Geometry and Topology of Finite-Gap Vortex Filaments

Abstract

In this talk, I will discuss the vortex filament flow, and its connection with the nonlinear Schrödinger equation (NLS), the geometry of solutions to the filament flow that correspond to finite-gap solutions of NLS, and in particular, the relationship between geometric properties of the filament to features of the Floquet spectrum of a periodic NLS potential. These will be illustrated by the example of elastic rod centerlines. I will also discuss the geometric implications of symmetric Floquet spectra, and the topological implications of certain isoperiodic deformations of the spectrum that open up real double points.