Final week, mathematician Claire Voisin was awarded the 2024 Crafoord Prize in Arithmetic — one of many high awards within the subject.

Voisin, who is predicated on the Jussieu Institute of Arithmetic in Paris, research algebraic geometry, a subject of analysis regarding geometric figures — referred to as varieties — which can be outlined by algebraic equations. The prototypical instance is the equation *x*^{2} + *y*^{2} = 1, which defines a circle.

She has been described because the world’s foremost professional on the still-unsolved Hodge conjecture, an algebraic-geometry downside that considerations the character of the varieties which can be contained inside a bigger selection. The conjecture is likely one of the Millennium Prize Issues — seven mathematical questions that every carry a US$1-million prize for the primary individual to resolve them.

Voisin has additionally labored on questions that arose from the speculative concepts from physics referred to as string idea. She spoke to *Nature* about a few of her best-known work.

## You’re the first girl to be awarded a Crafoord Prize in Arithmetic. Does this have a particular significance for you?

No. Since I do arithmetic, I’ve all the time been the primary girl to do that, or to do this. Generally I really feel that the media, every time they discuss me, say, ‘the primary girl who …’. Personally, I feel it’s not good to place emphasis on that. For me, I’m only a mathematician. I’m joyful if folks admire the arithmetic that I’m doing.

Bias definitely nonetheless exists. I definitely acknowledge that arithmetic, as a world, just isn’t encouraging to ladies at college and to younger girls. Personally — possibly due to my character, as a result of I don’t care what folks take into consideration me — I didn’t undergo from this.

## Did you want arithmetic as a toddler?

I did. I learnt some from my father. He was an engineer, so he had a really sensible model, and taught me very conventional arithmetic. It was very totally different after I went to highschool. In French faculties on the time, there was the style of ‘trendy arithmetic’, which was an try to show summary arithmetic, resembling set idea. We needed to do fully loopy issues, like compute the event of numbers in base 2.

Later, after I was in preparatory faculty, I didn’t suppose I needed to be a mathematician. In reality, I used to be occupied with philosophy, as a result of I believed that arithmetic was too mechanical. If you do arithmetic at school, at no level are you supposed to supply actually new concepts. Solely a lot in a while, I found that arithmetic has this depth on the degree of ideas.

## Folks usually discuss mathematical theories having depth. How do you outline ‘deep’?

I may give you an instance: the Cartesian coordinates of the aircraft. You possibly can clarify to a toddler you can affiliate two numbers, *x* and *y*, with every level on a aircraft — a flat floor — which suggests you could have two capabilities outlined on the aircraft. It’s quite simple, however it’s completely deep — it’s near having a philosophy of area. And that is as a result of seventeenth-century mathematician René Descartes.

My subject of arithmetic was revolutionized by the late French mathematician Alexander Grothendieck within the Sixties. And the place to begin was a type of revolution in the way in which of understanding geometry: what’s an area? If you outline what an area is, you give whole precedence to the examine of capabilities.

## And is arising with the definition of a ‘deep’ idea a artistic act?

I think about it essentially the most artistic a part of our work. I might distinction this to the technical developments of a idea — the place you should still want some creativity, however it’s extra like a Lego sport, the place you place collectively all of the technical particulars. However essentially the most artistic half is to place down the suitable definition that offers you a brand new solution to assault an issue. For me, it’s merely extraordinary.

## What are the prospects for fixing the Hodge conjecture?

I might say it’s a catastrophe! We have now numerous proof to recommend that the Hodge conjecture is true. However I might say this proof is predicated on arguments that all the things occurs as if the Hodge conjecture have been true.

The issue with the Hodge conjecture is that to show it, you must invent a manner of establishing attention-grabbing varieties. And we have now completely zero concepts on how to do this. So, at current, I see no hope.

## A few of your most celebrated work has been on the mirror-symmetry conjecture, which was impressed by string idea. Are you continue to engaged on physics-inspired issues?

I labored on mirror symmetry possibly for 3 or 4 years. Then I left, as a result of I didn’t really feel I used to be doing my greatest. I used to be making an attempt to grasp what these folks had in thoughts, however I gave up rapidly. The issue is, physicists have extraordinary concepts — type of like magic. However they don’t work on the similar scale of time as us. We mathematicians want numerous time to supply the suitable definitions and to show theorems. And we’re not joyful if the statements aren’t proved rigorously.

If you begin doing that, and also you come again three years later, and inform the physicists, “now I’ve proved your formulation rigorously”, they already went in one other route.

Some mathematicians have stayed involved with physics and have performed extraordinary issues. However for me it was not good, as a result of I wish to work alone and to ask my very own questions.

This interview has been edited for size and readability.