Claire Voisin took a long time to develop a passion for mathematics. However, she never disliked the subject. Growing up in France as the 10th of 12 children, she enjoyed solving math problems with her engineer father. At the age of 12, she began reading a high school algebra textbook independently, captivated by its definitions and proofs. “There was all this structure,” she said. “Algebra is really a theory of structures.”
During her university years, Voisin came to appreciate the depth and beauty of mathematics and realized her potential to make new discoveries. Before fully committing to mathematics, she pursued various interests such as philosophy, painting, and poetry. However, by her early 20s, mathematics became her primary focus, although painting and poetry continued to influence her, seeing math as an art form and a medium to challenge the limits of language.
Despite her dedication to mathematics, Voisin still finds time for painting and sculpting with clay. Yet, mathematics remains her main pursuit, as she delves into its “different world” where “it’s like you are dreaming.”
As a senior researcher at the French National Center for Scientific Research in Paris, Voisin specializes in algebraic geometry, studying shapes defined by sets of polynomial equations, akin to how a circle is defined by the polynomial x² + y² = 1. She is renowned for her expertise in Hodge theory, a tool mathematicians use to analyze fundamental properties of algebraic varieties.
Voisin’s contributions to mathematics have earned her numerous awards, including the Clay Research Award in 2008, the Heinz Hopf Prize in 2015, and the Shaw Prize for mathematics in 2017. In January, she made history as the first woman to receive the Crafoord Prize in Mathematics.
In an interview with Quanta, Voisin discussed the creative aspects of mathematics. The interview has been condensed and edited for clarity.
Voisin stresses the importance of language in mathematics for grasping old concepts and for forging new ones. “You could liken a mathematical theorem to a poem,” she remarked.
Laurence Geai for Quanta Magazine
Introduction
You liked math when you were young but didn’t envision pursuing it further. Why not?
There’s the wonder of a proof — the feeling you get when you comprehend it, when you realize its strength and how it empowers you. As a child, I could sense this already. And I appreciated the focus that mathematics demands. It’s something that, as I get older, I find increasingly fundamental to mathematical practice. The rest of the world fades away. Your entire mind is focused on exploring a problem. It’s an extraordinary experience, one that holds great significance for me — the act of immersing yourself in a realm detached from practical matters. Maybe that’s why my son is so drawn to playing video games.
But what kept me from diving into mathematics earlier, in a way, is that I have no interest in games whatsoever. It’s not my cup of tea. And during high school, mathematics felt like a game to me. It was challenging for me to take it seriously. I couldn’t see the profound depths of mathematics initially. Even as I started encountering very intriguing proofs and theorems after high school, I never considered that I could originate something myself, that I could claim it as my own.
I yearned for something deeper, more profound, something I could truly possess.
Before discovering that in mathematics, where did you seek it?
I found pleasure in philosophy and its emphasis on the concept. Also, until I was about 22, I devoted a lot of time to painting, particularly figurative pieces inspired by geometry. And I had a deep affection for poetry — the works of Mallarmé, Baudelaire, René Char. I was already inhabiting a different world, as is common when you’re younger.
But mathematics became increasingly significant. It truly engages your entire mind. Even when you’re not actively working on a specific problem at your desk, your mind remains occupied. So the more I delved into mathematics, the less I painted. I only recently resumed painting, now that my children have all moved out and I have more free time.
What led you to ultimately channel most of your creative energy into mathematics?
Mathematics grew more and more captivating to me. As a master’s and Ph.D. student, I discovered that 20th-century mathematics was profoundly rich and extraordinary. It was a realm of ideas and concepts. In algebraic geometry, there was the renowned revolution spearheaded by Alexander Grothendieck. Even before Grothendieck, there were remarkable achievements. So it’s a modern field, with ideas that are not only beautiful but also immensely potent. Hodge theory, which I specialize in, was part of that.
It became increasingly apparent that my life belonged there. Of course, I had a family life — a husband and five children — and other obligations and interests. But I realized that with mathematics, I could create something. I could devote myself to it because it was so elegant, so spectacular, so intellectually stimulating.
You were attracted to math because of Grothendieck’s breakthroughs in algebraic geometry. He essentially developed a new language for this type of math.
Are there ways in which the mathematical language you’re using now might still need to change?
Mathematicians continuously update their language. It’s unfortunate because it can make older papers difficult to understand. However, we revise past mathematics to gain better insights. It helps us improve how we write and prove theorems. Grothendieck’s work, particularly his use of sheaf cohomology in geometry, exemplifies this.
It’s crucial to become deeply acquainted with the subject you’re studying, to the extent that it feels like a native language. When a theory is emerging, it takes time to determine the right definitions and simplify everything. It may still be complex, but we become more comfortable with the definitions and concepts; they become more intuitive to use.
It’s an ongoing process. We continuously refine and simplify, theorizing about what’s important and which tools to provide.