Determinantal polynomials, vortices and random matrices: an exercise in experimental mathematical physics.
Inspired by the interpretation of eigenvalues of random matrices as a gas of interacting particles we develop a toy model of vortex or particle dynamics using determinantal polynomials of random matrices. By introducing quaternionic structures we generate vortex/anti-vortex systems, and from studying the phase surface of the associated wavefunction, we identify topological rules governing the creation and annihilation of the vortices. We also discuss an interpretation of annihilation events in terms of quaternionic states. This exploratory project is very much in the vein of experimental mathematics, and so contains many avenues for further investigation.
Dr Anthony Mays, The University of Melbourne