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In this paper we prove global well - posedness and scattering for thefocusing, energy - critical nonlinear Schr\"odinger initial value problem infour dimensions. Previous work proved this in five dimensions and higher usingthe double Duhamel trick. In this paper, using long time Strichartz estimateswe are able to overcome the logarithmic blowup in four dimensions.

Global well - posedness and scattering for the focusing, energy -  critical nonlinear Schr\"odinger problem in dimension $d = 4$ for initial  data below a ground state threshold