Professor Viazovska, this year you were honored with the Fields Medal, the highest award in mathematics, for your proof of the densest packing of spheres in 8 and 24 dimensions. Here in Berlin you talked about unsolved math problems. The most famous of these are undoubtedly the six outstanding issues, for which the Clay Foundation has bid $1 million each. Which of these would you most like to experience?
Perhaps the question of the solutions to the Navier-Stokes equations, the basic equations of fluid mechanics. Finally, how water or air flows is an important question, even if it is far from my own research.
Is it not the proof of the so-called Riemann conjecture, the basis of so many other statements in pure mathematics?
The Riemann hypothesis has ruined the lives of many. Because it is so famous, many have tried it, many false solutions have been presented. Such a thing is unpleasant, so I am personally afraid of it.
So you wouldn’t try the Riemann hypothesis?
If I had a really good idea, yes. But I haven’t done that yet, and I’m not going to spend the next fifty years of my life on it just because the problem is so famous. Of course it would be great if someone could fix this. But it may not be solvable. Anyway, the closest millennium problem to my own field of research is the Birch and Swinnerton-Dyer conjecture, a nice statement of number theory. But here, too, the evidence seems out of reach at the moment.
Is there an important issue that you are missing from the Clay Foundation’s list, such as Goldbach’s conjecture?
There are many interesting problems, and you should study them because you feel compelled to do so, not because they are on a list. Such lists are good for popularizing math among the public, but they are very subjective.
Notably, half of the outstanding problems on the Clay list are related to fields outside of pure mathematics. Does that bother you as a pure mathematician?
I think the distinction between pure and applied mathematics is a bit artificial. Interesting problems can be caused by application questions as well as by pure intellectual curiosity. Take the theory of elliptic curves. It has important applications in cryptography. But people started doing it long before there were computers.
Your work on the densest packing of spheres in higher dimensions was also purely driven by curiosity, wasn’t it? In three spatial dimensions, it was initially a practical problem to pack oranges or cannonballs particularly tightly.
Yes, but finding such packages was not difficult. It was hard to show that it couldn’t get any better. In this sense, the problem of the densest packing of spheres is one of pure mathematics, even in three spatial dimensions. You can also stack oranges at the grocery store without knowing there is no better way.
But is there any connection at all with practical applications for bulb packaging in more than three dimensions?
Yes, but not as proof that the packaging is as tight as possible. There is the idea, going back to Claude Shannon, that spherical packings in higher dimensions – and in types of spaces other than Euclidean position space – can be used for error correction of messages. Knowing how to wrap orbs in these cases gives us a fun way to send data over noisy channels so we can reconstruct them.
Finally a very different question. You have held the chair of number theory at the École polytechnique fédérale de Lausanne in Switzerland since 2018, but you are originally from Ukraine. You recently dedicated your Alice Roth lecture at ETH Zurich to the memory of Yulia Zdanovska, a 21-year-old mathematician and computer scientist who decided to help defend her country after Putin’s attack on Ukraine and died in March and was killed by a missile attack on Kharkov. What is the situation for mathematics in your home country?
The situation in Ukraine is very difficult for all people and therefore also for mathematicians. But the university work continues, I myself gave an online lecture this morning at Kyiv University. The enthusiasm for mathematics among the students is unbroken and it is now important to keep it up despite the difficult times.