‘Once in a Century’ Proof Settles Math’s Kakeya Conjecture

And even in cases where the grains all conspired to provide maximum overlap, they found, the number of grains intersecting any given point couldn’t be too big. Starting from the 2.5 bound, they were able to prove that the grains couldn’t overlap enough to result in a dimension slightly above that bound either. Then, starting from the higher bound, they showed that the same computational steps could be applied to nudge the bound even higher. And so on.

“It’s like perfecting a perpetual-motion machine. It’s magical,” Tao said. “They’re getting more at the output than the input.” Their machine took them all the way to a Minkowski (and Hausdorff) dimension of three, proving the three-dimensional Kakeya conjecture.

Tower of Dreams

The conjecture’s resolution is a seismic shift for the field of harmonic analysis, which studies the details of the Fourier transform.

A tower of three monumental conjectures in harmonic analysis rests atop the Kakeya conjecture. Each story in the tower needs to be sturdy for the stories above it to stand a chance themselves. If the Kakeya conjecture had been proved false — if Wang and Zahl had found a counterexample — the entire tower would have come tumbling down.

But now that they’ve proved it, mathematicians might be able to work their way up the tower, using Kakeya to build up proofs of these successively more ambitious conjectures. “All these problems that [mathematicians] dreamed about someday solving, they all look approachable now,” Guth said.

That process has already begun. Wang recently co-authored a separate paper reducing the next conjecture in the tower to a stronger version of the Kakeya conjecture, a step toward bridging the two levels.

It’s also a dimensional leap for this entire area of math that’s been somewhat stuck in 2D. “People understood what’s going on [in Kakeya-adjacent problems] really well in two dimensions, but we lacked the tools to study higher dimensions,” Wang said. “So I feel like this was necessary. It needed to be done.”

The four-dimensional Kakeya conjecture remains open, with a tower of four-dimensional conjectures above it as well. New difficulties will arise, Guth said, but he thinks that the jump from two dimensions to three was the hardest, and that Wang and Zahl’s proof can likely be adapted to that tower, and beyond.

“When I got excited about the Kakeya problem as a younger mathematician, it just felt so simple and geometrical that it was surprising to me that it was hard,” Guth said. Years later, Wang, his doctoral student, was motivated by the same deceptive simplicity.

“You have these concrete things you can visualize. It’s not as scary as other math theories,” Wang said. “I just wanted to understand why it’s hard.”

Now, thanks to Wang and Zahl’s efforts, that understanding is closer than ever. “I really think there’s a critical mass of ideas to really revolutionize the whole field coming from here,” Hickman said. “It’s a very, very exciting time.”