Wang Hong becomes a hot candidate for the Fields Medal. Why is the Kakeya conjecture so important?

A hundred years ago, Japanese mathematician Soichi Kagyani proposed an interesting plane geometry puzzle. When the plane conditions in the question were extended to general n-dimensional space, an important proposition called "Kageya Conjecture" was developed. Analysts later realized that the Kagyani Conjecture is closely related to one of the most important unsolved problems in contemporary mathematics, so it became the core of geometric measure theory, opening up a broad research space for this emerging field. However, for a long time, the mathematical community has only solved the Kagyani conjecture in 2-dimensional space. 3-dimensional space is like an indestructible fortress, and the mathematical weapons developed by mankind so far have been difficult to achieve full success - until the end of February 2025, Wang Hong and Joshua Zare published a paper that shocked the mathematical community.

Written by | Jiawei

The Japanese samurai who was attacked while using the toilet and the Riemann integral

Mathematics is a science that expresses difficult ideas in simple words.

—Edward Kasner and James Newman

This is like a spoiler. There are only a few days left in the second month of 2025, and the mathematics community seems to have seen the most important mathematical achievement of the year: Wang Hong, an associate professor at the Courant Institute of Mathematics at New York University, who is only 34 years old, collaborated with Joshua Zahl of the University of British Columbia to submit a 127-page paper on the preprint website arXiv, announcing the proof of the three-dimensional Kakeya Conjecture.

Wang Hong and Joshua Zare published a preprint of the paper on arXiv, claiming to have proved the 3D Kakeya conjecture | Image source: [2502.17655] Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions

The Kakeya conjecture was born out of the famous needle problem. In 1917, Japanese mathematician Soichi Kakeya (1886-1947) proposed a famous problem that was later named after him:

Which of these shapes has the smallest area? Note that the line segment does not sweep any area when moving forward and backward along the line segment.

It is said that Kakeya was inspired by a Japanese samurai who was attacked by surprise. He abstracted the samurai sword into an ideal long needle that does not take up space. At the same time, for the sake of convenience, he limited the problem to a two-dimensional plane. Kakeya commented:

The samurai had a long sword for self-defense, and he needed to be able to wield it freely in any space, no matter how large or small—even a toilet.

Note: Regarding the original statement and related history of the Kakeya needle problem, the records given in different documents are slightly different. For the sake of convenience, the statement in Julian Havil's book "Unbelievable: Counterintuitive Puzzles and Their Amazing Solutions" in the "Discovering Mathematics Series" published by Shanghai Science and Technology Education Press is adopted here.

A rare personal portrait of Soichi Kakeya | Source: Image collection of the Graduate School of Mathematical Sciences, University of Tokyo

A circle with a radius of 0.5 is the easiest shape to think of that satisfies the conditions, but it is obviously not the shape with the smallest area among all shapes that satisfy the conditions.

Kakeya and his colleagues and others initially speculated that an equilateral triangle with a height of 1 would be the convex shape with the minimum area that satisfies the condition in the question. Julius Pál, a very talented and ambitious Hungarian mathematician who moved from Pozsony, Hungary (now Bratislava, the capital of Slovakia), published a proof in 1921 that an equilateral triangle with a height of 1 is the convex shape with the minimum area that satisfies Kakeya's condition.

An equilateral triangle is the smallest convex shape that satisfies Kakeya's condition. | Image source: Chapter 13 of "Incredible: Counterintuitive Puzzles and Their Amazing Solutions"

The man who funded Bohr’s move to Copenhagen was mathematician Harald Bohr, brother of physicist and quantum theory pioneer Niels Bohr. It was most likely this mathematician who introduced Bohr to Kakeya’s spinning needle puzzle.

On the other hand, Kakeya and early researchers guessed that for non-convex figures, the answer pointed to the tricuspid line, a special member of the hypocycloid family. But it wasn't long before people realized that there were even smaller figures.

