In 2025, a team led by Zaher Hani at the University of Michigan solved one of David Hilbert's longstanding problems, seamlessly linking the mathematical descriptions of fluids across different scales. This breakthrough connects microscopic particle behavior to macroscopic flows like water in a sink. The achievement draws on techniques from quantum field theory and promises insights into atmospheric and oceanic dynamics.
David Hilbert, a prominent mathematician, outlined 23 key problems in 1900 that shaped the field's future. Among them, the sixth challenged researchers to derive the laws governing fluid behavior from fundamental mathematical axioms. For over a century, physicists relied on three separate frameworks: one for the microscopic scale of individual particles, another for the mesoscopic realm of particle collections, and a third for macroscopic fluids such as flowing water.
Zaher Hani and his colleagues at the University of Michigan finally bridged these descriptions in 2025, 125 years after Hilbert's list. Their solution unifies the equations that dictate particle motion in fluids, creating a cohesive mathematical foundation. The team adapted a diagram-based method originally developed by physicist Richard Feynman for quantum field theory, applying it to fluid dynamics.
This effort capped a five-year project, with key papers released earlier in 2025. "We heard from a lot of people regarding the result, especially from the leaders in the field who have checked the work very carefully," Hani noted. The preprint is now slated for publication in a leading mathematics journal.
Beyond its mathematical significance, the work could enhance models of complex fluids in the atmosphere and oceans. Hani's team is extending the approach to quantum fluids, where particle behaviors grow even more intricate.