In 1900, mathematician David Hilbert presented a list of mathematical problems that captured both the current state and future trajectory of mathematics. Now, 125 years later, Dr. Zahel Hani and his colleagues at the University of Michigan have successfully solved one of Hilbert’s enduring puzzles, significantly unifying various physical laws in the process.

Hilbert advocated for deriving all physical laws from mathematical axioms—assertions regarded as fundamental truths by mathematicians. His sixth problem sought to derive laws governing fluid behavior from such axioms.

Until 2025, physicists characterized fluids through three distinct paradigms based on scale: the microscopic scale of individual particles, the mesoscopic world of particle clusters, and the macroscopic scope of full-fledged fluids, such as water flowing in pipes. Despite advances in linking these scales, a seamless unification remained elusive until Hani and his team devised a solution.

The researchers’ breakthrough hinged on adapting diagram-based techniques pioneered by physicist Richard Feynman for the seemingly unrelated field of quantum field theory. This endeavor culminated in a published paper reflecting a five-year research initiative.

“We received validation of our results from numerous experts in the field,” Hani asserts. The study, currently available as a preprint, will soon appear in a highly regarded mathematics journal.

The findings represent not only a monumental achievement in mathematics but also offer the potential to enhance our understanding of complex fluid dynamics in natural systems, such as the Earth’s atmosphere and oceans. Hani notes they are also exploring a quantum variant of this issue, where microscale mathematics can reveal even more complex and intriguing particle behaviors.