Tricuspid line丨Source: Kakeya set - Wikipedia

Also in 1917, a seemingly different problem was solved by Abram Besicovitch, a mathematician from Russia who, coincidentally, had also gone to Denmark to seek a research position, and whose main sponsor was also Harald Bohr.

It was several more years before Besicovitch heard of Kakeya's puzzle, "with its fascinating intuitive formulation," and provided a completely unusual and unexpected solution.

In 1917, Besicovitch was thinking about the following Riemann integral problem:

If there is a Riemann integrable function f on a plane, then does there necessarily exist a rectangular coordinate system (x, y) such that for every fixed y, f(x, y) is Riemann integrable as a function of x, and the quadratic integral of f is equal to the double integral ∫∫f(x, y)dxdy?

In order to answer the above question, Besicovitch constructed a set: a figure that contains unit line segments pointing in all directions, but whose area (strictly speaking, Lebesgue measure) is 0.

This set is named Besicovitch set. Because of the existence of this set of line segments, the answer to the above question is no.

From the perspective of later generations, we can see that this problem is essentially an exploration and excavation of the important theorem in real analysis, the Fubini theorem. However, the charm of mathematics often lies in its unexpectedness. By applying the so-called "Bauer connection" (the Bauer mentioned above) on the Besicovitch set, it can be proved that the answer to the Kakeya needle problem is: there is no figure with the smallest area, because the area swept by the needle can be arbitrarily close to 0.

In this way, the integration problems in analysis and the geometric problems on the plane establish a wonderful connection, and the Besicovitch set is therefore called the Kakeya set.

It is also worth mentioning that popular science articles and some publications on the Internet make a mistake. Many people mistakenly believe that the existence of a Besicovitch set with an area of ​​0 can directly lead to the conclusion that "the minimum area swept by a needle can be 0". However, the figure obtained by placing needles pointing in all directions together and the figure swept by a needle continuously rotating 180° are two different things. In fact, the needle cannot continuously change from one position in the Besicovitch set to another position and make the swept area 0. That is why the "Bauer connection" technique is needed. At the same time, the conclusion is that the area swept by the needle can be arbitrarily small, but it is not 0.

Kakeya conjecture - from area to dimension

In the real numbers, an object can be very close to zero and yet not actually be zero. Somehow, this is the crux of the technique.

--Joshua Zare (University of British Columbia)

There are many ways to construct Besicovitch sets, the most classic of which is a technique called the "Perron tree", which simplifies Besicovitch's original construction and is named after Oskar Perron.

Imagine an equilateral triangle with a height of 1 (remember that this triangle is the smallest convex shape that satisfies the Kakeya problem), split it in half, and then slightly overlap the two right triangles, as shown below. This new figure has a smaller area than the triangle, but in it, each of the two upper corners can find a line segment with a length ≥ 1.

Now start again, divide the triangle into 8 equal parts, stack them up two by two, and then stack them up two by two. This kind of figure is called a Perron tree. If you repeat this step and divide the triangle into 16, 32, ..., 2^n, it is obvious that the area of ​​the entire figure can be smaller and smaller, and it can be proved that as the number of steps increases, the area of ​​the figure approaches 0 infinitely.

The construction process of the Peron tree | Source: Kakeya set - Wikipedia

In this way, Kakeya's needle problem seems to have been completely solved. So this is another very interesting and difficult entertaining math problem. But is that all? Wrong!

After solving the needle problem by applying the Besicovitch set on the plane, mathematicians began to pay attention to the properties of the set itself, especially its fractal dimension.

Since it contains unit vectors in any direction, intuitively, the dimension of this set should not be less than 2. This is the original Kakeya conjecture. In fact, there is more than one definition of dimension for a set of zero measure. In general, the most commonly used ones are Hausdorff dimension and Minkowski dimension. We will only introduce the former here.

