Hong Wang Twirls in Fractal Dimensions of Three
Hong Wang, one of this year’s winners of the Fields Medals, which honor the top mathematicians under the age of 40, once considered giving up on math and becoming an architect instead.
After graduating from Peking University in 2011 with a degree in mathematics, she went to École Polytechnique in France to pursue a master’s degree, but had her doubts. “I don’t know if I have enough talent,” she recalled thinking.
For a semester, she even avoided math and did a one-month internship at an architecture firm. Then she returned to mathematics. “I realized that studying architecture from zero is not easy, either,” she said.
Today, Dr. Wang, 35, is a professor of mathematics at New York University and the Institut des Hautes Études Scientifiques in France. Of the 68 recipients of Fields Medals, she is only the third woman to be honored.
“You just stop thinking about it, stop questioning yourself and focus on the work,” she said of overcoming her earlier career indecision.
Dr. Wang is best known for a proof that she and Joshua Zahl, now a professor at Nankai University in China, posted online last year involving a problem known as the Kakeya conjecture.
Think of a needle or a pencil — or a chopstick, says Dr. Wang, who was born in Guilin, China — and spin it around.
How much area does it sweep? The obvious thing to do is to twirl it around the middle, sweeping out the shape of a circle.
But when Sōichi Kakeya, a Japanese mathematician, posed this question in 1917, he realized there was a way to sweep a smaller area if the center of the pencil slid in a circular motion as it rotated.
The area of this shape, called a deltoid, is half that of the circle. Kakeya wondered: Is there a way to sweep even less area?
A couple of years later, Abram Besicovitch, a Russian mathematician, came up with a surprising answer. Through a complex series of moves, the area could be as little as one wanted, approaching zero.
A variation of Kakeya’s question popped up decades later in a seemingly unrelated field of mathematics: harmonic analysis, the study of oscillating motions, like waves. Dr. Wang’s earlier work in harmonic analysis led her to the Kakeya problem.
Kakeya originally imagined an infinitesimally thin needle. In the variation, the needle has some thickness, and its motion is not constrained to spinning in just two dimensions.
The problem can be recast in a way that is equivalent but easier for mathematicians to tackle. Instead of spinning one needle, imagine a set of ghostly needles pointing in all possible directions. You cannot change the direction that a needle points in, but you can slide it around, including through the other needles.
“I have to think about a large collection of long, thin tubes,” like uncooked spaghetti, Dr. Wang said. “Then you want to know how much they can overlap. Of course, for the uncooked spaghetti in real life, they cannot really overlap.”
The shape of the overlapping needles, or uncooked spaghetti, could be a fractal — an infinitely complex pattern with shapes that repeat as you magnify it — and thus its calculated dimension could be a fractional number.
But the Kakeya conjecture — a conjecture is a statement that mathematicians think is true but which has not yet been proved — stated needles pointing in all possible directions in the three dimensions, the fractal dimension of this shape of the needes had to be exactly three.
Proving that it had to be three turned out to be difficult.
Three decades ago, a mathematician showed that the shape had to have a dimension of at least 2.5. Then other mathematicians showed that it was slightly more than that.
“How small can it be?” Dr. Wang said. “So the goal is to show that it cannot be too small.”
Dr. Wang and Dr. Zahl made progress in 2022, and last year, they showed that the Kakeya conjecture was true in three dimensions; the fractal dimension of the collection of needles was indeed three.
For higher dimensions, the conjecture remains open.
