Unlike most of the work that mathematicians do, the research of Yu Deng, 37, a professor of mathematics at the University of Chicago and one of the winners of this year’s Fields Medals, which recognizes mathematicians under the age of 40, directly addresses how the universe works.
The equations of physics generally fall into one of two categories.
Some equations, like Newton’s second law of motion, describe the behavior of individual particles — billiard balls bouncing off one another is one example. But it is impossible to calculate the precise interactions of billions or trillions of particles.
Other equations offer a statistical approximation of particles’ collective behavior.
Back in 1900, David Hilbert, a German mathematician, laid out what he regarded as the biggest unsolved problems in the field. No. 6 was broad and vague: deriving the laws of physics from the simplest axioms.
That is, how can the statistical behavior of the whole be calculated, ground up, from the motion of the individual particles? Dr. Hilbert offered a specific example: gases.
Mathematicians did manage to show how they could, through a progression of logic and calculation, derive the Boltzmann equation, which describes changes in the density of particles, from the basic laws of motion. The derivation was valid for only extremely short periods of time, so it was a start, but insufficient.
Dr. Deng, who grew up in Shenzhen, China, said he first became interested in partial differential equations when he was an undergraduate. He started his studies at Peking University before transferring to the Massachusetts Institute of Technology, where he graduated with a bachelor’s degree in mathematics.
But he put aside partial differential equations, because he did not feel there were enough mathematical tools to make progress with them. He returned to the subject around 2018, he said.
In 2021, Dr. Deng and Zaher Hani, a mathematician at the University of Michigan, came up with a full derivation of the wave kinetic equation, which describes phenomena like turbulence.
He then turned to the Boltzmann equation. “That was really kind of an analogue of what we just did in the last few years on this wave kinetic turbulence theory,” Dr. Deng said. “It was a major open problem.”
Dr. Deng, Dr. Hani and another mathematician, Xiao Ma, figured out how break up the big complex problem into smaller, manageable pieces. In 2024, they posted their solution to deriving the Boltzmann equation from Newton’s laws of motion.
In a follow-up paper last year, they claimed they had solved Hilbert’s sixth problem, at least for the Boltzmann equation.
“I’m very happy,” Dr. Deng said of the Fields Medal, “not just because of myself, but also because the field I’m representing is getting significant recognition.”
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