When Hong Wang was six years old, her father, a schoolteacher in the emerald-hilled city of Guilin, China, showed her a puzzle at the end of a math textbook chapter. How should you plant seven trees to get the greatest number of rows of three trees? The problem is a classic exercise in incidence geometry: arranging points so that lines pass through as many of them as possible.
Wang solved it faster than her father. The answer: plant the trees in the shape of an equilateral triangle, with one at each corner, one at the midpoint of each side, and the seventh in the center. This produces six rows of three trees. She was six years old.
Nearly three decades later, on July 23, 2026, the International Mathematical Union awarded Wang, now 35 and a professor at New York University’s Courant Institute, the Fields Medal. She is the third woman in the award’s 90-year history to receive mathematics’s highest honor, following Maryam Mirzakhani in 2014 and Maryna Viazovska in 2022. Her achievement: solving the three-dimensional Kakeya conjecture, a problem that had resisted the world’s best mathematicians for more than half a century.
From Earth Sciences to Math’s Summit
Wang’s path to the Fields Medal was anything but linear. She entered Peking University in 2007 as an earth sciences major, not a mathematics one. The university had reserved just one spot in its math program for a student from her province, and she had not earned the highest test score. So she enrolled in geophysics instead, hoping to transfer later.
Her geophysics professor encouraged the switch, pointing out how deeply mathematical the science of seismic waves really is. Wang worked hard, earned the transfer, and began studying the subject she had loved since childhood. But even then her commitment wavered. After moving to France to study at the Ecole Polytechnique, she struggled with the language and briefly switched to architecture for a semester before returning to mathematics for good.
“I decided not to worry about whether I would be good or not,” she told Quanta Magazine, “and just work on understanding better.”
She earned her PhD at MIT under Larry Guth from 2014 to 2019, then completed a postdoc at the Institute for Advanced Study in Princeton. After faculty positions at UCLA and then NYU, she began producing results that would reshape the field.
Along the way, Wang racked up a string of landmark proofs. In 2019, she proved the 2D local smoothing conjecture with Guth and Ruixiang Zhang. In 2023, she and Kevin Ren proved the 2D Furstenberg set conjecture. But her crowning achievement came in February 2025, when she and collaborator Joshua Zahl of the University of British Columbia posted a 127-page proof of the 3D Kakeya conjecture.
The Problem That Wouldn’t Break
The Kakeya conjecture traces back to 1917, when Japanese mathematician Soichi Kakeya asked a deceptively simple question: What is the smallest area in which you can rotate a needle to face every possible direction? A few years later, Abram Besicovitch showed that the answer is zero. An infinitely thin needle can be maneuvered so that it sweeps out an area smaller than any positive number you name.
But the modern form of the problem, formulated in 1971 by Charles Fefferman, is far more consequential. Instead of an infinitely thin needle, imagine tubes of a fixed small thickness, pointing in every direction. The question becomes: do the points where these tubes overlap fill up the full three-dimensional space, or can they be compressed into something thinner? The Kakeya conjecture asserts they must fill the whole space. The 2D case was proved in 1971. The 3D case remained open for decades, a towering obstacle at the intersection of harmonic analysis and geometric measure theory.
The conjecture’s importance extends far beyond geometry. It sits at the base of a tower of major unsolved problems in Fourier analysis. Proving it false would have collapsed the entire structure. Proving it true, as Wang and Zahl did, opens the way to the next set of challenges.
The Proof Strategy: Sticky and Non-Sticky
Wang and Zahl’s strategy was elegant. They started by assuming a counterexample exists: a Kakeya set whose dimension is less than 3. Then they analyzed its structure. The tubes in such a set must fall into two categories. Either they are “sticky,” meaning that tubes with similar orientations cluster together, or they are “non-sticky,” with tubes spreading apart.
If the tubes are sticky, Wang and Zahl proved, the set must actually be three-dimensional. If they are non-sticky, the mathematics forces the set into a higher dimension still. Both paths lead to a contradiction. Therefore no counterexample can exist. Every Kakeya set in three dimensions must be full-dimensional.
