Fields day for Hong Wang

07.22.2026, by

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Hong Wang, 2026 Fields Medal laureate, giving a lecture at the IHES in November 2025.

The Chinese mathematician Hong Wang is the co-winner of the 2026 Fields Medal (the equivalent of the Nobel prize for the discipline). This distinction recognises work on solutions for the Kakeya conjecture in three dimensions, an extension from a century-old problem at the interface of mathematics and geometry. CNRS News provides insights.

The Chinese mathematician Hong Wang, a researcher at the Alexander Grothendieck Laboratory1 at the IHES institute for scientific studies, has just been awarded the prestigious Fields Medal [7] for solving the Kakeya conjecture in three dimensions, alongside Joshua Zahl2.

After earning a Bachelor’s degree in mathematics in 2011 from Peking University, the young Wang went further afield, eventually joining École Polytechnique, in France. While there, she further developed a taste for mathematics, earned her Master’s degree before once again leaving, this time for the United States and the legendary Massachusetts Institute of Technology (MIT), where she completed her PhD in 2019.

It was at that time, thanks in particular to her thesis supervisor Larry Guth, that Wang discovered the Kakeya conjecture. What exactly is it? Vincent Borrelli3, a mathematician and senior lecturer at Université Claude Bernard Lyon 1, enlightens us.

Where did the Kakeya conjecture originate?

Vincent Borrelli: It all started in the early 20th century, some 100 years ago. The Japanese mathematician Soichi Kakeya asked a question that may seem absurd at first, but that has proved amazingly deep: what is the smallest area of a plane in which a needle can be rotated in every possible direction?

The geometric form that immediately comes to mind is a disk. However, further exploration of the problem shows that other, more economic geometric figures are also possible. An especially promising one ultimately emerges: a triangle with three sides that are curved instead of straight, as if it had gone on a diet. Such triangles are known as ‘deltoids’. By switching from a disk (whose diameter is equal to the needle’s length) to a deltoid, the area is halved.

Aiguille en rotation à l’intérieur d’une deltoïde

Needle rotating inside a deltoid (Kakeya set). At each step of its rotation, the needle is in contact with the curve in three points: the two edges (in blue) and a point of tangency (in black). The middle of the needle (in red) outlines a circle with a diameter equal to half the needle’s length.

The mathematical community quickly became convinced that this deltoid provided the solution for the Kakeya problem. All that remained was to find a demonstration for it, an exercise that resisted all attempts, thereby lending the question a global reputation.

Yet the problem – and this was recognised retrospectively – had already been solved by Abram Besicovitch, a Russian mathematician who was Kakeya’s contemporary. His response was totally unexpected: there is no smaller figure! More specifically, there are flat geometric forms that allow for the needle to be rotated, with areas as small as desired. The mental barrier that prevented many scientists from finding the solution to the problem resided in the statement itself, which stipulates that “a minimal area” must exist.

Besicovitch went even further. The needle’s movement put aside, he showed that there are flat geometric shapes with no area (equal to 0) that can still contain the needle in every one of the plane’s directions. These improbable geometric forms are now known as ‘Besicovitch sets’.

How is it possible to build flat objects with no surface area?

V. B.: Therein lies the crux of the Kakeya problem. It is also from that point forward that the problem transformed into a purely geometric one, whereas originally it had a dynamic, kinetic component.

Let’s set aside Kakeya’s needle for a moment and build a zero-area object. To begin with, take a full triangle, which obviously does not have zero area, and then remove an inverted triangle from its centre, leaving three smaller triangles in the corners of the initial figure, whose area is thus reduced. By continuing the procedure indefinitely, it is possible to mathematically demonstrate that something will always remain of the initial shape. The figure obtained at the limit of the process is known as a Sierpiński triangle.

©ENS Éditions 2014 / Vincent Borrelli et Jean-Luc Rullière

Series of figures whose area shrinks each time they are hollowed out, until they disappear. All kinds of zero-area objects can be produced, such as this one made from a triangle.

It is a fractal figure, of which there are many, such as the Koch snowflake, the Apollonian gasket, and the dragon curve. While these objects have an area exactly equal to 0, a hierarchy can nevertheless be established among them according to their apparent ‘thickness’, as some are indeed visually thinner than others.

In mathematics, there is a number to measure this apparent thickness, known as the ‘fractal dimension’. A one-dimensional object can be thought of as a straight line or a curve; a two-dimensional one as a surface. The objects I have just mentioned have an intermediary dimension between 1 and 2 depending on their apparent thickness – close to 1 if they resemble curves, close to 2 if they are like surfaces.

Let’s go back to the Kakeya problem. Since we can find figures with zero area that contain the needle in all directions, a new question arises: what is the fractal dimension of such figures? Can we build a Besicovitch set that is so thin that its fractal dimension is less than 2? The question was solved in 1971 by Roy Davies, and the answer is no.

If the solutions to the Kakeya problem were found long ago, what work is being recognised by the Fields Medal?

V. B.: Kakeya’s question clearly extends to the third dimension. It involves finding the sets that contain the needle in every direction of space, and whose fractal dimension is the smallest possible.

Since the work of Davies, the mathematical community has been convinced that this dimension cannot be smaller than 3, without being able however to find a demonstration or refutation – that is the Kakeya conjecture. Wang and her colleague Joshua Zahl have precisely solved this conjecture.

The area of these six objects – which equals zero – does not account for their differences in density. The term “fractal dimension” is used to capture this visual impression. An object that is close to 1 dimension is filiform, and close to 2 is a surface.

While the second article4 (127 pages – Ed’s note) has not yet been validated by the community, specialists unanimously recognise the remarkable and innovative work of the Wang and Zahl duo. It is the crowning achievement of decades of effort interspersed with successive advances made by Jean Bourgain, Thomas Wolff, Nets Katz, Izabella Laba and Terence Tao, Larry Guth, and others I am forgetting.

The starting point is the following: supposing that there is an extremal Besicovitch set, in other words, one whose fractal dimension is less than 3. What would its geometric properties be? The central issue is to show that these properties must be exceptionally numerous, and ultimately contradictory. The two articles reveal the key role played by the ‘stickiness’ (to use the discipline’s jargon) of such a set. This property forces the invariance of its geometry at different scales, and makes it possible to arrive at the desired contradiction.

The Kakeya conjecture is connected to other important conjectures or questions from various fields of mathematics, in particular harmonic analysis and arithmetic combinatorics. It remains open for dimensions greater than 3.

Beyond its importance, Wang and Zahl’s work opens up many prospects that will no doubt lead to other advances, for the ripple effect and enthusiasm sparked by the solving of a conjecture are always quite compelling.