Kent

We study a spectral problem which is related to the propagation of electromagnetic waves in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a periodic medium. The defect is infinitely extended and aligned with one of the coordinate axes. Under certain geometrical assumptions, the underlying Maxwell operator reduces to an elliptic operator and we study the effect of the perturbation by the waveguide on its spectrum. We show that the perturbation introduces guided mode spectrum inside the band gaps of the fully periodic, unperturbed spectral problem and use variational arguments to prove that guided mode spectrum can be created by arbitrarily small perturbations.

We consider spaces of entire functions that are $p$-integrable

with respect to a radial weight. Such spaces are usually called

Fock spaces, and a classical example is provided by the Gaussian

weight. It turns out that a function belongs to some Fock

space if and only if its derivative belongs to a Fock space

with a (possibly) different weight. Furthermore, we characterize

the Borel measures $\mu$ for which a Fock space is continuously

embedded in $L^q(\mu0)$ with $q>0$. We then illustrate the

applicability of these results to the study of properties such as

boundedness, compactness, Schatten class membership and the invariant

subspaces of integration operators of Volterra type acting on Fock spaces.

(joint work with Jose Angel Pelaez)

Much is now known about exp-log series, and their connections with o-

minimality and Hardy fields. However applied mathematicians who work with

differential equations, almost invariably want series involving

trigonometric functions which those theories exclude. The seminar looks at

one idea for incorporating oscillating functions into the framework of

Hardy fields.