This video claims to list the 3 biggest breakthroughs in math this year:

https://www.youtube.com/watch?v=hRpcWpAeWng

But the first paper discussed - supposedly "one of the most significant advances in mathematical physics in decades" - looks problematic to me.

Hilbert posed the problem of rigorously deriving equations of fluid flow from the classical mechanics of elastic hard spheres. This paper claims to solve that problem:

• Yu Deng, Zaher Hani, and Xiao Ma, "Hilbert's sixth problem: derivation of fluid equations via Boltzmann's kinetic theory": https://arxiv.org/abs/2503.01800

They claim to derive a version of the Navier-Stokes equations starting from a system of hard spheres undergoing elastic collisions, going through the Boltzmann equation as an intermediate step.

But this paper raises objections that look valid to me:

• Shan Gao, "Comment on 'Hilbert's Sixth Problem: Derivation of Fluid Equations via Boltzmann's Kinetic Theory' by Deng, Hani, and Ma": https://arxiv.org/abs/2504.06297

Gao's main objection is that the paper takes a limit that forces the fraction of volume occupied by spheres to approach zero, so the system they're studying is really an infinitely dilute gas. Gao argues that they're studying a "rescaled gas model" rather than a realistic fluid. Furthermore, it's been known for a long time that the "molecular chaos assumption" underlying Boltzmann's equation - the assumption that the motion of molecules is random and *uncorrelated* - breaks down in an actual fluid, though Deng, Hani and Ma claim to have proved it in the limit of an infinitely dilute gas.

So, even if the paper is mathematically valid, it hasn't solved Hilbert's problem.

It's still a step toward a solution.