The focus of this workshop is the development and structure of singular structures in solutions to nonlinear partial differential equations. These structures arise from many contexts: vortices in superconductors, Bose-Einstein Condensates or liquid crystals, domain walls in micromagnets or copolymers, phase boundaries in materials, and patterns in chemotaxis, for example. These problems are especially appealing since they lie at a meeting place between analysis and geometry. The mathematics of singular solutions is challenging and requires a diverse collection of analytical and geometrical tools, including variational methods, geometric measure theory, implicit function theorems, and PDE regularity theory. In this meeting we bring together leading mathematicians who bring different perspectives and methods to bear on many of these questions in order to see the power of these new mathematical ideas, with the expectation that this will create a wealth of new directions, problems, methods and solutions.