String Theory Inspires a Brilliant, Baffling New Math Proof

In particular, Katzarkov wanted to break the mirror image’s curve count into pieces, then use the mirror symmetry program to show that there was a corresponding way to break up the four-fold’s Hodge structure. He could then work with these pieces of the Hodge structure, rather than the whole thing, to show that four-folds can’t be parameterized. If any one of the pieces couldn’t be mapped to a simple 4D space, he’d have his proof.

But this line of reasoning depended on the assumption that Kontsevich’s mirror symmetry program was true for four-folds. “It was clear that it should be true, but I didn’t have the technical ability to see how to do it,” Katzarkov said.

He knew someone who did have that ability, though: Kontsevich himself.

But his friend wasn’t interested.

Digging In

For years, Katzarkov tried to convince Kontsevich to apply his research on mirror symmetry to the classification of polynomials — to no avail. Kontsevich wanted to focus on the whole program, not this particular problem. Then in 2018, the pair, along with Tony Pantev of the University of Pennsylvania, worked on another problem that involved breaking Hodge structures and curve counts into pieces. It convinced Kontsevich to hear Katzarkov out.

Katzarkov walked him through his idea again. Immediately, Kontsevich discovered an alternative path that Katzarkov had long sought but never found: a way to draw inspiration from mirror symmetry without actually relying on it. “After you’ve spent years thinking about this, you see it happening in seconds,” Katzarkov said. “That’s a spectacular moment.”

Kontsevich argued that it should be possible to use the four-fold’s own curve counts — rather than those of its mirror image — to break up the Hodge structure. They just had to figure out how to relate the two in a way that gave them the pieces they needed. Then they’d be able to focus on each piece (or “atom,” as they called it) of the Hodge structure separately.

This was the plan Kontsevich laid out for his audience at the 2019 conference in Moscow. To some mathematicians, it sounded as though a rigorous proof was just around the corner. Mathematicians are a conservative bunch and often wait for absolute certainty to present new ideas. But Kontsevich has always been a little bolder. “He’s very open with his ideas, and very forward-thinking,” said Daniel Pomerleano, a mathematician at the University of Massachusetts, Boston, who studies mirror symmetry.

There was a major ingredient they still had no idea how to address, Kontsevich warned: a formula for how each atom would change as mathematicians tried to map the four-fold to new spaces. Only with such a formula in hand could they prove that some atom would never reach a state corresponding to a properly “simplified” four-fold. This would imply that four-folds weren’t parameterizable, and that their solutions were rich and complicated. “But people somehow got the impression that he said it was done,” Pomerleano said, and they expected a proof soon.

When that didn’t come to pass, some mathematicians began to doubt that he had a real solution. In the meantime, Tony Yue Yu, then at the French National Center for Scientific Research, joined the team. Yu’s fresh insights and meticulous style of proof, Kontsevich said, turned out to be crucial to the project.

When lockdowns began during the Covid pandemic, Yu visited Kontsevich at France’s nearby Institute for Advanced Scientific Studies. They relished the quiet of the deserted institute, spending hours in lecture halls where there were more blackboards, Yu recalled.

Meeting regularly with Pantev and Katzarkov over Zoom, they quickly completed the first part of their proof, figuring out precisely how to use the number of curves on a given four-fold to break its Hodge structure into atoms. But they struggled to find a formula to describe how the atoms could then be transformed.

What they didn’t know was that a mathematician who had attended Kontsevich’s lecture in Moscow — Hiroshi Iritani of Kyoto University — had also started pursuing such a formula. “He was enchanted by my conjecture,” Kontsevich said. “I didn’t know, but he started to work on it.”

In July 2023, Iritani proved a formula for how the atoms would change as four-folds were mapped to new spaces. It didn’t give quite as much information as Kontsevich and his colleagues needed, but over the next two years, they figured out how to hone it. They then used their new formula to show that four-folds would always have at least one atom that couldn’t be transformed to match simple 4D space. Four-folds weren’t parameterizable.

Still Processing

When the team posted their proof in August, many mathematicians were excited. It was the biggest advance in the classification project in decades, and hinted at a new way to tackle the classification of polynomial equations well beyond four-folds.

But other mathematicians weren’t so sure. Six years had passed since the lecture in Moscow. Had Kontsevich finally made good on his promise, or were there still details to fill in?

And how could they assuage their doubts, when the proof’s techniques were so completely foreign — the stuff of string theory, not polynomial classification? “They say, ‘This is black magic, what is this machinery?’” Kontsevich said.

“Suddenly they come with this completely new approach, using tools that were previously widely believed to have nothing to do with this subject,” said Shaoyun Bai of the Massachusetts Institute of Technology. “The people who know the problem don’t understand the tools.”

Bai is one of several mathematicians now trying to bridge this gap in understanding. Over the past few months, he has co-organized a “reading seminar” made up of graduate students, postdoctoral researchers and professors who hope to make sense of the new paper. Each week, a different mathematician digs into some aspect of the proof and presents it to the rest of the group.

But even now, after 11 of these 90-minute sessions, the participants still feel lost when it comes to major details of the proof. “The paper contains brilliant original ideas,” Bai said, which “require substantial time to absorb.”

Similar reading groups have been congregating in Paris, Beijing, South Korea and elsewhere. “People all over the globe are working on the same paper right now,” Stellari said. “That’s a special thing.”

Hassett likens it to Grigori Perelman’s 2003 proof of the Poincaré conjecture, which also used entirely new techniques to solve a famous problem. It was only after other mathematicians reproduced Perelman’s proof using more traditional tools that the community truly accepted it.

“There will be resistance,” Katzarkov said, “but we did the work, and I’m sure it’s correct.” He and Kontsevich also see it as a major win for the mirror symmetry program: While they’re not closer to proving it, the result provides further evidence that it’s true.

“I’m very old, and very tired,” Katzarkov said. “But I’m willing to develop this theory as long as I’m alive.”