The Boltzmann equation has been a cornerstone of statistical physics for about 140 years, but because of the extremely singular nature of the Boltzmann collision operator, the tools necessary for rigorous study of this equation (without relying on the so-called "Grad cutoff" assumption) have only recently emerged. This central equation provides a basic example where a wide range of geometric fractional derivatives occur in a physical model of the natural world. 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 we prove that they decay rapidly in time to equilibrium as predicted by celebrated the Boltzmann H-theorem. This is joint work with P. Gressman An overview of our proof can be found at: http://www.pnas.org/content/107/13/5744 The full proof is posted on the arXiv at: http://arxiv.org/abs/0912.0888 http://arxiv.org/abs/1002.3639