[2026 Fields Medal Candidates] ④ Yu Deng uses mathematics to explain gas molecules’ motion and resolves a 125-year problem

Yu Deng, a professor at the University of Chicago, is regarded as a Fields Medal candidate for his core achievements on Hilbert’s Sixth Problem. Courtesy of the University of Chicago

[Editor’s Note] From July 23 to 30, 2026, the International Congress of Mathematicians (ICM) will be held in Philadelphia, USA. At the opening ceremony, the winners of the highest honor for young mathematicians under the age of 40, the Fields Medal, will be announced. In 2022, June Huh, a professor at Princeton University, became the first Korean-descended scholar to receive the Fields Medal, drawing nationwide attention. Dong-A Science introduces mathematicians who are widely considered strong contenders for this year’s awards at the ICM, convening four years after Huh’s victory.

In 2024 and 2025, there was a line of research that left mathematical physicists thrilled and unable to hide their excitement. The main protagonist of this work is Yu Deng (37), a professor at the University of Chicago, who is now regarded as a leading candidate for this year’s Fields Medal. Deng produced a key result related to the sixth of the 23 famous problems presented by David Hilbert at the International Congress of Mathematicians in 1900. His work is considered to have provided an important answer to a 125-year-old challenge of formulating the fundamental laws of physics in a mathematically rigorous way.

In August 2024, Deng, together with Zaher Hani, a professor at the University of Michigan, and Ma Xiaoq, also a professor at the University of Michigan, posted their findings on the preprint server arXiv. They proved that when numerous particles move and collide according to Newtonian mechanics, the collective motion of the system can be described over long times by the Boltzmann equation. Their study focused on the case in which particles are dispersed throughout infinite space.

In March 2025, they extended their results to a more realistic gas model in which particles repeatedly collide inside a box, and released this work on arXiv as well. The mathematical community has hailed these two papers as a major breakthrough on the core of Hilbert’s Sixth Problem.

Rha Junhyun, a professor at the Korea Institute for Advanced Study (KIAS) in the School of Mathematics, said, “They made a central contribution in proving the long-time validity of the Boltzmann equation, which had remained one of the hard parts of Hilbert’s Sixth Problem,” adding, “The mathematical community is extremely excited and delighted by the fact that they extended proofs, which had been possible only for short times over several decades, to long times.” 

● The task of casting physics in the language of mathematics

To understand Deng’s work, one must first look at Hilbert’s Sixth Problem. Hilbert’s Sixth Problem asks whether physics can be built on a rigorous framework like mathematics. In particular, it poses the task of mathematically deriving how the motion of individual molecules in a gas gives rise to the motion of the gas as a whole.

Air and other gases are made up of countless molecules. In the microscopic world, each molecule moves according to Newton’s laws of motion. In principle, if we knew the position and velocity of every molecule, we could compute the motion of the entire gas.

In reality, however, this is nearly impossible. A real gas contains an uncountable number of molecules that are constantly colliding with one another. If one tried to track each molecule individually all the way through, the number of possible configurations would explode.

Physicists therefore use different equations to describe gases. The motion of individual molecules is described by Newtonian mechanics, while the overall distribution of particles is described by the Boltzmann equation. When treating the gas as a continuous medium, they use fluid equations such as the Navier–Stokes equations. These describe how quantities like density, velocity, and pressure of the gas change over time.

The task for mathematicians was to rigorously derive the sequence of models: starting from Newtonian mechanics, passing through the Boltzmann equation, and arriving at the Navier–Stokes equations.

In 1975, American mathematician Oscar Lanford was the first to successfully derive the Boltzmann equation from Newtonian mechanics. However, his result held only for very short times. As time progressed, particles underwent repeated collisions and their influences became entangled, making it impossible with existing methods to prove validity over long times.

The main obstacle was recollisions. For the Boltzmann equation to hold, most particles must move essentially independently. Over long times, however, the likelihood that the same particles meet and collide again and again becomes large. One must, in principle, track which particles collide with whom and when, and how the impact of one collision propagates to others. Mathematicians were unable to solve this problem for decades.

Deng, Hani, and Ma analyzed these complex collisions in a new way. They first systematically classified all possible collision patterns. They then discarded cases in which there were excessively many recollisions and computed the remaining cases individually. By finely estimating the probability that each collision scenario actually occurs, they proved that the Boltzmann equation remains valid even over long times.

In this proof, a “molecular” methodology that Deng and Hani had developed in their earlier work on wave equations played a decisive role. They newly applied and extended a technique for decomposing complex interactions into manageable units to the particle collision problem.

