On [GHKK] by Seán Keel

I had been trying for many years, together with Paul Hacking, to find a canonical toroidal compactification of the moduli space of K3 surfaces, following a vague but persistent idea that K3 surfaces are like toric surfaces, but with the lattice polytope replaced by a sphere. Spheres in this connection had already appeared in a stunning 2004 paper of Kontsevich and Soibelman, that I was studying, without much understanding.

It was towards the end of my first visits to IHES in 2007, that I first heard of related, and for me easier to follow, ideas of Gross and Siebert, from Mark Gross who was there on an overlapping visit. Paul and I began to work with Mark, and our interest in the moduli space was gradually superseded by something more fundamental and basic, that has occupied my thinking since: in order for what we wanted to do to work, Calabi-Yaus (and their simpler open cousins, log CYs) had to admit a vast generalization of the theta functions on Abelian varieties (or the monomials on their open cousins, algebraic tori). It was around 2012 that we formalized this conjecture. I was encouraged by Maxim, who very enthusiastically embraced the idea, and by Yan Soibelman, who first explained to me how it was very natural from the point of view of Maxim’s Homological Mirror Symmetry conjecture.

Some years later Goncharov was at IHES for a short visit, and was going to give a talk about his joint conjectures and results with Fock on canonical bases of functions on cluster varieties. I had never heard of cluster varieties, but a canonical basis was just what we were conjecturing and I was looking forward to Goncharov’s lecture. Just before It started, Maxim came into my office, as he commonly does, to show off some interesting thing that had occurred to him. I asked him, didn’t he want to go to the lecture and he said no, because he had already heard it. So I stayed in my office (you don’t walk away from Maxim) and listened (probably with very little understanding) to whatever it was he wanted to show me.

It was only several months later that I saw a preprint of Fock and Goncharov, and quickly realized that their conjectured bases of functions on cluster varieties was a very special case of our conjecture on canonical theta functions for log CYs. The exciting thing was that Fock and Goncharov had really stunning potential applications for their (and thus for our) conjectures. I immediately began to bug Mark about cluster varieties — I really wanted to prove our conjectures in that special case. Mark was (and still is) very very busy with an ambitious project with Siebert (the so called Gross-Siebert Program), and he dismissed me, getting about as close to rude as he ever does.

But then, maybe a year later, Maxim gave a series of lectures in Paris, that Mark attended, and at one he gave ideas and conjectures connecting mirror symmetry with the main open conjectures in the cluster variety world — including really interesting conjectures of Fomin-Zelevinski, and Fock-Goncharov. This got Mark very interested and, together with Paul, we looked seriously into the questions, which in fact were a near perfect simple case of what we were trying to do in much greater generality.

In quite short order we were able to prove all the main conjectures, and I agreed to give a series of lectures on our results at that years instance of Maxim’s Miami Mirror Symmetry conference. There was one basic conjecture that we still had not proven — positivity in the so called Laurent phenomenon.

In Miami, I realized there was actually a gap in our work. For our results, we actually needed this positivity. I knew Maxim had thought about it, so I asked him. He said he did not know how to show it, but that experimental evidence convinced him that it was absolutely true. I remember I was at dinner, with Maxim, Soibelman, Goncharov, Kapranov — Mark was not there, as he was preparing his lecture for the next day. I suggested to Maxim that we work together on this positivity, and he agreed. I was pretty excited to be at dinner with such hot shots, and to have a formal collaboration with Maxim.

The next morning, I ran into Mark, and told him about the arrangement. He started laughing, he was very excited because that night he had figured out how to prove the necessary positivity. We went ahead and wrote the paper. As Maxim will tell you, he had almost nothing to do with it. But certainly having his name on it has had absolutely fabulous promotional benefit.

A central role in our work was played by a combinatorial gadget of Gross and Siebert — so called broken lines — which are piecewise straight paths on the spheres above (or rather their analogs for the log CY case). It bothered me that we did not know what these things “really” were. A while later, I learned, in a really inspiring conversation with Mattias Jonsson, that the Berkovich version of an analytic disk (with one marked point), contains a canonical such piecewise straight path.

Then, in a long conversation with Maxim, I remember it was in the small lecture hall, the one in the main IHES building, we realized that these broken lines must indeed be just these canonical paths that Jonsson had mentioned, and that the structure constants in the mirror algebra must be counts of these analytic disks (explaining why they are positive integers, they are naive counts of something). This was one of the most exciting, and important, mathematical conversations of my life.

To make use of the idea would require a lot of foundational work in Berkovich geometry (about which I knew, at that time, essentially nothing). Back then, I had a very promising graduate student, who had just arrived at UT from China. I gave him this problem. He was a very smart student, but this was a very tough lift — especially as I knew way too little about the subject to be able to help him much. He ended up doing something different, also very nice, instead (he has since left mathematics and now has a very nice job at Google). Unknown to me, Maxim also had a new and promising student, coincidently, also from China. He gave him essentially the same problem.

The result was Tony Yu’s quite spectacular PhD thesis. Tony and I have been building on this work since, reformulating much of the Gross-Siebert program in Berkovich geometric terms. At this point, we have realized many of the wild hopes from that (for me) momentous afternoon with Maxim in the small lecture room. Tony and I have since gone back and redone much of GHKK (Gross-Hackin-Keel-Kontsevich) — the paper that received the award — using the Berkovich formulation. Maxim may not have had much to do with the original GHKK. But his ideas were foundational in this reformulation (which I personally like a lot better).

Photo credit: © Simons Foundation