Mathematician has solved the Kakeya conjecture

There are problems that seem childlike on the surface but conceal a monstrous mental labyrinth in whose dead ends some of the most gifted minds in human history have gotten lost. In 1917, Japanese mathematician Soichi Kakeya posed one such deceptively simple problem. You just need to place a needle or a pen against a wall, pointing its tip upward. If you want to turn it around so the tip points downward, what is the smallest surface area the tip’s path would cover?

The intuitive answer is that the needle would trace a perfect circle. But if moved skillfully, it can draw a sort of triangle with concave sides, covering a smaller area.

Mathematician Hong Wang describes a devilish variation of the Kakeya problem. She holds a golden pen in the air and begins to rotate it delicately. What would be the smallest possible volume that allows the pen to point in all directions?

Wang and her colleague Joshua Zahl are the first people to emerge from this mental labyrinth. They’ve solved the Kakeya conjecture in three dimensions.

Hong Wang was born 34 years ago in Guilin, a Chinese city surrounded by mountains so sharp and lush they seem unreal. The landscape, central to legends of dragons and demons, is so beautiful that in China there’s a famous phrase attributed to a poet: “I’d rather be born in Guilin than be a god.”

Wang moves her pen through the air in a garden in the Madrid town of El Escorial, where she’s come to present her findings during a three-day conference organized by Spain’s Institute of Mathematical Sciences (ICMAT). The researcher traces volumes in the air, as if in a trance. Her work has opened the door to an unknown abstract world and stunned her colleagues. “It stands as one of the top mathematical achievements of the 21st century,” said her Israeli colleague Eyal Lubetzky.

The solution to the Kakeya problem isn’t a 3D drawing, but a 127-page study filled with formulas. One attendee at the El Escorial conference joked that only two people in the world can fully understand those 127 pages: the authors themselves.

“I never set out to solve the Kakeya problem,” says Wang, a professor at New York University. She doesn’t even remember the first time she heard about the needle spinning in the air, but she does recall the moment she discovered her real objective: the restriction conjecture. “It was while reading a paper by a Spanish mathematician, Luis Vega,” she recalls.

The restriction conjecture is one of the most significant open problems in harmonic analysis — a branch of mathematics that studies how to break down a signal, such as sound, into more basic components. The main technique, called the Fourier transform after its creator — French mathematician Joseph Fourier (1768–1830) — now makes it possible to compress digital audio and video files.

It’s one of the hottest areas in the field, with applications that save millions of lives by enabling medical imaging technologies such as MRI scans and electrocardiograms. The restriction conjecture deals with how the Fourier transform behaves differently when limited to a curved surface, like a sphere.

Chinese mathematician Hong Wang, explaining the Kakeya problem with a pen in El Escorial, on June 12.Pablo Monge

Wang speaks of her assault on the restriction conjecture as if she had just set up base camp at the foot of a hostile, never-before-climbed mountain in her native Guilin. “Kakeya’s conjecture is the starting point; it’s at the base of a tower of conjectures,” she explains. “The restriction conjecture is more powerful. To make progress, you need to understand Kakeya’s conjecture very well,” adds Wang, who understood it so well that she solved it.

When many lines — or needles — overlap in space, they can form a configuration of wave packets. That’s why, in the words of Australian-American mathematician Terence Tao, one of the greatest living minds in mathematics, the restriction conjecture implies the Kakeya conjecture.

Spanish mathematician Antonio Córdoba, 76, dedicated his 1977 doctoral thesis to the Kakeya challenge. In a public article published in EL PAÍS in March, following the conjecture’s resolution, he explained that the needles in the original problem become parallelepipeds, cylinders, or tubes in higher dimensions.

Córdoba, a former director of ICMAT, praised the work of Wang and Zahl: “They use — following the path of my thesis — complex calculations of how parallelepipeds overlap in space, based on classical Euclidean geometry, but with such combinatorial complexity that their development requires more than 120 pages of intricate reasoning,” he wrote. “It’s an example of what I like to call suprematism in harmonic analysis — due to the use of rectangles and tubes, reminiscent of the works of the Russian Suprematism art movement — but, in their case, it’s a baroque suprematism, if you’ll allow the oxymoron,” he added.

Luis Vega, the Spanish mathematician who inadvertently introduced Wang to the restriction conjecture, is a disciple of Córdoba and former scientific director of the Basque Center for Applied Mathematics in Bilbao. Four years ago, he won Spain’s National Research Prize, awarded by the Ministry of Science.

His answers to this newspaper’s questions offer a glimpse into the complexity of the achievement. “I haven’t worked on these problems in a while. In fact, I’ve been following them from afar. Very sophisticated techniques have been developed that require time and the ability to understand them,” he explains. “It’s clear that Hong Wang and Joshua Zahl are now the standard to follow — and, as I say, a very difficult one to follow. The path they take and wherever it leads will, without a doubt, be fascinating,” he says.

Wang comes across as an extremely humble person, refusing even to mention the possibility of winning a Fields Medal — the highest honor of the International Mathematical Union, reserved for geniuses under 40. The gold of the medal is engraved with a Latin inscription: “Transire suum pectus mundoque potiri” — which can be translated as: “To pass beyond your understanding and make yourself master of the universe."

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