In this thesis we present a range of different knot theories and then generalise them. Working with this, we focus on biquandles with linear and quadratic biquandle functions (in the quadratic case we restrict ourselves to functions with commutative coefficients). In particular, we show that if a biquandle is commutative, the biquandle function must have non-commutative coefficients, which ties in with the Alexander biquandle in the linear case. We then describe some computational work used to calculate rack and birack homology.

In PG(2; q), the projective plane over the field Fq of q elements, a (k; n)-arc is a set K of k points with at most n points on any line of the plane. When n = 2, a (k; 2)-arc is called a k-arc. A fundamental question is to determine the values of k for which K is complete, that is, not contained in a (k + 1; n)-arc. In particular, what is the largest value of k for a complete K, denoted by mn(2; q)? This thesis focusses on using some algorithms in Fortran and GAP to find large com- plete (k; n)-arcs in PG(2; q). A blocking set B is a set of points such that each line contains at least t points of B and some line contains exactly t points of B. Here, B is the complement of a (k; n)-arc K with t = q +1 - n. Non-existence of some (k; n)-arcs is proved for q = 19; 23; 43. Also, a new largest bound of complete (k; n)-arcs for prime q and n > (q-3)/2 is found. A new lower bound is proved for smallest size of complete (k; n)-arcs in PG(2; q). Five algorithms are explained and the classification of (k; n)- arcs is found for some values of n and q. High performance computing is an important part of this thesis, where Algorithm Five is used with OpenMP that reduces the time of implementation. Also, a (k; n)-arc K corresponds to a projective [k; n; d]q-code of length k, dimension n, and minimum distance d = k - n. Some applications of finite geometry to operational research are also explained.

In this thesis, we classify the cubic surfaces with twenty-seven lines in three dimensional projective space over small finite fields. We use the Clebsch map to construct cubic surfaces with twenty-seven lines in PG(3; q) from 6-arcs not on a conic in PG(2; q). We introduce computational and geometrical procedures for the classification of cubic surfaces over the finite field Fq. The performance of the algorithms is illustrated by the example of cubic surfaces over F13, F17 and F19.

A major theme in geometric measure theory is establishing global properties, such as rectifiability, of sets or measures from local ones, such as densities or tangent measures. In establishing sufficient conditions for rectifiability it is useful to know what local properties are possible in a given setting, and this is the theme of this thesis. It is known, for 1-dimensional subsets of the plane with positive lower density, that the tangent measures being concentrated on a line is sufficient to imply rectifiability. It is shown here that this cannot be relaxed too much by demonstrating the existence of a 1-dimensional subset of the plane with positive lower density whose tangent measures are concentrated on the union of two halflines, and yet the set is unrectiable. A class of metrics are also defined on R, which are functions of the Euclidean metric, to give spaces of dimension s (s > 1), where the lower density is strictly greater than 21-s, and a method for gaining an explicit lower bound for a given dimension is developed. The results are related to the generalised Besicovitch 1/2 conjecture. Set functions are defined that measure how easily the subsets of a set can be covered by balls (of any radius) with centres in the subset. These set functions are studied and used to give lower bounds on the upper density of subsets of a normed space, in particular Euclidean spaces. Further attention is paid to subsets of R, where more explicit bounds are given.

The projective plane of order 11 is the dominant focus of this work. The motivation for working in the projective plane of order 11 is twofold. First, it is the smallest projective plane of prime power order such that the size of the largest (n, r)-arc is not known for all r ∈ {2,...,q + 1}. It is also the smallest projective plane of prime order such that the (n; 3)-arcs are not classified. Second, the number of (n, 3)-arcs is significantly higher in the projective plane of order 11 than it is in the projective plane of order 7, giving a large number of (n, 3)-arcs for study. The main application of (n, r)-arcs is to the study of linear codes. As a forerunner to the work in the projective plane of order eleven two algorithms are used to raise the lower bound on the size of the smallest complete n-arc in the projective plane of order thirty-one from 12 to 13. This work presents the classification up to projective equivalence of the complete (n, 3)- arcs in PG(2, 11) and the backtracking algorithm that is used in its construction. This algorithm is based on the algorithm used in [3]; it is adapted to work on (n, 3)-arcs as opposed to n-arcs. This algorithm yields one representative from every projectively inequivalent class of (n, 3)-arc. The equivalence classes of complete (n, 3)-arcs are then further classified according to their stabilizer group. The classification of all (n, 3)-arcs up to projective equivalence in PG(2, 11) is the foundation of an exhaustive search that takes one element from every equivalence class and determines if it can be extended to an (n′, 4)-arc. This search confirmed that in PG(2, 11) no (n, 3)-arc can be extended to a (33, 4)-arc and that subsequently m4(2, 11) = 32. This same algorithm is used to determine four projectively inequivalent complete (32, 4)-arcs, extended from complete (n, 3)-arcs. Various notions under the general title of symmetry are defined both for an (n, r)-arc and for sets of points and lines. The first of these makes the classification of incomplete (n; 3)- arcs in PG(2, 11) practical. The second establishes a symmetry based around the incidence structure of each of the four projectively inequivalent complete (32, 4)-arcs in PG(2, 11); this allows the discovery of their duals. Both notions of symmetry are used to analyze the incidence structure of n-arcs in PG(2, q), for q = 11, 13, 17, 19. The penultimate chapter demonstrates that it is possible to construct an (n, r)-arc with a stabilizer group that contains a subgroup of order p, where p is a prime, without reference to an (m < n, r)-arc, with stabilizer group isomorphic to ℤ1. This method is used to find q-arcs and (q + 1)-arcs in PG(2, q), for q = 23 and 29, supporting Conjecture 6.7. The work ends with an investigation into the effect of projectivities that are induced by a matrix of prime order p on the projective planes. This investigation looks at the points and subsets of points of order p that are closed under the right action of such matrices and their structure in the projective plane. An application of these structures is a restriction on the size of an (n, r)-arc in PG(2, q) that can be stabilized by a matrix of prime order p.

