Last spring Maryna Viazovska solved a fascinating unsolved problem about the most efficient way to pack 8-dimensional spheres. Erica Klarreich wrote a great story about the problem and Viazovska’s result in Quanta magazine:

Reading the article I was surprised to learn that a former classmate of mine – Henry Cohn – played a role in the story. This little connection made me follow the story more closely than I’ve followed other math stories. So, I was super excited when I saw that Viazovska was speaking at Harvard:

Her talk last week was fantastic. I’ve been out of academic math for too long to follow every detail, but I’d say that the talk was pitched at a 1st year graduate student level which meant that I could follow enough of it to get the general idea of what she did.

Also, as an aside, while probably any talk in the Harvard math department has impressive people in the audience, the various math folks sitting around would have reduced me to a pile of jello as a speaker. One sort of surprising thing, though, was that there weren’t a lot of people there aside from the Harvard faculty members. The last time I’d been at a math talk at Harvard was in the mid 90’s when Cliff Taubes was doing a series of lectures on the new Seiberg-Witten theory. Those lectures (I think) were in the same classroom and were standing room only. I’d guess maybe 30 total people were there for Viazovska’s lecture, which was a little disappointing to me.

The talk started off with a couple of small surprises to me. First, it actually isn’t totally obvious how to define the best packing of spheres (in any dimension), so you need to be careful and get the definitions right. Second, the best packing aren’t necessarily nice symmetric lattices.

Next up was a discussion of the background work that Henry Cohn and Noam Elikes (who were both there for the talk) had done. That work included a theorem giving a bound on the sphere packing density that used a function with certain peculiar properties. I think (but am not totally sure) that people knew that this function must exist, but no one knew how to find it or describe it in any simple, closed-form way. Viazovska’s result was writing down the formula for that function in 8 dimensions.

She didn’t show the entire procedure for how to write down the specific function (but very dryly and hilariously assured the audience that she’d checked the calculation at home and it worked so there was nothing to worry about 🙂 ), but instead showed a case that she said was a bit easier. The easier case still seemed like an impossible problem to me, though sort of reminded me of this famous problem:

Let be the positive, real solutions to the equation . Show that

Unfortunately I had to leave the talk before the end to get to a dinner, but I’m really happy that I got to see it. I hope there’s more publicity around this result because that students who are interested in math will find it to be fascinating. There are also several nice techniques from Fourier analysis and complex analysis in the result that would make great examples in intro graduate level courses in those subjects.

Anyway, I thought her talk was the end of the story, but I was pleasantly surprised to see a poster for a second talk – this time at BU – earlier this week.

I happened to have a meeting in Boston at 2:30 and was able to take the T over to BU to catch Henry’s talk. His talk was more of an overview of the sphere packing results over the years with a sketch of Viazovska’s result. For me it was a nice complement to the talk at Harvard because now I had a much better feel for the historical context of the results. He also mentioned a little bit more about the work that Tom Hales did on the 3 dimensional result, which is an incredible story all by itself.

I think this talk that he gave at the Berlin Mathematical School this summer is close to the talk he gave at BU:

I wish that I had some good ideas for how to share the 8 dimensional sphere packing results with kids or with the public. Right now I don’t, but it feels like the problem is accessible even if the solution isn’t. For now my best high dimensional project is Bjorn Poonen’s sphere problem:

Sharing Bjorn Poonen’s n-dimensional sphere problem with kids

At least this problem shows off some of the really strange properties of high dimensional spheres.

Anyway, even though I don’t really have any good ideas about how to share this new result, it was a really nice couple of weeks. It was fun getting a little peek back inside of academic math.

## Comments

You say “I think (but am not totally sure) that people knew that this function must exist, but no one knew how to find it or describe it in any simple, closed-form way.” Actually, my understanding is that, while there was enormous circumstantial evidence in favor of the existence of the function, nobody could prove it existed — indeed, if someone had proved the function’s existence, that would’ve settled the packing problem, with or without a formula for the function. The pivotal role that can be played by a mere assertion of existence is one of the mnd-blowing things I love about math.

Thanks – hadn’t thought of it that way.

I wanted to ask Henry about it in his talk, but the questions were focusing on some of the specifics of the modular forms in Viazovska’s paper and I didn’t want to detail that conversation.

I was able to convince a science magazine for middle schoolers to let me write something for them about sphere-packing. Hopefully I’ll be able to share that in a few months!

Can’t wait!