Lifting in $W {1, p}$.- The Geometry of $J(u)$ and $\Sigma(u)$ in 2D; Point Singularities and Minimal Connections.- The Geometry of $J(u)$ and $\Sigma(u)$ in 3D (and higher); Line Singularities and Minimal Surfaces.- A Digression: Sphere-Valued Maps.- Lifting in Fractional Sobolev Spaces and in $VMO$.- Uniqueness of Lifting and Beyond.- Factorization.- Applications of the Factorization.- Estimates of Phases: Positive and Negative Results.- Density.-...