On the vortex filament in 3-spaces and its generalizations

In this article,we devote to a mathematical survey on the theory of the vortex filament in 3-dimensional spaces and its generalizations.We shall present some effective geometric tools applied in the study,such as the Schrodinger flow,the geometric Korteweg-de Vries(KdV)flow and the generalized bi-Schrodinger flow,as well as the complex and para-complex structures.It should be mentioned that the investigation in the imaginary part of the octonions looks very fascinating,since it relates to almost complex structures and the G2 structure on S^(6).As a new result in this survey,we describe the equation of generalized bi-Schr?dinger flows from R1 into a Riemannian surface.