We provide the rigorous derivation of the Boltzmann equation from hard sphere dynamics, for arbitrarily long times that cover the full lifespan of the Boltzmann equation. This extends Lanford's landmark theorem (1975), and is a major step towards solving Hilbert's sixth problem.

The main ingredients of the proof include (1) a time layering argument, (2) a cumulant ansatz that memorizes the full collision history on time [0,t], (3) analysis of an integral constructed from such collision history, and (4) a carefully designed combinatorial algorithm that allows to control such integrals. This is joint work with Zaher Hani and Xiao Ma.