Fractal structure is a self-similar geometric figure, that is, no matter how many times you zoom in or out, you can see the same or similar shape. Fractal structure has infinite complexity and details, and often appears in nature, such as snowflakes, leaves, coastlines, etc.
Studies have shown that when the shapes of things repeat themselves, they evoke a sense of relaxation in our brains, which reduces stress and makes us feel relaxed. Human civilization has subconsciously learned to use this feature since its infancy: think about the patterns on your bed sheets, wallpaper, and curtains at home. This is called "fractal fluency." The principle is probably that repeated visual elements can reduce the computing resources used by the brain to process visual information, so it makes people feel relaxed and happy.

The self-repeating structure of the fractal Gosper curve | Source: Fractal curve - Wikipedia

The dimension of a fractal is not an integer. We can define it using the Hausdorff dimension. No matter how complex the figure being measured is, we can always cover it with small circles of radius e (allowing partial overlap) (because we can cover the entire plane with small circles, we can of course cover any figure on the plane).

Demonstration of the approximate calculation method of fractal dimension | Image source: Hausdorff dimension - Wikipedia

We consider the number of small circles N(e) covering the measured figure, and then consider the logarithmic operation [LogN(e)/Log(1/e)]. The limit when e approaches 0 is the Hausdorff dimension of the fractal. It is easy to verify that when the measured figure is regular enough, the Hausdorff dimension is the dimension in the usual sense.

Back to the Kakeya conjecture, after half a century of exploration, Roy Davies successfully proved in 1971 that the Hausdorff dimension and Minkowski dimension of the Besicovitch set on the plane are exactly 2. Since the 1-dimensional case is trivial, mathematicians further conjectured that for any positive integer n,

In n-dimensional Euclidean space, does the set of unit vectors in all directions have both Minkowski dimension and Hausdorff dimension equal to n?

This is the complete statement of the Kakeya conjecture. For the case of 3 dimensions and higher dimensions, it is like an impenetrable fortress, blocking the offensive of generations of mathematicians. Because n-dimensional space itself must contain unit vectors in all directions, the difficulty here is actually whether the dimension of the high-dimensional Besicovitch set (Kakeya set) - a set that occupies a "volume" of 0 but contains unit vectors in all directions - is "large" enough to be equal to the space itself. Although mathematicians including Thomas Wolff, Jean Bourgain (1994 Fields Medal winner), Nets Katz, Terence Tao and others have achieved important phased results in this field, they are still unable to cope with even the simplest special case of n=3.

Turning the infinite into the finite: the true value of the Kakeya conjecture

If you think this is easy, you have misunderstood the problem.

——Bjarne Stroustrup, the father of C++ language

Although the study of the Kakeya conjecture gave birth to the modern branch of mathematics, geometric measure theory, it was not until 1971 that Charles Fefferman (1978 Fields Medal winner) pointed out the connection between the conjecture and the field of modern harmonic analysis in his paper "The Multiplier Problem for the Ball" that the mathematical community generally regarded it as an extremely difficult and interesting problem - that is, it lacked seriousness and connotation, and was studied simply to satisfy scholars' curiosity and for the aesthetic purpose of appreciating the pure beauty of mathematics.

In fact, noting that the integral problem that Besicovitch was thinking about in 1917 is related to Kakeya's needle problem, it would not be surprising if there is a connection between Kakeya's conjecture and the most important topic in modern harmonic analysis.

According to the book "10,000 Scientific Problems: Mathematics", the Kakeya set (Besicovitch set) has profound connections with many branches such as harmonic analysis, number theory, partial differential equations, etc. For example, it plays an important role in the oscillation integral theory of harmonic analysis and the distribution of Dirichlet series, and is closely related to the local smoothness of solutions to wave equations.

In fact, the information about the dimension of the Besicovitch set will determine the life and death of a series of mathematical conjectures. These mathematical conjectures are little known and have no reputation in public opinion. But in the eyes of mathematicians, their importance is no less than the famous Riemann hypothesis. It can be said that the geometry of the Besicovitch set supports a large number of topics in partial differential equations, harmonic analysis and other fields. The most striking of these is that this conjecture is a necessary condition for the validity of the three central conjectures in analysis.

