In this talk, we establish quantitative bounds for the defocusing
energy-supercritical Nonlinear Schr\"odinger equation (NLS) and use
these to give a universal blow-up criteria which goes below the
scaling invariant threshold. These results are in line with a recent
breakthrough construction of finite-time blow-up solutions, and in
particular give the first generic result distinguishing potential
defocusing blow-up phenomena from many of the known examples of
blow-up in the focusing setting. The argument is based on a delicate
refinement of induction on scales techniques. At the end of the talk,
we will briefly describe applications to related models.