Cracking a 125-Year-Old Math Problem

A breakthrough in Hilbert's sixth problem is a major step in grounding physics in math

- JACK MURTAGH

WHEN THE GREATEST MATHEMATICIAN alive unveils a vision for the next century of research, the math world takes note. That's exactly what happened in 1900 at the International Congress of Mathematicians at Sorbonne University in Paris. Legendary mathematician David Hilbert presented 10 unsolved problems as ambitious guideposts for the 20th century. He later expanded his list to include 23 problems, and their influence on mathematical thought over the past 125 years cannot be overstated.

Hilbert's sixth problem was one of the loftiest. He called for “axiomatizing” physics, or determining the bare minimum of mathematical assumptions behind all its theories. Broadly construed, it's not clear whether mathematical physicists could ever be sure they had fully met this challenge. Hilbert mentioned some specific subgoals, however, and researchers have since refined his vision into concrete steps toward its solution.

In March mathematicians Yu Deng of the University of Chicago and Zaher Hani and Xiao Ma, both at the University of Michigan, posted a paper to the preprint server arXiv.org in which they claim to have achieved one of these goals. If their work withstands scrutiny, it will mark a major stride toward grounding physics in math and may open the door to analogous breakthroughs in other areas of physics.

In the paper, the researchers suggest they have figured out how to unify three physical theories that explain the motion of fluids. These theories govern a range of engineering applications from aircraft design to weather prediction— but until now, they rested on assumptions that hadn’t been rigorously proven. This breakthrough won’t change the theories themselves, but it mathematically justifies them and strengthens our confidence that the equations work the way we think they do.