Knots in dynamical systems with applications to electromagnetism and quantum systems

Mathematical knot theory plays an increasingly important role in biology, chemistry and physics. In this project we aim to study the time evolution of knots in different dynamical systems. We are particularly interested in differential equations that are motivated by electromagnetism and quantum mechanics.

For such differential equations we aim to prove the existence of solutions that contain knots, which evolve as desired, and explicitly construct such solutions.

In the case of electromagnetic fields this refers to vector fields, representing the electric and magnetic part of such a field, that satisfy Maxwell's equations and have closed flow lines in the shape of a given knot for all time.

In particular, we want the knot type of this closed flow line to be stable, i.e., it is not allowed to change over time.In the case of quantum wavefunctions we are concerned with complex-valued functions that satisfy linear or non-linear Schrödinger equations and whose nodal set is knotted at a moment in time.

We plan to develop a construction of such functions for which the time evolution of such a quantum vortex knot is determined by a prescribed surface, embedded in 4-dimensional space representing space and time.We also study relations between topological properties of knots and the corresponding functions.

For example, we investigate the connection between the fibration property of a knot K and the non-vanishing of a magnetic field induced by an electric current through a knotted wire in a shape that is isotopic to K.These mathematical problems are approached with techniques from differential geometry, low-dimensional topology and the theory of differential equations.

The proposal also discusses the two way transfer of knowledge between the host institute and the candidate.