THE BEAUTY OF MATHEMATICS: IT CAN NEVER LIETO YOU

STEFAN FALKE FOR QUANTA MAGAZINE

A FEW YEARS back, a prospective doctoral student sought out Sylvia Serfatywith some existential questions about the apparent uselessness of puremath. Serfaty, then newly decorated with the prestigious Henri PoincaréPrize, won him over simply by being honest and nice. “She was very warmand understanding and human,” said Thomas Leblé, now an instructor at theCourant Institute of Mathematical Sciences at New York University. “Shemade me feel that even if at times it might seem futile, at least it would befriendly. The intellectual and human adventure would be worth it.” ForSerfaty, mathematics is about building scientific and human connections. Butas Leblé recalled, Serfaty also emphasized that a mathematician has to findsatisfaction in “weaving one’s own rug,” alluding to the patient, solitarywork that comes first.

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Born and raised in Paris, Serfaty first became intrigued by mathematics inhigh school. Ultimately she gravitated toward physics problems,constructing mathematical tools to forecast what should happen in physicalsystems. For her doctoral research in the late-1990s, she focused on theGinzburg-Landau equations, which describe superconductors and theirvortices that turn like little whirlwinds. The problem she tackled was todetermine when, where and how the vortices appear in the static (time-independent) ground state. She solved this problem with increasing detailover the course of more than a decade, together with Étienne Sandier of theUniversity of Paris-East, with whom she co-authored the book Vortices in theMagnetic Ginzburg-Landau Model.

In 1998, Serfaty discovered an irresistibly puzzling problem about how thesevortices evolve in time. She decided that this was the problem she reallywanted to solve. Thinking about it initially, she got stuck and abandoned it,but now and then she circled back. For years, with collaborators, she builttools that she hoped might eventually provide pathways to the desireddestination. In 2015, after almost 18 years, she finally hit upon the right pointof view and arrived at the solution.

“First you start from a vision that something should be true,” Serfaty said. “Ithink we have software, so to speak, in our brain that allows us to judge that

ABOUTOriginal story reprinted with permission from Quanta Magazine, an editorially independent division ofthe Simons Foundation whose mission is to enhance public understanding of science by covering researchdevelopments and trends in mathematics and the physical and life sciences

moral quality, that truthful quality to a statement.”

And, she noted, “you cannot be cheated, you cannot be lied to. A thing is trueor not true, and there is this notion of clarity on which you can baseyourself.”

In 2004, at age 28, she won the European Mathematical Society prize for herwork analyzing the Ginzburg-Landau model; this was followed by thePoincaré Prize in 2012. Last September, the piano-playing, bicycle-ridingmother of two returned as a fulltime faculty member to the Courant Institute,where she had held various positions since 2001. By her count, she is one offive women among about 60 full-time faculty members in the mathdepartment, a ratio she figures is unlikely to balance itself out anytime soon.

Quanta Magazine talked with Serfaty in January at the Courant Institute. Anedited and condensed version of the conversation follows.

When did you find mathematics?

In high school, there was one episode that crystallized it for me: We hadassignments, little problems to solve at home, and one of them seemed very

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difficult. I had been thinking about it and thinking about it, and wanderingaround trying to find a solution. And in the end I came up with a solution thatwas not the one that was expected—it was more general than the problemwas calling for, making it more abstract. So when the teacher gave thesolutions, I proposed mine as an alternative, and I think everybody wassurprised, including the teacher herself.

I was happy that I’d found a creative solution. I was a teenager, and a little bitidealistic. I wanted to have a creative impact, and research seemed like abeautiful profession. I knew I was not an artist. My dad is an architect andhe’s really an artist, in the full sense of the word. I always compared myself tothat image: the guy who has talent, has a gift. That played a role in buildingmy self-perception of what I could do and what I wanted to achieve.

So you don’t think of yourself as having a gift—you weren’t a prodigy.

No. We do a disservice to the profession by giving this image of littlegeniuses and prodigies. These Hollywood movies about scientists can besomewhat counterproductive, too. They are telling children that there aregeniuses out there that do really cool stuff, and kids may think, “Oh, that’snot me.” Maybe 5 percent of the profession fits that stereotype, but 95percent doesn’t. You don’t have to be among the 5 percent to do interestingmath.

For me, it took a lot of faith and believing in my little dream. My parents toldme, “You can do anything, you should go for it”—my mother is a teacher andshe always told me I was at the top of my cohort and that if I didn’t succeed,who will? My first university math teacher played a big role and reallybelieved in my potential, and then as I pursued my studies, my intuition wasconfirmed that I really liked math—I liked the beauty of it, and I liked thechallenge.

So you have to be comfortable with frustration if you want to be amathematician?

That’s research. You enjoy solving a problem if you have difficulty solving it.The fun is in the struggle with a problem that resists. It’s the same kind ofpleasure as with hiking: You hike uphill and it’s tough and you sweat, and atthe end of the day the reward is the beautiful view. Solving a math problem isa bit like that, but you don’t always know where the path is and how far youare from the top. You have to be able to accept frustration, failure, your ownlimitations. Of course you have to be good enough; that’s a minimumrequirement. But if you have enough ability, then you cultivate it and build onit, just as a musician plays scales and practices to get to a top level.

