What is the smallest distance a needle can be rotated until its position is reversed? This mathematician has finally solved Kakeya's conjecture.

There are problems that appear childish, but which hide a monstrous mental labyrinth whose dead ends have led to the demise of some of humanity's most gifted minds. The Japanese mathematician Soichi Kakeya posed a very simple one in 1917. Just place a needle or a pen against a wall and point the tip toward the ceiling. If we want to turn it upside down and point it toward the floor, what is the smallest surface area that the trajectory will trace? The intuitive answer is that by inverting the needle, it will form a perfect circle, but if moved skillfully, it will compose a kind of triangle with concave sides, covering a smaller area. Mathematician Hong Wang explains a devilish variation on Kakeya's problem. She takes a golden pen in the air and begins to gently spin it. What would be the smallest volume to point everywhere? Wang and his colleague Joshua Zahl are the first people to emerge alive from this labyrinth. They have solved Kakeya's 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, the subject of legends of dragons and demons, is so beautiful that a lapidary phrase attributed to a poet circulates in China: "I'd rather be born in Guilin than be a god." Wang moves her pen in the air in a garden in the Madrid town of El Escorial, where she came to explain her results for three days in June at a conference organized by the Institute of Mathematical Sciences (ICMAT). The researcher draws volumes in the air, as if in a trance. Her work has opened the door to an unknown abstract world and has shocked her colleagues. "It is one of the greatest mathematical achievements of the 21st century," proclaimed her Israeli colleague Eyal Lubetzky .

The solution to Kakeya's problem isn't a three-dimensional drawing, but a 127-page study filled with formulas. An attendee at the El Escorial conference jokes that only two people in the world are capable of fully understanding those 127 pages: the authors themselves. "I had no ambition to solve Kakeya's problem," says Wang, from New York University (USA). The professor doesn't even remember the first time she heard about the spinning needle, but she does remember the day she learned about her true goal: the restriction conjecture. "It was while reading a study by a Spanish mathematician, Luis Vega ," she recalls.

The constraint conjecture is one of the most important open problems in harmonic analysis, a branch of mathematics that studies how to break down a signal, such as sound, into its most basic components. The main technique, called the Fourier transform after its creator, the Frenchman Joseph Fourier (1768–1830), now allows for the compression of digital audio and video files. It is one of the hottest areas of the discipline, and its applications save millions of lives by also enabling the creation of medical diagnostic images , such as magnetic resonance imaging and electrocardiograms. The constraint conjecture addresses the different behavior of the Fourier transform when restricted to a curved surface, such as a sphere.

Wang speaks of his assault on the Restriction Conjecture as if he had just set up base camp at the foot of a hostile, never-before-climbed mountain in his native Guilin. “Kakeya’s conjecture is the starting point; it’s at the base of a tower of conjectures,” he notes. “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 he solved it. When many lines—or needles—overlap in space, they can give rise to a configuration of wave packets, which is why the Restriction Conjecture implies Kakeya’s conjecture, in the words of American Terence Tao , one of the greatest living mathematicians.

Antonio Córdoba , 76, a Spaniard, dedicated his 1977 doctoral thesis to Kakeya's challenge. In a popular text published in EL PAÍS in March, following the resolution of the conjecture, he explained that the needles in the initial proposal become parallelepipeds, cylinders, or tubes in larger dimensions. Córdoba, former director of the ICMAT, applauded the work of Wang and Zahl. "They employ—in the wake of my thesis—complicated calculations of the overlapping of parallelepipeds in space, based on classical Euclidean geometry, but of such combinatorial complexity that their development requires more than 120 pages of intricate reasoning," Córdoba explained. "It is an example of what I like to call Suprematism in harmonic analysis—due to the use of rectangles and tubes, similar to those observed in works of the Russian painting movement —but, in their case, it is a Baroque Suprematism, if you'll pardon the oxymoron," he added.

Luis Vega, the Spaniard who unwittingly revealed the restriction conjecture to Wang, is a disciple of Córdoba and former scientific director of the Basque Center for Applied Mathematics in Bilbao. Four years ago, he won the National Research Award granted by the Ministry of Science. His answers to this newspaper's questions give an idea of the complexity of the achievement. "I haven't worked on these things for 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 acknowledges. "It's clear that Hong Wang and Joshua Zahl are currently the trail to follow, and, as I say, very difficult to follow. The path they follow and the end of the path, whatever it may be, will undoubtedly be exciting," he opines.

The New York University professor behaves like an extremely humble person, refusing to even mention the possibility of winning a Fields Medal, the International Mathematical Union's highest award, reserved for geniuses under 40. The gold medal features a Latin inscription: " Transire suum pectus mundoque potiri ," which roughly translates as "Transcend oneself and conquer the world."