This monograph contains a study of the global Cauchy problem for the Yang-Mills equations on (6+1) and higher dimensional Minkowski space, when the initial data sets are small in the critical gauge covariant Sobolev space. (H) over dot(A)((n-4)/2). Regularity is obtained through a certain "microlocal geometric renormalization" of the equations which is implemented via a family of approximate null Cronstrom gauge transformations. The argument is then reduced to controlling some degenerate elliptic equations in high index and non-isotropic L-p spaces, and also proving some bilinear estimates in specially constructed square-function spaces.