way. You can substitute the word “entropy” by the word “disorder” and so on. And the other one is showing everything, instead of giving explanations. Even though the words will be technical and full of things that are impossible to understand, that’s where you will be able to see which are the moments of tension, which are the moments of relationships, the moments of anger, happiness. All these parts of the job can be visible. This explains why I chose to preserve the technical parts so that it can feel like you are in the world of mathematicians and observing and witnessing everything that is going on without understanding what exactly it is that all these people are searching for.

What is of interest is the creative process—which is the basis of a mathematical theorem, but also the basis of all other creative jobs. In the book, I wanted to make sure all of the elements of the life of the mathematician were included: when you travel, when you submit a paper, when a paper is rejected, as well as the music I listen to while working, the tea I drink.

One of my favorite aspects of the book is the way it shows the back and forth between you and your collaborator, Clément Mouhot, that was so integral to formulating the theorem. Are mathematical partnerships like this common?
Collaborations are very popular; most mathematics is done by two or three authors at the same time. People think the job of mathematician is a solitary activity, but here is an example where you can see how much it depends on collaboration between the coauthors. It was important to pay tribute in the best way I could in this respect. The book is a big thank you to my collaborator.

I enjoyed reading about the meandering path you took to arrive at the subject of your theorem. How do you choose a problem to work on?
In mathematics the process may be more difficult than some other subjects because you are not guided by experiments. In pure mathematics, everything is possible. But which is the interesting thing? Deciding is half of the job really.

It can be seen in the book that the choice of subject is in part due to chance. In the first chapter, when we were thinking of working on a certain problem, we end up working on another. This was really a result of the discussion between us. A good problem should be not too simple or else it would be easy, but also not too difficult. Usually mathematicians are thinking of many more problems than they eventually try to do. It’s a complicated process in which it’s a mixture of what you hope to do, what you think you can do, and your interactions with other people.

What new problems are you working on these days?
There are lots of things I want to explore. I turned recently to a problem that is at the same time simpler in appearance and more geometric, more pure, related to problems of deformations of surfaces. I thought it was good to change subjects and go in a direction that was less in the spotlight, but still has the possibility of something that is quite interesting. It is not something on the list of mathematicians’ most important things to do. On my list I did not think of this as an important and rich problem. It came as a surprise that it turned out to be so. It’s very difficult to know how rich a problem can turn out to be. In mathematics the way a problem is solved is at least as important as the answer it solves.

Has winning the Fields Medal changed much in your life?
Very much so, in extreme proportion. I did not realize this when it occurred, but the change would be enormous. It gave me access to the media, giving me a possibility to connect with many, many people, and also gave me a lot of new duties, so to speak. I have given hundreds of public lectures, published books, appeared on dozens of TV and radio programs, so I have been in the media in a way that would not have been possible without the Fields Medal.

I received a large amount of funding, and I’ve been striking partnerships with a number of companies and my university. All of this buzz surrounding the medal, I was able to convert it into some real progress in my institute and for the community. I’m very happy about how this has turned out. The medal is given by a committee. It is connecting the whole world community of mathematicians. I really owe this reward to the community.

Source: Scientific America

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