How do you tackle a problem?

One of the first pieces of advice I got as I was starting my Ph.D. was fromTristan Rivière (a previous student of my adviser, Fabrice Béthuel), who toldme: People think that research in math is about these big ideas, but no, youreally have to start from simple, stupid computations—start again like astudent and redo everything yourself. I found that this is so true. A lot ofgood research actually starts from very simple things, elementary facts,basic bricks, from which you can build a big cathedral. Progress in mathcomes from understanding the model case, the simplest instance in whichyou encounter the problem. And often it is an easy computation; it’s just thatno one had thought of looking at it this way.

Do you cultivate that perspective, or does it come naturally?

This is all I know how to do. I tell myself that there are always very brightpeople who have thought about these problems and made very beautiful andelaborate theories, and certainly I cannot always compete on that end. But letme try to rethink the problem almost from scratch with my own little basicunderstanding and knowledge and see where I go. Of course, I have builtenough experience and intuition that I sort of pretend to be naive. In the end,I think a lot of mathematicians proceed this way, but maybe they don’t wantto admit it, because they don’t want to appear simple-minded. There is a lotof ego in this profession, let’s be honest.

Does the ego help or hinder mathematical ambition?

We do math research because we like the problems, and we enjoy findingsolutions, but I think maybe half of it is because we want to impress others.Would you do math if you were on a desert island and there was no one toadmire your beautiful proof? We prove theorems because there is anaudience to communicate it to. A lot of the motivation is presenting the workat the next conference and seeing what colleagues think. And then peopleappreciate it and provide positive feedback, and this feeds the motivation.And then you may get prizes, and if so, maybe you get even more prizesbecause you already have prizes. And you get published in good journals, andyou keep track of how many papers you published and how many citationsyou got on MathSciNet, and you inevitably get in the habit of sometimescomparing yourself to your friends. You are constantly judged by your peers.

This is a system that increases people’s productivity. It works very well topush people to publish and to work, because they want to maintain theirranking. But it also puts a lot of ego into it. And at some point I think it’s toomuch. We need to put more focus on the real scientific progress, rather thanon the signs of wealth, so to speak. And I certainly think this aspect is notvery female-friendly. There’s also the nerd stereotype—I don’t think ofmyself as a nerd. I don’t identify with that culture. And I don’t think thatbecause I’m a mathematician I have to be a nerd.

Would more women in the field help shift the balance?

I’m not super-optimistic, in terms of women in the field. I don’t think it’s aproblem that is going to naturally resolve itself. The numbers over the last 20years are not a great improvement, sometimes even decreasing.

The question is: Can you convince men that it would really be better forscience and math if there were more women around? I’m not sure they are allconvinced. Would it be better? Why? Would it make their life better, would itmake the math better? I tend to think it would be better.

In what way?

It’s good to have a diversity of frames of mind. Two different mathematiciansthink in two slightly different ways, and women do tend to think a little bitdifferently. Math is not about everybody staring at a problem and trying tosolve it. We don’t even know where the problems are. Some people decidethey are going to explore over here, and some people explore over there.That’s why you need people with different points of view, to think ofdifferent perspectives and find different roads.

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In your own work over the past two decades, you’ve specialized in one area ofmathematical physics, but this has led you in a variety of directions.

It’s really beautiful to observe, as you progress in your mathematicalmaturity, how everything is somehow connected. There are so many thingsthat are related, and you keep building connections in your intellectuallandscape. With experience you develop a point of view that is pretty muchunique to yourself—somebody else would come at it from a different angle.That’s what’s fruitful, and that’s how you can solve problems that maybesomebody smarter than you wouldn’t solve just because they don’t have thenecessary perspective.

And your approach has unexpectedly opened doors to other fields—how didthat come about?

One important question I had from the beginning was to understand thepatterns of the vortices. Physicists knew from experiments that the vorticesform triangular lattices, called Abrikosov lattices, and so the question was toprove why they form these patterns. This we never completely answered, butwe have made progress. A paper we published in 2012 rigorously connectedthe Ginzburg-Landau problem of vortices with a crystallization problem forthe first time. And this problem, as it turns out, arises in other areas of math,such as number theory and statistical mechanics and random matrices.

What we proved was that the vortices in the superconductor behave likeparticles with what’s called a Coulomb interaction—essentially, the vorticesact like electric charges and repel each other. You can think of the particlesas people who don’t like each other but are forced to stay in the same room—where should they stand to minimize their repulsion to others?

Was it difficult to cross over into a new area?

It was a challenge, because I had to learn the basics of a new subject area andnobody knew me in that field. And initially there was some skepticism aboutour results. But arriving as newcomers allowed us to develop a new point ofview because we weren’t burdened by any preconceived notions—ignoranceis helpful in this instance.

Some mathematicians, they start with something, they know how to do it,and then they create variants, like derivative products: You make the film

and then you sell the T-shirts, and then you sell the mugs. I think the way thatyou can distinguish good mathematicians is that they are constantly movingfurther and forward and advancing onto new ground.

Original story reprinted with permission from Quanta Magazine, aneditorially independent publication of the Simons Foundation whosemission is to enhance public understanding of science by covering researchdevelopments and trends in mathematics and the physical and life sciences.