Daniel Peralta-Salas

Tying Knots in Light Fields

We construct analytically, a new family of null solutions to Maxwells equations in free space whose field lines encode all torus knots and links. The evolution of these null fields, analogous to a compressible flow along the Poynting vector that is shear free, preserves the topology of the knots and links. Our approach combines the construction of null fields with complex polynomials on S3. We examine and illustrate the geometry and evolution of the solutions, making manifest the structure of nested knotted tori filled by the field lines.

BELTRAMI FIELDS WITH A NONCONSTANT PROPORTIONALITY FACTOR ARE RARE

We consider the existence of Beltrami fields with a nonconstant proportionality factor f in an open subset U of

When do knots in light stay knotted

{mathbb{R}^3}

Foliated vector fields without periodic orbits

R3. By reformulating this problem as a constrained evolution equation on a surface, we find an explicit differential equation that f must satisfy whenever there is a nontrivial Beltrami field with this factor. This ensures that there are no nontrivial regular solutions for an open and dense set of factors f in the Ck topology,

A note on the Liouvillian integrability and the qualitative properties of the mass rate equation for black holes

{kgeqq 7}

Submanifolds that are level sets of solutions to a second-order elliptic PDE

k≧7. In particular, there are no nontrivial Beltrami fields whenever f has a regular level set diffeomorphic to the sphere. This provides an explanation of the helical flow paradox of Morgulis etxa0al. (Commun Pure Appl Math 48:571–582, 1995).