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