Adjoints of ideals in regular local rings
  The adjoint of an ideal I in a regular local ring R is the R-ideal adj(I):=H^0(Y, I\omega_Y), where f:Y -> Spec(R) is a proper birational map with Y nonsingular and IO_Y invertible, and \omega_f is a canonical relative dualizing sheaf. (Such an f is supposed to exist.) The basic conjecture is that I.adj(I^n)=adj(I^{n+1}) whenever n is >= the analytic spread of I. This is a strong version of the Briancon-Skoda theorem, implying all other known versions. It follows from the (conjectural) vanishing of H^i(Y,\omega_Y) for all i>0. When R is essentially of finite type over a char. 0 field, that vanishing has been deduced by Cutkosky from Kodaira vanishing. It also holds whenever dim.R = 2; and in this case we can say considerably more about adjoints, tying in e.g., with classical material on adjoint curves and conductors. 
Joseph Lipman