The additive and multiplicative formulations of the Deligne--Simpson problem ask, respectively, if a collection of complex matrices with prescribed conjugacy classes has sum zero or product the identity. Both versions may be restated as a single question about the existence of a connection with regular singularities on the projective line. We approach this question by generalizing Crawley-Boevey's moment map construction for quivers and by using a technical property coming from Beilinson and Drinfeld's work on the geometric Langlands correspondence.