Wu, Dan. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 62-63). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Calibrated Geometry --- p.9 / Chapter 2.1 --- Theory of calibrations --- p.9 / Chapter 2.2 --- Two classical examples --- p.14 / Chapter 2.3 --- Calibrations and the VCP-forms --- p.19 / Chapter 3 --- Constructing Calibrations --- p.23 / Chapter 3.1 --- Clifford algebra and Spin groups --- p.23 / Chapter 3.2 --- Calibrations and spinors --- p.30 / Chapter 4 --- Calibrations in Exceptional Geometry --- p.39 / Chapter 4.1 --- G2 calibration --- p.40 / Chapter 4.2 --- Cayley calibration --- p.49 / Bibliography --- p.62

by Lo Yiu Ming. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 89-90). / Abstract also in Chinese. / Abstract --- p.iii / Acknowledgements --- p.iv / Chapter Chapter 1. --- Introduction --- p.1 / Chapter 1.1. --- Main Idea --- p.5 / Chapter 1.2. --- Organization --- p.8 / Chapter Chapter 2. --- Geometric and Analytic Preliminaries --- p.9 / Chapter 2.1. --- A Review on Measure Theory --- p.9 / Chapter 2.2. --- Submanifolds in Rn --- p.11 / Chapter 2.3. --- Several Results from PDEs --- p.17 / Chapter 2.4. --- Biharmonic Comparison Lemma --- p.20 / Chapter Chapter 3. --- Approximate Graphical Decomposition --- p.24 / Chapter 3.1. --- Some Preliminaries --- p.24 / Chapter 3.2. --- Approximate Graphical Decomposition --- p.30 / Chapter Chapter 4. --- Existence & Regularity of Measure-theoretic Limits of Minimizing Sequence --- p.41 / Chapter 4.1. --- Willmore Functional and Area --- p.41 / Chapter 4.2. --- Existence of Measure-theoretic Limit of Minimizing Sequence --- p.45 / Chapter 4.3. --- Higher Regularity at Good Points --- p.54 / Chapter 4.4. --- Convergence in Hausdorff Distance Sense --- p.62 / Chapter 4.5. --- Regularity near Bad Points --- p.64 / Chapter Chapter 5. --- Existence of Genus 1 Minimizers in Rn --- p.83 / References --- p.89

This dissertation contains work from the author's papers [35] and [36] with coauthor Eyvindur Palsson. The classic Mattila-Sjolin theorem shows that if a compact subset of $mathbb{R}^d$ has Hausdorff dimension at least $frac{(d+1)}{2}$ then its set of distances has nonempty interior. In this dissertation, we present a similar result, namely that if a compact subset $E$ of $mathbb{R}^d$, with $d geq 3$, has a large enough Hausdorff dimension then the set of congruence classes of triangles formed by triples of points of $E$ has nonempty interior. These types of results on point configurations with nonempty interior can be categorized as extensions and refinements of the statement in the well known Falconer distance problem which establishes a positive Lebesgue measure for the distance set instead of it having nonempty interior / Doctor of Philosophy / By establishing lower bounds on the Hausdorff dimension of the given compact set we can guarantee the existence of lots of triangles formed by triples of points of the given set. This type of result can be categorized as an extension and refinement of the statement in the well known Falconer distance problem which establishes that if a compact set is large enough then we can guarantee the existence of a significant amount of distances formed by pairs of points of the set

Many problems in geometric measure theory are centered around finding conditions and structures on a set to guarantee that its distance set must be large. Two notions of structure that are of importance in this work are Hausdorff dimension and thickness. Recent progress has been made on generalizing the notion of thickness so part of this work also generalizes previous results using this new upgraded version of thickness. We also show why a famous conjecture about distance sets does not hold on the real line and thus, why this conjecture needs to happen in higher dimensions. Furthermore, we give explicit distance set and thickness calculations for a special class of self-similar sets. / Master of Science / Part of the study of geometric measure theory is centered around creating interesting structures to place on a set and determining what sort of threshold on that structure allows you to guarantee that some interesting geometric property exists for that set. An example of this is determining when you can guarantee that a set contains many unique distances between elements in that set. This work presents various types of structures that help to investigate the problem of when you can guarantee that a set has the previously mentioned geometric property.

The vector-valued coarea formula provides a relationship between the integral of the Jacobian of a map from high dimensions down to low dimensions with the integral over the measure of the fibers of this map. We explore minimizers of this functional, proving existence using both a variational approach and an approach with currents. Additionally, we consider what properties these minimizers will have and provide examples. Finally, this problem is considered in metric spaces, where a third existence proof is given.

In my opinion, compactness is the most important concept in mathematics. We 'll track it from the one-dimensional real line in calculus to infinite dimensional spaces of functions and surfaces and see what it can do.

In this paper we prove structure results for the singular sets of stationary harmonic maps and mean curvature flows local to particular singularities. The original work is contained in Chapter 5 and Chapter 8. Chapters 1-5 are concerned with energy minimising maps and stationary harmonic maps. Chapters 6-8 are concerned with mean curvature flows and Brakke flows. In the case of stationary harmonic maps we consider a singularity at which the spine dimension is maximal, and such that the weak tangent map is homotopically non-trivial, and has minimal density amongst singularities of maximal spine dimen- sion. Local to such a singularity we show the singular set is a bi-Hölder continuous homeomorphism of the unit disk of dimension equal to the maximal spine dimension. A weak tangent map is translation invariant along a subspace, and invariant under dilations, so it completely defined by its values on a sphere. Such a map is said to be homotopically non-trivial if the mapping of a sphere into some target manifold cannot be deformed by a homotopy to a constant map. For an n-dimensional mean curvature flow we consider a singularity at which we can find a shrinking cylinder as a tangent flow, that collapses on an (n−1)-dimensional plane. Local to such a singularity we show that all singularities have such a cylindrical tangent, or else have lower Gaussian density than that of the shrinking cylinder. The subset of cylindrical singularities can be shown to be contained in a finite union of parabolic (n − 1)-dimensional Lipschitz submanifolds. In the case that the mean curvature flow arises from elliptic regularisation we can show that all singularities local to a cylindrical singularity with (n − 1)-dimensional spine are either cylindrical singularities with (n − 1)-dimensional spine, or contained in a parabolic Hausdorff (n − 2)-dimensional set.

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