Zaher Hani: Hilbert’s Sixth Problem: From Particles to Waves (October 16, 2025)

Video Description

In his sixth problem, Hilbert called for the derivation of the equations of fluid mechanics—such as the Euler and Navier-Stokes equations—by way of rigorously justifying Boltzmann’s kinetic theory for particle systems. The scope of this program, now known as Hilbert’s program, was precisely framed in the mid-20th century through the works of Grad and Cercignani, who identified the correct limiting process involved: the Boltzmann-Grad limit. In his celebrated work, Lanford (1975) gave the first rigorous derivation of Boltzmann’s equations, albeit only valid for short times. However, Hilbert’s sixth problem requires a long-time extension of Lanford’s result, which remained open for decades. In recent joint work with Yu Deng (University of Chicago) and Xiao Ma (University of Michigan), we extend Lanford’s theorem to long times—specifically for as long as the solution of Boltzmann’s equation exists. This allows for the full execution of Hilbert’s program and the derivation of the fluid equations in the Boltzmann-Grad limit. The underlying strategy follows an earlier joint work with Yu Deng that resolved a parallel problem, in which colliding particles are replaced by nonlinear waves; thus, establishing the mathematical foundations of wave turbulence theory. In this talk, we will review this progress and discuss some future directions. Zaher Hani is the Frederick W. and Lois B. Gehring Professor of Mathematics at the University of Michigan, Ann Arbor. He works in the field of nonlinear PDE and mathematical physics. Before moving to Michigan in 2018, Hani held positions at Georgia Tech (2014-2018), and the Courant Institute of Mathematical Sciences, NYU (2011-2014). He graduated with a Ph.D. from the University of California, Los Angeles in 2011 under the supervision of Terence Tao. For more information go to: https://www.simonsfoundation.org/event/mps-annual-meeting-2025/