The team first resolved the case in which particles are dispersed in infinite space. They then extended their results to a realistic gas model, in which particles repeatedly collide inside a box. Through this process, they succeeded in deriving the Boltzmann equation from Newtonian mechanics over long times.

By connecting their work with existing results that derive the Navier–Stokes equations from the Boltzmann equation, they opened a path to explaining the crucial bridge from the motion of individual particles to the macroscopic flow of a gas.

Professor Rha said, “Hilbert’s Sixth Problem is part of a larger program that asks how rigorously we can describe the physical world using mathematics,” adding, “Depending on factors such as the scale of the phenomena we want to observe in the physical world, we use various models from Newtonian mechanics to continuum mechanics, and the central issue is whether we can rigorously justify the process of reducing complexity.”

He continued, “The work of these three mathematicians is an important example showing that one can rigorously derive relationships between different mathematical models that describe the physical world at different scales.”

The reason the mathematical community is paying close attention to Deng’s prospects for the Fields Medal is not only because he resolved a long-standing problem. Alongside his proof of the long-time validity of the Boltzmann equation, he also introduced a new methodology for analyzing complex interactions.

Regarding the possibility that Deng will receive the Fields Medal, Professor Rha said, “I think the chances are quite high.” He added, “He is highly regarded not only for solving a major open problem but also for presenting new methods that are likely to be applied to many future problems.”

● A quiet mathematician who loves Go and poetry

Deng grew up in Shenzhen, China. From a young age, he stood out in mathematics competitions. In 2006, he represented China at the International Mathematical Olympiad (IMO) and won a gold medal. He entered Peking University and later transferred to the Massachusetts Institute of Technology (MIT) in the United States, where he became a Putnam Fellow with the top distinction in the 2010 William Lowell Putnam Mathematical Competition, a premier contest for undergraduate students in North America. He earned his PhD from Princeton University in 2015. After positions at the Courant Institute of Mathematical Sciences at New York University and the University of Southern California (USC), he joined the University of Chicago.

Deng studies partial differential equations that describe natural phenomena through equations. This branch of mathematics expresses processes such as the spreading of waves, the flow of air, and the motion of gases in terms of changes over time and space. In a 2022 USC interview, Deng described the appeal of partial differential equations by saying, “The equations are very abstract, but even if we don’t find explicit formulas, we can often learn a lot about the solutions,” and added, “What fascinates me most is that we can explore scientific questions that are hard to approach directly through mathematics.”

Researchers remember Deng as a quiet yet tenacious mathematician. Professor Rha said, “My impression was that Professor Deng is a very humble person,” and continued, “It’s hard to believe that someone so calm is the author of the most important result of our time; in his talks, he focuses on conveying the mathematical ideas rather than on self-promotion.”

Outside of mathematics, Deng enjoys the board game Go. As a child, he trained so hard that he once dreamed of becoming a professional Go player, juggling schoolwork and Go training. He later decided to devote himself to mathematics, but he still enjoys soccer, comics, and poetry. He is known in particular for his love of the works of the Tang dynasty poet Li Shangyin.

Recently, Deng has solidified his status as a Fields Medal contender by receiving a series of major international awards in mathematics. He has received the Gold Medal of the International Congress of Chinese Mathematicians (ICCM), a Clay Mathematics Institute Research Award supporting young mathematicians, and the Leonard Eisenbud Prize for Mathematics and Physics from the American Mathematical Society. He has also been selected as an invited speaker at this year’s International Congress of Mathematicians (ICM).

There are still some variables in the Fields Medal race. The results were made public only recently, and the papers are still undergoing verification and journal publication. Even so, many in the mathematical community list Deng as one of the strongest contenders for this year’s Fields Medal. 

<Profile of Yu Deng>
Nationality China
Year of birth 1989 (37 years old)
Research fields Partial differential equations, fluid mechanics, harmonic analysis, statistical physics
Degrees BSc, Massachusetts Institute of Technology (MIT); PhD, Princeton University

Awards Sloan Research Fellowship (2021); ICBS Frontiers of Science Award (2024); MCA Prize (2025); ICCM Gold Medal (2025); Antonio Ambrosetti Medal (2025); Oberwolfach Prize (2025); Leonard Eisenbud Prize for Mathematics and Physics (2026); Clay Research Award (2026)
Lectures Invited speaker, International Congress of Mathematicians (2026)

<Reference>
arxiv.org/abs/2408.07818
arxiv.org/abs/2503.01800