This thesis is about development of students' geometrical reasoning, in particular of inclusive relations between 2 dimensional shapes, e.g. the rhombus as a special case of the kite. Students in the study worked with a dynamic figure constructed using Dynamic Geometry Software. The figure is a quadrilateral whose diagonals are constructed so that they are of fixed length and perpendicular. All students in the study were observed to use a strategy of 'dragging' to keep one diagonal as the perpendicular bisector of the other. This generated a 'family of shapes' which was comprised of an infinite number of kites, arrowheads (i.e. concave kites), one rhombus and two isosceles triangles. I have called this strategy 'Dragging Maintaining Symmetry' (DMS) and I claim it has the potential to mediate the understanding of the rhombus as a special case of the kites (and the isosceles triangle in the context of dynamic geometry). However students in the study typically perceived the shapes, generated using DMS, according to a partitional view i.e. as different shapes, albeit with common properties such as line symmetry. When asked how many kites it would be possible to make by dragging the figure some students reported that there were four kites (one typical kite in each of four relative positions). It appears that they perceived the dragging activity as a journey to a discrete end position rather than as an action that resulted in a continuously changing figure. To address this problem I showed the students an animation of the figure under DMS. This proved to be the catalyst which moved their reasoning towards perceiving inclusive relations between the rhombus and kite.

This dissertation describes the rigidity theory of bar joint frameworks, especially infinite ones. The first chapter revises some of the well established results for finite frameworks. We then look at how this can be extended to the infinite case, specifically from the analysis point of view. In particular, we look at vanishing flexibility that is observed in some specific examples. Then we look at a proof of the sufficient condition for the existence of a flex in an infinite framework as described in Owen and Power [6]. In the fourth chapter we establish that the rigidity operator arising from the infinite matrix is bounded. 'Ve then observe its structure for specific examples. As decribed in [8], we describe the representation of the rigidity operator as a matrix valued function on the torus. Finally we look at the decomposition of the space of infinitesimal flexes for crystal frameworks in terms of a product basis.

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In this thesis three main problems are studied. The first is a generalization of a well known question by P. McMullen on convex polytopes: 'Determine the largest number v(d, k) such that any set of u(d, k) points lying in general position in M.d can be mapped, by a permissible projective transformation, onto the vertices of a k-neighbourly polytope.7 Bounds for u(d, k) are obtained. The upper bound is attained using oriented matroid techniques. The lower bound is proved indirectly, by considering a partition problem equivalent to McMullen's question. The core partition problem, mentioned above, can be modified in the following manner: 'Let X be a set of n points in general position in Rd then, what is the minimum k such that for all A, B partition of X there is always a set {x ,..., Xk) C X, such that conv(A {x ,... Xk}) n conv(B {x ,... Xk}) = 0' For this question, through an asymptotical analysis, a relationship between the number of points in the set (n) , and the number to be removed (k) , is shown. Finally, another problem in convex polytopes proposed by von Stengel is considered: 'Consider a polytope, V, in dimension d with 2d facets, which is simple. Two vertices form a complementary pair, (x,y), if every facet of V is incident with x or y. The d cube has 2d l complementary vertex pairs. Is this the maximal number among the simple d polytopes with 2d facets', It is shown that the conjecture stated above holds up to dimension seven and extra conditions, under which the theorem holds in general, are exposed. A nice interpretation of von Stengel's question, in terms of coloured Radon partitions, is also introduced.

We study orthogonal modular varieties associated with the moduli of generalised Kummer manifolds. We are particularly interested in understanding the singularities that arise in certain toroidal compactifications. Throughout, we place particular emphasis on the application of these results to problems involving the birational classification of moduli spaces.

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The main theme of this thesis is to classify subsets in PG(2, 13) of k elements where no three are collinear. Such sets are called k-arcs. In particular, we are interested in the values of k for which the k-arc is complete. Chapter 1 is devoted to basic definitions of projective spaces of n dimensions, subspaces and the fundamental theorem of projective geometry. Chapter 2 is a preliminary to the main task. It contains the construction of PG(1, q) and the classification of all subsets of PG(1, 13). Chapter 3 is devoted to the construction of PG(2, q), derivation of some general equations and a few small results used in the plane. Chapter 4 is the main one and it contains the complete classification of k-arcs in PG(2, 13). In Chapter 5 the work is extended to projective spaces of order 17 and 19 as a test for the computer programs. Finally, there is a list of the programs used as well as tables of the points of the planes discussed throughout the work.

This thesis is concerned with the application of operadic methods, particularly modular operads, to questions arising in the study of moduli spaces of surfaces as well as applications to the study of homotopy algebras and new constructions of ‘quantum invariants’ of manifolds inspired by ideas originating from physics. We consider the extension of classical 2–dimensional topological quantum field theories to Klein topological quantum field theories which allow unorientable surfaces. We generalise open topological conformal field theories to open Klein topological conformal field theories and consider various related moduli spaces, in particular deducing a Möbius graph decomposition of the moduli spaces of Klein surfaces, analogous to the ribbon graph decomposition of the moduli spaces of Riemann surfaces. We also begin a study, in generality, of quantum homotopy algebras, which arise as ‘higher genus’ versions of classical homotopy algebras. In particular we study the problem of quantum lifting. We consider applications to understanding invariants of manifolds arising in the quantisation of Chern–Simons field theory.

Geometricians often say that a geometric mode of cognition is more effective than algebra in many tasks. This dissertation investi- gates geometrisation as a mathematical approach: its main epistemic constituents and benefits. The aim is to explain why geometrisation can be effective and how this effectiveness may be achieved. The use of the term 'geometric' in mathematical practice seems to be applied not only for mathematical concepts and methods. Of- ten it simply indicates the use of diagrams. The analysis of various examples draws the distinction between these two meanings. It also clarifies the impact of the two aspects into mathematical advance. The central argument of this thesis is that visual 'geometry' can ease the application of geometric concepts and methods. The argument is based on a case study drawn from a promising contemporary mathematical area - geometric group theory. This ap- proach involves the diagrammatic representations of groups by Cayley graphs. They are not only attractive objects in themselves but also useful constructions which can allow the endorsement of an interesting and sophisticated geometry of groups.