Defect dynamics in nonlinear Hamiltonian partial differential equations

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This proposal will investigate mathematical issues related to the behaviour of vortex filaments in fluids. This is a topic that has fascinated scholars dating back at least to the days of Leonardo da Vinci, and over the past 60 years, physicists have realized that vortex-like objects are present not only in everyday fluids such as water and air, but also in a wide range of other physical phenomena, ranging from very small scales (quantum mechanical fluids such Bose-Einstein condensates, superconductors, micromagnetic materials) to extremely large scales (hypothetical cosmic strings). In real fluids, vortex structures have finite thickness, but they can sometimes be described by simpler, 1-dimensional models. For example, under some circumstances, one might be able to construct a reasonable mathematical model of a very long, thin tornado by neglecting its thickness and treating it as a 1-dimensional curve that evolves in time. Such reduced models for vortices in superfluids have been used by physicists for close to 50 years, but a rigorous mathematical justification of the approximation involved in neglecting the vortex thickness has never been supplied. A main goal of this proposal is to supply such a justification. Another goal is to develop a precise mathematical understanding of what happens when vortex filaments collide. Here, too, physicists have carried out a huge amount of research, which suggests that colliding filaments undergo a process called "reconnection", and moreover that this process has important implications for a range of physical phenomena. But from a mathematical point of view, this process is very poorly understood. We aim to build the foundations of the mathematical theory of vortex reconnection. The mathematical techniques that will be developed to carry out this research will have important implications for a range of problems that extends far beyond the specific questions that we address here.