Chris Kottke

Accurate finite-difference time-domain simulation of anisotropic media by subpixel smoothing

Finite-difference time-domain methods suffer from reduced accuracy when discretizing discontinuous materials. We previously showed that accuracy can be significantly improved by using subpixel smoothing of the isotropic dielectric function, but only if the smoothing scheme is properly designed. Using recent developments in perturbation theory that were applied to spectral methods, we extend this idea to anisotropic media and demonstrate that the generalized smoothing consistently reduces the errors and even attains second-order convergence with resolution.

An index theorem of Callias type for pseudodifferential operators

We prove an index theorem for families of pseudodifferential operators generalizing those studied by C. Callias, N. Anghel and others. Specifically, we consider operators on a manifold with boundary equipped with an asymptotically conic (scattering) metric, which have the form D + i Φ, where D is elliptic pseudodifferential with Hermitian symbols, and Φ is a Hermitian bundle endomorphism which is invertible at the boundary and commutes with the symbol of D there. The index of such operators is completely determined by the symbolic data over the boundary. We use the scattering calculus of R. Melrose in order to prove our results using methods of topological K-theory, and we devote special attention to the case in which D is a family of Dirac operators, in which case our theorem specializes to give family versions of the previously known index formulas.

A Callias-Type Index Theorem with Degenerate Potentials

A generalization of Callias’ index theorem for self adjoint Dirac operators with skew adjoint potentials on asymptotically conic manifolds is presented in which the potential term may have constant rank nullspace at infinity. The index obtained depends on the choice of a family of Fredholm extensions, though as in the classical version it depends only on the data at infinity.

Vortex core identification in viscous hydrodynamics

We describe a software package designed for the investigation of topological fluid dynamics with a novel algorithm for locating and tracking vortex cores. The package is equipped with modules for generating desired vortex knots and links and evolving them according to the Navier–Stokes equations, while tracking and visualizing them. The package is parallelized using a message passing interface for a multiprocessor environment and makes use of a computational steering library for dynamic user intervention.

Dimension of monopoles on asymptotically conic 3-manifolds

The virtual dimensions of both framed and unframed SU(2) magnetic monopoles on asymptotically conic 3-manifolds are obtained by computing the index of a Fredholm extension of the associated deformation complex. The unframed dimension coincides with the one obtained by Braam for conformally compact 3-manifolds. The computation follows from the application of a Callias-type index theorem.