Abstract: The binormal flow, which models the evolution of vortex filaments, and the cubic nonlinear Schrödinger equation are deeply connected through the Hasimoto transform, linking vortex filament dynamics with dispersive partial differential equations. In this talk, I will give an overview of this connection and discuss the formation of corners in binormal flow, as well as the role of Dirac masses in the nonlinear Schrödinger equation. I will present several results on corner formation in vortex filaments and singular NLS solutions, illustrating some of the main ideas used to study these problems.