Visually complex objects with infinitesimally fine features, naturally call for mathematical representations. The geometrical property of self-similarity - the whole similar to its parts - when iterated to infinity generates such features. Finite sets of affine contractions called Iterated Function Systems (IFS), with their compact attractors IFS fractals, can be applied to represent detailed self-similar shapes, such as trees or mountains. The fine local features of such attractors prevent their straightforward geometrical handling, and often imply a non-integer Hausdorff dimension. The main goal of the thesis is to develop an alternative approach to the geometry of IFS fractals in the classical sense via bounding sets. The results are obtained with the objective of practical applicability. The thesis thus revolves around the central problem of determining bounding sets to IFS fractals - and the convex hull in particular - emphasizing the fundamental role of such sets in their geometry. This emphasis is supported throughout the thesis, from real-life and theoretical applications to numerical algorithms crucially dependent on bounding.

Orientation – the process by which animals determine their position in an environment – can be accomplished by using the visually distinct properties of objects or surfaces, known as features (i.e., colour or pattern) or the relationship among objects and surfaces, known as geometry (i.e., wall length or angular information). Although features have been shown to facilitate the encoding of geometry, little is known as to whether restricting one’s viewpoint to include fewer features will still facilitate the encoding of geometry. During this experiment, men and women were trained to search near either an acute or an obtuse corner of a virtual parallelogram-shaped room that contained either three or four discrete and distinctive features. Participants were subsequently tested for their encoding of wall length and angles when the cues were presented in isolation, together, or in conflict. Results showed that the number of features present during training did not influence the encoding of geometry. However, the discrete and distinctive properties of the features overshadowed the encoding of angles by women as well as by participants who were trained with the obtuse corner. Although some groups of participants did not encode angular information when this was the only available geometric cue, all groups weighed angles more heavily than wall length when the cues provided conflicting information. This result suggests that one type of geometric cue (i.e., wall length) can facilitate the encoding of another (i.e., angles).

The aim of this thesis is to give a self-contained introduction to the hyperbolic geometry and the theory of discrete subgroups of PSL(2,R), and to generalize the results on equivariant functions. We show that there is a deep relation between the geometry of the group and some analytic and algebraic properties of these functions. In addition, we provide some applications of equivariant functions consisting of new results as well as providing new and simple proofs to classical results on automorphic forms.

Knot theory is an important branch of mathematics with applications in other branches of science. In this paper, we explore invariants on a special class of knots, known as virtual knots. We find new invariants by taking quotients of quandles, and introducing the fundamental Latin Alexander quandle and its Grobner basis. We also demonstrate examples of computations of these invariants.

The central question which will be addressed in this paper is: can we know the true geometry of space? My answer will be in the negative, but not first without heavy qualification. The thesis concerns the notion of truth in mathematical science, i.e. physical science for which mathematics (particularly geometry) is integral, and will ask whether we can know with certainty, or via some empirical test, which geometry is an accurate description of the actual universe. It will be a fairly historical approach, but hopefully not entirely so. We will begin with a 17th century debate on the nature of space between Newton and Leibniz and how Kant proposed to resolve the debate, and then move on to the views of the late 19th century mathematician Poincaré, but we will end with Einstein's Theory of Relativity - a theory which uses a very different geometry to which most of us are perhaps accustomed. In general, the goal will be to better understand the nature of geometry and its role in scientific theory; specifically, however, it will be an attempt to answer, in the negative, the central question before us. / Graduate

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Distance geometry problems (DGPs) are concerned with the construction of structures given partial information about distances between vertices. I present a generalisation which I call the angle and distance geometry problem (ADGP), in which partial angle information may be given as well. The work is primarily concerned with the algebraic and theoretical aspects of this problem, although it contains some information on practical applications. The embedding space is typically real three dimensional space for applications such as computer aided design and molecular chemistry, although other embedding spaces are possible. I show that both DGP and ADGP are NP-hard, but that in some sense the ADGP is more expressive than the DGP. To combat the problems of NP-hardness I present some graph theoretic heuristics which may be applied to both DGP and ADGP, and so reduce the time required by general purpose algorithms for their solution. I discuss the general purpose algorithms Cylindrical Algebraic Decomposition and Gröbner bases and their application to this field. In addition, I present an O(n) parallel algorithm for computing convex hulls in three dimensions, using O(n²) processors connected in a mesh-like topology with no shared memory.

In this thesis we apply methods from homological algebra to the study of finite $p$-groups. Let $G$ be a finite $p$-group and let $\mathbb{F}_p$ be the field of $p$ elements. We consider the cohomology groups $\operatorname{H}^1(G,\mathbb{F}_p)$ and $\operatorname{H}^2(G,\mathbb{F}_p)$ and the Massey product structure on these cohomology groups, which we use to deduce properties about $G$. We tie the classical theory of Massey products in with a general method from deformation theory for constructing hulls of functors and see how far the strictly defined Massey products can take us in this setting. We show how these Massey products relate to extensions of modules and to relations, giving us cohomological presentations of $p$-groups. These presentations will be minimal pro-$p$ presentations and will often be different from the presentations we are used to. This enables us to shed some new light on the classification of $p$-groups, in particular we give a `tree construction' illustrating how we can `produce' $p$-groups using cohomological methods. We investigate groups of exponent $p$ and some of the families of groups appearing in the tree. We also investigate the limits of these methods. As an explicit example illustrating the theory we have introduced, we calculate Massey products using the Yoneda cocomplex and give 0-deficiency presentations for split metacyclic $p$-groups using strictly defined Massey products. We also apply these methods to the modular isomorphism problem, i.e. the problem whether (the isomorphism class of) $G$ is determined by $\F_pG$. We give a new class $\mathcal{C}$ of finite $p$-groups which can be distinguished using $\mathbb{F}_pG$.

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No description available.