Published on: 2026-07-27
Source: People’s Republic of China — Translation
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On the morning of July 23 local time in Philadelphia (USA), the International Congress of Mathematicians 2026 opened. At the opening ceremony, the International Mathematical Union officially announced the names of the laureates of the twenty-first Fields Medal. Huáng Wān and Yú Dèng became recipients of the award — this is the first time that Chinese citizens have received this prize.
The Fields Medal is awarded once every four years, and the number of laureates does not exceed four people. The award is given to scientists under the age of 40 at the time of the award for outstanding achievements in mathematics. It is one of the highest honors in the international mathematical community and is often called the “Nobel Prize of Mathematics.”
Van Houn, together with co-author Joshua Zahl, applied a subtle multiscale induction method and proved Kakeya’s three-dimensional conjecture, which had remained unresolved for over a hundred years. Furthermore, in collaboration with Larry Guth, Alex Iosevich, and Ou Yumen, she achieved a major breakthrough in the two-dimensional Falconer distance problem, and together with Ren Kaiyuan, fully solved the Furstenberg set conjecture.
Dan Yu, together with co-authors Zaher Hani and Ma Xiao, performed a rigorous derivation from the system of hard spheres on large time scales to the Boltzmann equation, constructing a derivation chain from Newtonian dynamics of hard spheres to the Boltzmann equation and then to the Navier–Stokes equations, which represents a key progress in solving Hilbert’s sixth problem.
Proof of Kakeya’s three-dimensional hypothesis
During the development of concepts such as fractal dimension, mathematicians formulated the modern version of Kakeya’s conjecture: for any Kakeya set in n-dimensional space, both of its fractal dimensions — the Hausdorff dimension and the Minkowski dimension — must be strictly equal to n.
This latest version asserts that even if the volume (measure) of a Kakeya set can be compressed to an arbitrarily small value, its geometric structure inevitably remains so dense that in terms of fractal dimension it cannot be “fitted” into any space of dimension lower than n.
For this formulation of the hypothesis, the two-dimensional case was proven back in 1971, but the three-dimensional case remained unsolved for a long time and was considered one of the most difficult problems in mathematics.
It was precisely in the three-dimensional case that Van Hoorn and her co-author Joshua Zale made a historic breakthrough. The traditional approach considered intersecting needles as infinitely thin “tubes” and tried to calculate the volume of their intertwinings, which led to hopelessly complex calculations. They proposed a completely new perspective: to break these intricately intertwined tubes at different scales into “finest grains” and proved that, no matter how skillfully you arrange the intersections of the needles, no point in space can be excessively compressed or overlapped. The limitation on overlaps once and for all “locks” the upper bound of the figure’s deformation, which proves Kakei’s three-dimensional hypothesis.
In 2022–2025, Wang Hong and Zhal published a series of three articles, in which they gradually presented the complete proof. The final work, published in 2025, contains 127 pages and caused a sensation in the global mathematical community immediately after appearing online.
The proof of Kakeya’s three-dimensional conjecture became the central work for which Wang Hong was awarded the Fields Medal. The conjecture itself stimulated the development of a branch of mathematics known as geometric measure theory. The theory is closely connected with a number of key problems in harmonic analysis and serves as an important foundation for their study.
Harmonic analysis plays a crucial role in modern technologies such as signal processing and image compression. Behind the high-quality videos and sharp images we see every day are the mathematical tools of harmonic analysis — in particular, Fourier analysis — which provide efficient data compression and recovery.
From Newton to Boltzmann
Wan Hung proved a hypothesis that had remained unsolved for several decades, and the achievements of Deng Yu are also related to one of the old mathematical problems. In order to promote the development of mathematics in the 20th century, David Hilbert formulated 23 key mathematical problems at the International Congress of Mathematicians in 1900, later known as the “Hilbert problems.” Over more than a hundred years, only a small portion of them have been fully solved, while most still remain unresolved.
The main direction of Dan Yu’s research is related to Hilbert’s sixth problem, the central theme of which is the mathematical axiomatization of physics. By the end of the 19th century, humanity already had a deep understanding of classical mechanics, laid down by Newton, and was able to describe collisions of individual microparticles using it. At the same time, statistical physics began to develop, designed to characterize the macroscopic properties manifested in vast assemblies of particles. The physicist Boltzmann derived the equation bearing his name, which describes the evolution of the statistical distribution of a large number of gas molecules, thereby creating a bridge between the micro- and macro-levels.
However, the question soon arose: how, based on classical Newtonian mechanics, can the Boltzmann equation be derived?
Classical mechanical equations can be directly applied to describe the collision of two individual particles. But when the number of particles in a system reaches astronomical magnitudes, mathematicians long struggled to find a consistent and rigorous mathematical foundation that would allow deriving macroscopic laws from microscopic ones.
A more serious problem lies in the fact that at the micro level, Newtonian mechanics equations are time-reversible — the collision of two particles, even if played backward, does not violate the laws of physics. However, in the macro world, there is an arrow of time, and processes are irreversible: for example, after sublimation of dry ice, carbon dioxide spontaneously fills the entire room and never reassembles back into a piece of dry ice.
From this arises the main question of Hilbert’s sixth problem: is it necessary that a microscopic reversible equation converges to a macroscopic irreversible statistical equation as the number of particles tends to infinity?
It was in this direction that Den Yu, together with co-authors – Zaher Hani, Ma Xiao, and others – made a historic breakthrough. As a model, they chose the system of a rarefied gas of hard spheres, representing each gas molecule as a tiny elastic billiard ball that collides and bounces inside the container. Developing entirely new combinatorial and analytical tools, they first, in 2024, in a 164-page article, rigorously proved for the first time over large time scales that at sufficiently low particle density, the statistical behavior of the hard sphere system exactly converges to the evolution described by the Boltzmann equation.
On this basis, in 2025 the same three authors published a second summarizing article, in which they extended the entire chain of reasoning even further: by taking the mesoscopic Boltzmann equation as an intermediate link and performing a macroscopic limiting transition, they rigorously derived the Euler equations for compressible fluids, as well as the Navier–Stokes–Fourier equations for incompressible fluids, thereby fully achieving the objectives of the narrow subsection of Hilbert’s sixth problem.
It was precisely thanks to their work that a breakthrough was made in an important direction—to overcome the long-standing strict mathematical gap between classical particle systems, the Boltzmann equation, and macroscopic physical systems, as well as to propose new tools for studying other complex particle systems. This result is recognized as one of the most significant achievements in 125 years related to Hilbert’s sixth problem.
Milestone in history
Dan Yu and Wang Hong became the first Chinese citizens to be awarded the Fields Medal, with Wang Hong becoming the third woman in the history of this award. Their success is of epochal significance for the development of mathematics in China.
It is noteworthy that Van Hong and Deng Yu enrolled at Peking University in the same year — 2007. Van Hong entered the Faculty of Geophysics and transferred to Mathematics only in her second year; Deng Yu won a gold medal at the International Mathematical Olympiad (IMO) while still in high school and was admitted to the university without entrance exams.
Different starting points, different paths — yet Dan Yu and Wang Hong have risen to the highest peaks in their scientific careers. There is no doubt that many unexplored horizons still remain in the world of mathematics, waiting for their Chinese researchers.
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