Solution to a 125-Year-Old Math Problem: A Historic Convergence of Physics and Mathematics

Researchers have unified the explanation of three levels of fluid dynamics into a single framework, fulfilling a chapter of Hilbert’s dream

Once there was a dream—to explain the world’s complexity in a single language. A formula that would bring everything, from the atoms and molecules of physics to the vastness of the cosmos, within the bounds of logic.
In 1900, at the International Congress of Mathematicians in Paris, the legendary mathematician David Hilbert outlined this dream. He created a list of 23 unsolved problems—the sixth of which was the most ambitious: to establish the mathematical foundations of the fundamental theories of physics.

It was like a vast ocean, where for centuries scientists tried to understand the flow and behavior of fluids at three levels: microscopic, mesoscopic, and macroscopic. But at each level, uncertainty persisted—there was no reliable proof of how well the explanation at one level could be logically translated to another.

Finally, a solution!
In March 2025, mathematician Yu Deng of the University of Chicago, together with Zhihua Han and Xiao Ma of the University of Michigan, published a research paper on arXiv.org. They claimed that the three major theories of fluid dynamics—Newton’s particle-based model (microscopic), the Boltzmann equation (mesoscopic), and the Euler and Navier-Stokes equations (macroscopic)—can be unified. This is not only a solution to a mathematical problem; it stands as a historic milestone in establishing the foundations of physics.

Three perspectives, one reality
These three theories analyze fluid motion on different scales. At the microscopic level, the motion of each particle is calculated separately using Newton’s laws. At the mesoscopic level, the Boltzmann equation analyzes the average behavior of hundreds of particles. On the macroscopic scale, fluids are treated as continuous matter, with the Euler and Navier-Stokes equations being applied. These equations are used in everything from modern airplane design to weather forecasting.

Although these theories explain the same phenomena in the real world, the mathematical bridges between them had long been missing. Hilbert’s dream was for these layers to transform from one to another in a logically consistent way.

Building the mathematical bridge: Deng, Han, and Ma
The researchers first demonstrated the process by which the microscopic level transitions to the mesoscopic: how the behavior of an infinite number of particles is expressed in terms of averages in the Boltzmann equation. Then they made the transition from the mesoscopic to the macroscopic, where this average behavior is captured by the large-scale fluid dynamic equations.

One of their key contributions is mathematical analysis over long time periods. Previous research could only validate these transitions for short durations. But real fluids are not momentary; each flow arises from countless collisions, histories, and paths of particles. By factoring in the effects of this history, the researchers showed that, over time, the impact of collisions remains bounded and does not create inconsistencies with the main equations.

Global reactions
“This is a triumph of pure mathematics and, at the same time, the fulfillment of a dream in physics,” remarked Dr. Lucy Hunter, physicist at the University of Cambridge.
A reader at Bangladesh University of Engineering and Technology, student Sabbir Rahman, said, “This kind of research inspires us to think of new ideas. It’s not just about knowledge; it’s about a change in perspective.”

Final words
This achievement is not merely a mathematical proof—it’s the answer to a philosophical quest that lasted more than a century. Where David Hilbert once dreamed, that dream has now taken shape at the hands of Deng, Han, and Ma.

The question now is—will Hilbert’s remaining problems be solved in the same way? Today, a new chapter is written in the pages of history, one that gives hope that, through the combination of mathematics, physics, and imagination, new kinds of triumphs in knowledge are possible.