Specifically, in Fourier analysis there are the so-called restriction conjecture and the Bochner-Riesz conjecture, and in a larger field there is the local smoothness conjecture. The relationship between inclusion and difficulty is as follows:

Kakeya conjecture ⊂ restriction conjecture ⊂ Bochner-Riesz conjecture ⊂ local smoothness conjecture

This also means that once the Kakeya conjecture is not true, then the subsequent conjectures will all be false. Modern analysts can rest in tears.

The importance of this set of mathematical conjectures essentially stems from the importance of the Fourier transform. The Fourier transform can express almost any function as a sum of sine waves. It is the most powerful mathematical tool for physicists and engineers, perhaps only comparable to matrix theory; the only thing more important than it is the basic common sense such as the four arithmetic operations of addition, subtraction, multiplication and division.

The Fourier transform is the most important tool for solving differential equations and is the mathematical basis behind quantum mechanics ideas such as the uncertainty principle. It is also of unparalleled value in practical applications and plays an important role in analyzing and processing signals, making things like modern mobile phones possible.

From the restriction conjecture to the local smoothness conjecture, each restricts the "error" of the Fourier transform to varying degrees, so mathematicians and engineers would certainly prefer to live in a world where these conjectures are true, because in such a world, the "error" caused by the Fourier transform can always be "controlled", at least not worse than expected.

Furthermore, mathematicians were surprised to discover that the techniques used in harmonic analysis for the above conjecture could also be used to prove major results in the seemingly unrelated field of number theory (which could facilitate the proof of the Riemann hypothesis).

As time goes by, the Kakeya conjecture has gained more attention after it was linked to the central topic of analysis. Unfortunately, it is too difficult. Just to talk about the special case when n=3, until 1995, Thomas Wolfe could only prove that the Hausdorff and Minkowski dimensions of the Besicovitch sets in 3D space must be at least 2.5. This lower limit is difficult to improve. It was not until 1999 that Terence Tao and his collaborators broke through the Minkowski dimension and obtained a new lower bound: 2.500000001. Although it was only an improvement of 0.000000001, it was an achievement from scratch. Therefore, their paper was published in Annals of Mathematics, one of the four top journals in the field of mathematics.

After this paper by Terence Tao et al., another breakthrough was made by the two scholars mentioned at the beginning of this article - Wang Hong and Zaer - who in 2022 followed the framework of Terence Tao et al. and creatively introduced projection theory, and finally proved the Kakeya conjecture in 3D space on a special type of Besicovitch set! They are therefore considered to be the people who have the deepest understanding of Besicovitch sets.

The person who understands most deeply

(Choosing a research direction) It depends on your interest. If you are interested, then study. If you are not interested, then there is no need to study...

—Wang Hong, Courant Institute of Mathematics at New York University

In July 2023, Wang Hong gave an academic report at her alma mater | Source: Beijing International Center for Mathematical Research

Wang Hong, 34, was successfully admitted to the Department of Earth and Space Sciences at Peking University with a score of 653 in the college entrance examination at the age of 16. Later, due to her strong interest in mathematics, she transferred to the Department of Mathematics.

While studying for her doctorate at MIT, she studied under the famous mathematician Larry Guth (Larry Guth, like the aforementioned Tao, is a top authority in geometric measure theory and analysis) and began to delve into the field of harmonic analysis. Since July 2023, she has been an associate professor at the Courant Institute of Mathematics at New York University.

Zare is a student of Terence Tao, and he received his doctorate in 2013. Both of them have done extremely important work on various major issues. Their academic achievements have been highly recognized by the international mathematics community.

Joshua Zahl | Image source: personal.math.ubc.ca/~jzahl/

In 2020, Wang Hong, Gus and Ou Yumeng collaborated to promote Falconer's distance set problem (Inventions); in 2023, Wang Hong and Kevin Ren completely solved the Furstenberg set conjecture. Around 2020, scholars working on fractal geometry generally believed that the Furstenberg conjecture was difficult to solve or that there were currently no tools to solve it. Today, Wang Hong and Zare announced the solution to the 3D Kakeya problem; people can't help but look forward to the day when Falconer's distance set problem is completely solved! So much so that in the first half of last year, there was a call to nominate Wang Hong as a candidate for the Fields Medal.

