Mathematical Aspects of General Relativity

Mathematical general relativity, the subject of this workshop, is a remarkable confluence of different areas of mathematics. Einstein’s equation, the focus of mathematical relativity, is one of the most fruitful nonlinear hyperbolic PDE systems under study. As well, some of the most challenging geometric analysis problems in Reimannian geometry and elliptic PDE theory arise from the study of the initial data for Einstein’s equations. In addition, these studies play a crucial role in modeling the physics of astrophysical and cosmological systems. This workshop reflected the rapid progress seen in the field in recent years, and highlighted some of the most interesting questions under study in mathematical relativity

Advances in numerical bifurcation software : MatCont

The mathematical background of MatCont, a freely available toolbox, is bifurcation theory which is a field of hard analysis. Bifurcation theory treats dynamical systems from a high-level point of view. In the case of continuous dynamical systems this means that it considers nonlinear differential equations without any special form and without restrictions except for differentiability up to a sufficiently high order (in the present state of MatCont never higher than five.) The number of equations is not fixed in advance and neither is the number of variables or the number of parameters, some of which can be active and others not. The aim of bifurcation theory is to understand and classify the qualitative changes of the solutions to the differential equations under variation of the parameters. This knowledge cannot be applied to practical situations without numerical software, except in some artificially constructed situations. Matcont is a toolbox that computes bifurcation diagrams through numerical methods, namely continuation. This dissertation describes the advances and innovations that were made including the detection and continuation of new bifurcations in discrete-time systems

Forced symmetry breaking of Euclidean equivariant partial differential equations, pattern formation and Turing instabilities

Many natural phenomena may be modelled using systems of differential equations that possess symmetry. Often the modelling process introduces additional symmetries that are only approximately present in the real physical system. This thesis investigates how the inclusion of small symmetry breaking effects changes the behaviour of the original solutions, such a process is called forced symmetry breaking. Part I introduces the general equivariant bifurcation theory required for the rest of this work. In particular, we generalise previous techniques used to study forced symmetry breaking to a certain class of Euclidean invariant problems. This allows the study of the effects of forced symmetry breaking on spatially periodic solutions to differential equations. Part II considers spatially periodic solutions in two dimensions that are supported by the square or hexagonal lattices. The methods of Part I are applied to investigate how the translation free solutions, supported by these lattices, are altered when the perturbation term possesses certain symmetries. This leads to a partial classification theorem, describing the behaviour of these solutions. This classification is extended in Part III to three-dimensional solutions. In particular, the cubic lattices: simple, face centred, and body centred cubic, are considered. The analysis follows the same lines as Part II, but is necessarily more complex. This complexity is also present in the results, there are much richer dynamical possibilities. Parts II and III lead to a partial classification of the behaviour of spatially periodic solutions to differential equations in two and three dimensions. Finally in Part IV the results of Part III, concerning the body centred cubic lattice, are applied to the black-eye Turing instability. In particular, the model of Gomes [39] is cast in a new light where forced symmetry breaking is present, leading to several qualitative predictions. Nonlinear optical systems and the Polyacrylamide-Methylene Blue-Oxygen reaction are also discussed

Geometric manipulation of light : from nonlinear optics to invisibility cloaks

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 189-203).In this work, we study two different manipulations of electromagnetic waves governed by macroscopic Maxwell's equations. One is frequency conversion of such waves using small intrinsic material nonlinearities. We study conversion of an input signal at frequency w1 to frequency Wk due to second or third harmonic generation or four-wave mixing using coupled-mode theory. Using this framework, we show there is a critical input power at which maximum frequency conversion is possible. We study in depth the case of third harmonic generation, its solutions, and their stability analysis. Based on the dynamics of the system, we propose a regime of parameters that 100%- efficient frequency conversion is possible and propose a way of exciting this solution. We also look at same analysis for the case of degenerate four-wave mixing and come up with 2d and 3d designs of a device that exhibits high-efficiency second-harmonic generation. Second, we consider proposals for invisibility cloaks to change the path of electromagnetic waves in a certain way so that the object appears invisible at a certain frequency or a range of frequencies. Transformation-based invisibility cloaks make use of the coordinate invariance of Maxwell's Equations and require complex material configuration e and p in the cloak. We study the practical limitations of cloaking as a function of the size of the object being cloaked. Specifically, we study the bandwidth, loss, and scattering limitations of cloaking as the object gets larger and show that cloaking of objects many times larger than the wavelength in size becomes practically impossible.by Hila Hashemi.Ph.D