  We consider the 3D Boltzmann equation for the Maxwellian particle and soft
potential with an angular cutoff. We prove sharp global well-posedness with
initial data small in the scaling-critical space. The solution also remains in
$L^{1}$ if the initial datum is in $L^{1}$, even at such low regularity. The
key to existence, uniqueness and regularity criteria is the new bilinear
spacetime estimates for the gain term, the proof of which is based on novel
techniques from nonlinear dispersive PDEs including the atomic $U$-$V$ spaces,
multi-linear frequency analysis, dispersive estimates, etc. To our knowledge,
this is the first 3D sharp global result for the Boltzmann equation.
