This dissertation explores numerical solutions for the cohomogeneity one Einstein and Ricci soliton equations when the principal orbits are SU(3)/T^2 and Sp(3)/Sp(1)^3. We present new numerical evidence for steady, expanding solitons as well as Einstein metrics with positive scalar curvature. In the case of steady solitons we produced a one-parameter family of solutions. In the expanding case, we generated a two-parameter family of solutions and in particular in the negative Einstein case we generated a one-parameter family of solutions. In the compact Einstein case we found numerical evidence for an in nite number of Einstein metrics. / Thesis / Master of Science (MSc)

Wong, Chun Yin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 133-135). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese.

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This thesis presents two main results on analytic and topological aspects of scalar curvature. The first is a gluing theorem for scalar-flat manifolds with vanishing mean curvature on the boundary. Our methods involve tools from conformal geometry and perturbation techniques for nonlinear elliptic PDE. The second part studies bordisms of positive scalar curvature metrics. We present a modification of the Schoen-Yau minimal hypersurface technique to manifolds with boundary which allows us to prove a hereditary property for bordisms of positive scalar curvature metrics. The main technical result is a convergence theorem for stable minimal hypersurfaces with free boundary in bordisms with long collars which may be of independent interest.

This thesis presents through a number of applications a self-contained and robust methodology for exploring mathematical models of pattern formation from the perspective of a dynamical system. The contents of this work applies the methodology to investigate the influence of the domain-size and geometry on the evolution of the dynamics modelled by reaction-diffusion systems (RDSs). We start with deriving general RDSs on evolving domains and in turn explore Arbitrary Lagrangian Eulerian (ALE) formulation of these systems. We focus on a particular RDS of activator-depleted class and apply the detailed framework consisting of the application of linear stability theory, domain-dependent harmonic analysis and the numerical solution by the finite element method to predict and verify the theoretically proposed behaviour of pattern formation governed by the evolving dynamics. This is achieved by employing the results of domain-dependent harmonic analysis on three different types of two-dimensional convex and non-convex geometries consisting of a rectangle, a disc and a flat-ring.

Thesis (PH. D)--University of Chicago, 1916. / "Private edition, distributed by the University of Chicago Libraries, Chicago, Illinois." "Reprinted from Giornale di matematiche di Battaglini Vol. LIV, 1916." Includes bibliographical references. Also available on the Internet. Also issued online.

In this work, we investigate the mid-range behavior of geometry along a grafting ray in Teichm\"{u}ller space. The main technique is to describe the hyperbolic metric $\sigma_{t}$ at a point along the grafting ray in terms of a conformal factor $g_{t}$ times the Thurston (grafted) metric and study solutions to the linearized Liouville equation. We give a formula that describes, at any point on a grafting ray, the change in length of a sum of distinguished curves in terms of the hyperbolic geometry at the point. We then make precise the idea that once the length of the grafting locus is small, local behavior of the geometry for grafting on a general manifold is like that of grafting on a cylinder. Finally, we prove that the sum of lengths of is eventually monotone decreasing along grafting rays.

Finite frames are special collections of vectors utilized in Harmonic Analysis and Digital Signal Processing. In this thesis, geometric aspects and construction techniques are considered for the family of k-vector frames in Fn = Rn or Cn sharing a fixed frame operator (denoted Fk(E, Fn), where E is the Hermitian positive definite frame operator), and also the subfamily of this family obtained by fixing a list of vector lengths (denoted Fk µ(E, Fn), where µ is the list of lengths). The family Fk(E, Fn) is shown to be diffeomorphic to the Stiefel manifold Vn(Fk), and Fk µ(E, Fn) is shown to be a smooth manifold if the list of vector lengths µ satisfy certain conditions. Calculations for the dimensions of these manifolds are also performed. Finally, a new construction technique is detailed for frames in Fk(E, Fn) and Fk µ(E, Fn).

Most signal processing systems today need to estimate parameters of the underlying probability distribution, however quantifying the robustness of this system has always been difficult. This thesis attempts to quantify the performance and robustness of the Maximum Likelihood Estimator (MLE), and a robust estimator, which is a Huber-type censored form of the MLE. This is possible using diff erential geometric concepts of slope. We compare the performance and robustness of the robust estimator, and its behaviour as compared to the MLE. Various nominal values of the parameters are assumed, and the performance and robustness plots are plotted. The results showed that the robustness was high for high values of censoring and was lower as the censoring value decreased. This choice of the censoring value was simplifi ed since there was an optimum value found for every set of parameters. This study helps in future studies which require quantifying robustness for di fferent kinds of estimators.

Thesis (Ph. D.)--University of Chicago, 1935. / Vita. Lithoprinted. "Private edition, distributed by the University of Chicago libraries, Chicago, Illinois."