The Furstenberg set conjecture is a famous mathematical problem in the field of fractal geometry. Specifically, this conjecture is related to the Hausdorff dimension, which involves the complexity and size of a set. The conjecture proposes that for a specific geometric structure on a set (such as a set composed of straight lines in a plane), its Hausdorff dimension must satisfy certain conditions. It can also be seen from the description that it is very similar to the Kakeya conjecture. Specifically, if the projections of a set in different directions have a certain structure, then the overall dimension of the set should have a lower limit. This conjecture has been widely studied in the mathematical community because it not only involves set theory and geometry, but also is related to fields such as dynamical systems and number theory.

After Wang Hong published his paper, the prediction market platform believes that he is very likely to win the Fields Medal, the highest award in mathematics next year.

Wang Hong and Zare's latest paper is even more voluminous (127 pages). The authors call their 2022 (Sticky Kakeya sets and sticky Kakeya conjecture) and 2024 (The Assouad dimension of Kakeya sets in R³) papers a trilogy together with this one. In the introduction, the authors write that their proof is based on and implements the idea of ​​solving the Kakeya conjecture in the "Katz-Tao program" framework.

Since analytical techniques cannot handle unit vectors pointing in all directions, the application of harmonic analysis tools requires discretization of the problem first. The basic idea is to give unit vectors a "width" - no longer treat them as ideal geometric line segments without width, but as real solid needles. At the same time, they calculated the volume of the Besicovitch set after the vector elements of the set were given width. After comparing the numerical relationship between the width and the solid needle, a contradiction can be obtained by using the method of contradiction, and then the conjecture is established.

In the actual proof process, there are inevitably some exceptions, whose structures are very "ugly" and require a little bit of analysis. As can be seen from the length of the article, this process is not easy.

If Wang Hong and Zare's paper eventually passes peer review, their results can be said to be epoch-making. Many scholars believe that Wang Hong may become the first Chinese mathematician to win the Fields Medal, the highest honor in mathematics. Considering that the paper needs at least a year to be reviewed, I think it is very likely that she will win the award in 2030 (when she is still under the 40-year-old age limit for the award).

In the near future, Chinese mathematicians will rise rapidly with their unique innovative thinking and rigorous research methods, and gradually occupy an important position in the global mathematical landscape. At the same time, more and more talented female mathematicians who are brave enough to break through traditional limitations have emerged. They, like bright stars, use their extraordinary wisdom to inject new energy into the development of world mathematics.

Acknowledgements: We would like to thank Professor Jiu Ding of the University of Southern Mississippi and Professor Yi Ni of California Institute of Technology for their comments on this paper.

References

[1] Hong Wang, Joshua Zahl, The Assouad dimension of Kakeya sets in R³, arXiv:2401.12337

[2] Hong Wang, Shukun Wu, Restriction estimates using decoupling theorems and two-ends Furstenberg inequalities, arXiv:2411.08871

[3] Terry Tao, The three-dimensional Kakeya conjecture, after Wang and Zahl, What's new

[4] Kevin Ren, Hong Wang, Furstenberg sets estimate in the plane, arXiv:2308.08819

[5] Dr. Touno does not know PDE, Polynomial Methods in Combinatorics: Kakeya Conjecture on Finite Fields, Zhihu

[6] Skipping grades twice at the age of 5 - The secret of a 16-year-old girl getting into Peking University, Sina.com

[7] Fefferman, Charles. "The multiplier problem for the ball." Annals of mathematics 94.2 (1971): 330-336.

[8] Jordana Cepelewicz, A Tower of Conjectures That Rests Upon a Needle, Quanta Magazine

[9] Jordana Cepelewicz, New Proof Threads the Needle on a Sticky Geometry Problem, Quanta Magazine

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