Current seminars

Abstract: The study of existence and multiplicity of time-periodic solutions for semilinear Klein-Gordon equation has recently been proposed as a toy model to understand stability properties of Anti-de Sitter spacetime under certain perturbation, a question which is of great interest in general relativity. I will present a result on existence and multiplicity of Cantor families of small amplitude, analytic in time and periodic solutions for the completely resonant cubic nonlinear Klein-Gordon equation on S3 for an asymptotically full measure set of frequencies. The solutions are constructed by a Lyapunov- Schmidt decomposition and a Nash-Moser iterative scheme. We first find non-degenerate solutions of the resonant system, then, in view of small divisors problem, we solve the Range equation by a Nash-Moser iteration.

Abstract: According to Rytov's law, the polarization vector of light follows a Fermi-Walker transport equation in optical fibers. Recent advancements in theory propose a modification to Rytov's law due to fiber bending. The aim of this talk is to further extend these predictions from flat to curved space-time. This involves perturbatively solving Maxwell's equations under the assumption that the wavelength is significantly shorter than the fiber's radius of curvature, as well as the characteristic length-scales of the ambient space-time. This results in a coupling of the polarization vector to the spatial Riemann curvature tensor and second derivatives of the lapse function.