The speaker will show how to use the steepest descent method to obtain precise asymptotics for solutions of the Cauchy problem for the defocusing NLS equation on the line. Special emphasis is given to the case where the initial data q0 has minimal smoothness and decay. If the initial data is smooth and has rapid decay, absolute estimates suffice, but now one must use cancellations from oscillations. Precise asymptotics for solutions of the Cauchy problem in the case that q0 has minimal smoothness and decay, are needed,, in particular, in the solution of the perturbation problem for NLS. This is joint work with Xin Zhou.