The proof has been described as a “perpetual-motion machine” by Terence Tao, a Fields Medalist himself. “They’re getting more at the output than the input,” Tao said. Nets Katz of Rice University called it “a once-in-a-century kind of result.”
Starting from a known lower bound of 2.5 for the Kakeya dimension, proved by Thomas Wolff in 1995, Wang and Zahl showed that if no counterexample exists at dimension d, none can exist at dimension d plus any epsilon. By repeating this bootstrapping argument over and over, they pushed the bound all the way to 3.
Three Women in 90 Years
Wang joins an exclusive and, until recently, all-male club. Since the Fields Medal was first awarded in 1936, 64 mathematicians had received it before the 2026 ceremony. Of the first 52, all were men. Mirzakhani broke the barrier in 2014; Viazovska followed in 2022. Now Wang makes three.
The statistic is a stark measure of the structural barriers that women have faced in mathematics. Wang’s own story illustrates the problem in a different way. Her talent was obvious from childhood. Her father, a math teacher, gave her puzzles; she solved them faster than he did. She devoured textbooks before the semester started, then bought more and solved those too. Yet when she applied to Peking University, the math department had exactly one reserved slot for students from her province. She missed it. She got in through earth sciences instead.
The narrowness of the pipeline is not just a Chinese phenomenon. Worldwide, women earn roughly a third of mathematics PhDs but remain dramatically underrepresented among Fields Medalists, tenured faculty, and invited speakers at major conferences. Wang herself has spoken about the self-doubt that accompanies her work, describing each proof as lucky and worrying about falling behind her peers. “You have doubt,” she said of the Kakeya proof, “but also the argument feels natural.”
What Wang’s Path Reveals
Wang’s trajectory from earth sciences major to Fields Medalist challenges assumptions about how mathematical talent emerges and should be cultivated. Her path was not a straight line from childhood prodigy to elite PhD program. It detoured through geophysics, French language classes, and even a semester of architecture. Each detour could have ended her mathematical career. Instead, each one enriched it.
The geophysics professor who encouraged her to pursue math understood something fundamental: mathematical thinking is not confined to mathematics departments. It appears in physics, in engineering, in the way seismic waves move through the Earth. The architecture semester taught her about structure and form. Even the struggle with French gave her a different perspective on certainty; she came to appreciate mathematics precisely because, unlike language, “no one can come and say, ‘Oh, actually, this is not true.'”
Wang’s story suggests that the search for mathematical talent should look in unexpected places. The next Fields Medalist might be studying something else entirely. She might be in a geophysics lecture, or a design studio, or working through puzzles in her father’s textbook, not yet knowing what she will become.
For now, Wang plans to regain balance. She wants to read for pleasure, like her advisor Guth. She wants to avoid getting too invested in any single problem. The 4D Kakeya conjecture remains open. The restriction conjecture, the local smoothing conjecture in higher dimensions: the tower of problems that rests on the Kakeya needle still stands. But for the first time in decades, mathematicians can see how to climb it.
All because a six-year-old girl in Guilin planted seven trees and never stopped asking what else the lines could show her.
References
Wolchover, Natalie. “Living Fully in the Math World Means Threading the Needle.” Quanta Magazine, July 23, 2026. https://www.quantamagazine.org/hong-wang-wins-2026-fields-medal-the-third-woman-ever-20260723/
Howlett, Joseph. “‘Once in a Century’ Proof Settles Math’s Kakeya Conjecture.” Quanta Magazine, March 14, 2025. https://www.quantamagazine.org/once-in-a-century-proof-settles-maths-kakeya-conjecture-20250314/
Wang, Hong, and Joshua Zahl. “The 3D Kakeya Conjecture.” arXiv:2502.17655, February 2025.
NYU News. “NYU Professor Hong Wang Wins Fields Medal.” July 23, 2026. https://www.streetinsider.com/PRNewswire/NYU+Professor+Hong+Wang+Wins+Fields+Medal/26806147.html
Chang, Kenneth. “Hong Wang Twirls in Fractal Dimensions of Three.” The New York Times, July 23, 2026. https://www.nytimes.com/2026/07/23/science/hong-wang-fields-medal.html