In this talk we explain our recent proof of global stability for the Boltzmann equation (1872) with the physically important collision kernels derived by Maxwell 1867 for the full range of inverse power intermolecular potentials, $r^{-(p-1)}$ with $p > 2$ and more generally. Our solutions are perturbations of the Maxwellian equilibrium states, and they decay rapidly in time to equilibrium as predicted by Boltzmann's celebrated H-Theorem.This equation provides a basic example where a wide range of geometric fractional derivatives occur in a physical model of the natural world. We are able to characterize these non-isotropic fractional differentiation effects precisely using in part the "geometric Littlewood-Paley'' theory of Stein and Klainerman-Rodnianski. This is joint work with P. Gressman.