# One problem less in dimension 24

## HCM PhD student Danylo Radchenko was part of the working group that solved the sphere packing problem in dimension 24. Here, he explains to us how he got involved in this exciting research, what was the key to solving the problem finally, and what it is like to be a PhD student at the Max Planck Institute for Mathematics in Bonn.

**You are co-author of a paper about the sphere packing problem in dimension 24. Let’s start from the scratch: What’s the sphere packing problem about? **

This problem goes way back to 1611 when Johannes Kepler discussed the question how to stack cannon balls most efficiently. He found the answer, but he couldn’t prove it. This is why it’s called the “Kepler conjecture”. Since then, physicists have been working with his solution. But for mathematicians it’s not enough to know the answer, we also have to find a way to deduce logically why it is indeed correct. The solution is actually pretty simple. Almost everyone will find it intuitively if you give them a bunch of balls and some time. But the mathematical proof of this is highly complex. Even for a normal three dimensional room, the proof was found only in 1998 by Thomas Hales.

**And how did you get from there to dimension 24?**

We did not start working on this problem with any particular dimension in mind. However, it was known for some time, from the work of Cohn and Elkies that the sphere packing problem in dimensions 8 and 24 is rather special, and the solution could follow in these dimensions from the existence of certain mysterious functions. Therefore, we were always focused on these special dimensions and the properties of those mysterious functions.

**How did you solve this problem?**

We have been working on it for several years already. There were phases when we were more focused on it and phases when we were less involved, but we never stopped. It was a group of three people: Maryna Viazovska, Andrii Bondarenko,and myself. We did a lot of computations over the time, but somehow didn’t make the progress we wanted. Then, last year, Maryna had the brilliant idea of not constructing the functions we actually needed, but slightly different ones using the theory of modular forms. She discovered that there is a direct relation between these two different fields. She made the real breakthrough. At first, I was skeptic and thought this couldn’t be true. But then I realized quickly that this was going to be the answer. Still, it took us more than a year of hard work until the solution was complete in dimension eight. I helped Maryna on some minor details and with some of the computer calculations, but the solution in this case is rightfully hers.

**What was the next step?**

Several days after Maryna had submitted her paper on the sphere packing problem in dimension eight, she called me and asked if we could now join forces with Henry Cohn, Abhinav Kumar, and Stephen D. Miller, who were also working on this problem, to solve the problem in dimension 24, which is more challenging for technical reasons. Of course I was happy to do so. We then completed the paper in one week of very intense work. At peak moments our team exchanged about ten emails every hour. We were in quite a hurry, because now that the solution in dimension eight was out there, others could have done it for dimension 24 as well.

**Is this only of theoretical interest or are there some aspects of the sphere packaging problem that are related to practical issues? **

It is for example related to coding theory and combinatorics. Improvements made for the sphere packing problem also stimulate new developments in those fields. An object closely related to the solution in the 24-dimensional case, the so-called binary Golay code, was used by the NASA in the Voyager program to check if messages sent through long distances arrive correctly despite the high background noise in space.

In the end, I think that the most important result of our work is different: it is not the solution of the sphere packing problem itself, but the newly discovered connection to modular forms. This could lead to completely new developments. That’s very exciting, and we are currently trying to better understand this connection.

**Is the sphere packaging problem part of your PhD project, too?**

No. In my PhD, I’m working on the topic of functional equations for polylogarithms. That was a bit stressful because when the intense phase of working on the sphere packaging problem started, I was about to submit my PhD thesis. Of course, then I thought, the PhD could wait. But now I’ve submitted it, finally, and I will defend it soon.

**You’re originally from the Ukraine. How did you come to Bonn?**

Maryna invited me as a guest to the Max Planck Institute when she was a PhD student here herself and I was still a bachelor student. Back then I also met Don Zagier for the first time, who’s now my advisor. I was really impressed by the atmosphere here and wanted to return. It’s actually hard to find words for how nice it is. There just happens so much in mathematics in Bonn. You have really great colleagues to talk to, there are a lot of talks and events, and researchers from all over the world come to the Hausdorff Center for guest programs. That’s very inspiring. The mood here in Bonn definitely contributed significantly to my work.