Joint Princeton-Rutgers Seminar on Geometric PDEsReal-valued Sobolev functions are well-understood and play an immense role. By contrast, the theory of Sobolev maps with values into the unit circle is not yet sufficiently developed. Such maps occur in a number of physical problems. The reason one is interested in Sobolev maps, rather than smooth maps is to allow maps with point singularities, such as $x/|x|$ in 2-d, or line singularities in 3-d which appear in physical problems. It turns out that these classes of maps have a rich structure. Geometrical and topological effects are already conspicuous, even in this very simple framework. On the other hand, the fact that the target space is the circle (as opposed to higher-dimensional manifolds) offers the option to study their lifting and raises some tough questions in Analysis.