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Yiu, Fa Wai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 55-58). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Elementary properties and notations of Hyperbolic space --- p.9 / Chapter 3 --- Poisson kernel and Conformal densities --- p.16 / Chapter 3.1 --- Poisson kernel --- p.17 / Chapter 3.2 --- Conformal densities --- p.19 / Chapter 4 --- Patterson construction and decomposition --- p.27 / Chapter 4.1 --- Patterson construction --- p.27 / Chapter 4.2 --- Patterson decomposition --- p.33 / Chapter 5 --- Bonahon surfaces and Grided surfaces --- p.39 / Chapter 5.1 --- Bonahon surfaces --- p.40 / Chapter 5.2 --- Grided surfaces --- p.46 / Chapter 6 --- Margulis number of Hyperbolic Manifolds --- p.51 / Margulis Number for Hypcrbolic 3-manifolds --- p.5 / Chapter 6.1 --- Gcomertrically finite groups --- p.51 / Chapter 6.2 --- Margulis number of Closed Hyperbolic Manifolds --- p.53 / Bibliography --- p.55

Lo, Chi Yu. / On t.p. "n" is a superscript. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 92). / Abstracts in Chinese and English. / Chapter 0 --- Introduction --- p.5 / Chapter 1 --- "CR structures and the group SU(p, q)" --- p.7 / Chapter 1.1 --- Almost complex structure and CR manifolds --- p.7 / Chapter 1.2 --- Automorphism Groups of Ball and Polydisc --- p.17 / Chapter 1.3 --- "The group SU(p,q) and its Maurer Cartan form" --- p.24 / Chapter 2 --- Cartan´ةs construction on nondegenerate CR manifold --- p.33 / Chapter 2.1 --- A digression on the Frobenius Theorem and projective structure --- p.33 / Chapter 2.2 --- Cartan bundle and canonical forms --- p.45 / Chapter 2.3 --- Calculations of real hypersurface in C2 --- p.60 / Chapter 3 --- Geometric consequences and chain --- p.66 / Chapter 3.1 --- CR equivalence problem --- p.66 / Chapter 3.2 --- CR manifolds of dimension 3 --- p.71 / Chapter 3.3 --- Definition of chains --- p.78 / Chapter 3.4 --- Chains on a special kind of Reinhardt hyper surf ace in C2 --- p.87 / Bibliography --- p.92

When I design an object or a piece of furniture I do it with geometry in my mind. I use simple shapes like squares, rectangles and lines to create forms that then become an object. I organize the shapes in space thinking about the negative and positive space that they create. I also think of how each shape interacts with each other. The overall shape of an object contains other shapes inside, and these shapes create a relationship between each other in space. I also consider how geometry relates to the function of the object. In my designs, geometry and function shape the object. In this essay I describe the design and production process for a cheese tray, a night lamp and a candle holder. I talk about how I used geometry to design each one of these objects and the production process involved.

In this work, we study how solutions of certain non-compact geometric flows of fast-diffusion type interact with their asymptotic geometries at infinity. In the first part, we show the long time existence theorem to the inverse mean curvature flow for complete convex non-compact initial hypersurfaces. The existence and behavior of a solution is tied with the evolution of its tangent cone at infinity. In particular, the maximal time of existence can be written in terms of the area ratio between the initial tangent cone at infinity and the flat hyperplane. In the second part, we study the formation of type II singularity for non-compact Yamabe flow. Assuming the initial metric is conformally flat and asymptotic to a cylinder, we show the higher order asymptotics of the metric determines the curvature blow-up rates at the tip in its first singular time. We also show the singularities of such solutions are modeled on rotationally symmetric steady gradient solitons.