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Morrison has constructed a geometric relationship between K3 surfaces with large Picard number and abelian surfaces. In particular, this establishes that the period spaces of certain families of lattice polarized K3 surfaces (which are closely related to the moduli spaces of lattice polarized K3 surfaces) and lattice polarized abelian surfaces are identical. Therefore, we may study the moduli spaces of such K3 surfaces via the period spaces of abelian surfaces. In this thesis, we will answer the following question: from the moduli space of abelian surfaces with endomorphism structure (either a Shimura curve or a Hilbert modular surface), there is a natural map into the moduli space of abelian surfaces, and hence into the period space of abelian surfaces. What sort of relationship exists between the moduli spaces of abelian surfaces with endomorphism structure and the moduli space of lattice polarized K3 surfaces? We will show that in many cases, the endomorphism ring of an abelian surface is just a subring of the Clifford algebra associated to the N\'eron-Severi lattice of the abelian surface. Furthermore, we establish a precise relationship between the moduli spaces of rank 18 polarized K3 surfaces and Hilbert modular surfaces, and between the moduli spaces of rank 19 polarized K3 surfaces and Shimura curves. Finally, we will calculate the moduli space of E_8^2 + <4>-polarized K3 surfaces as a family of elliptic K3 surfaces in Weierstrass form and use this new family to find families of rank 18 and 19 polarized K3 surfaces which are related to abelian surfaces with real multiplication or quaternionic multipliction via the Shioda-Inose construction. / Thesis (Master, Mathematics & Statistics) -- Queen's University, 2011-08-12 14:38:04.131

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The character table of a finite group is a very powerful tool to study the groups and to prove many results. Any finite group is either simple or has a normal subgroup and hence will be of extension type. The classification of finite simple groups, more recent work in group theory, has been completed in 1985. Researchers turned to look at the maximal subgroups and automorphism groups of simple groups. The character tables of all the maximal subgroups of the sporadic simple groups are known, except for some maximal subgroups of the Monster M and the Baby Monster B. There are several well-developed methods for calculating the character tables of group extensions and in particular when the kernel of the extension is an elementary abelian group. Character tables of finite groups can be constructed using various theoretical and computational techniques. In this thesis we study the method developed by Bernd Fischer and known nowadays as the theory of Clifford-Fischer matrices. This method derives its fundamentals from the Clifford theory. Let G = N·G, where N C G and G/N = G, be a group extension. For each conjugacy class [gi]G, we construct a non-singular square matrix Fi, called a Fischer matrix. Once we have all the Fischer matrices together with the character tables (ordinary or projective) and fusions of the inertia factor groups into G, the character table of G is then can be constructed easily. In this thesis we apply the coset analysis technique (this is a method to find the conjugacy classes of group extensions) together with theory of Clifford-Fischer matrices to calculate the ordinary character tables of seven groups of extensions type, in which four are non-split and three are split extensions. These groups are of the forms: 21+8 + ·A9, 37:Sp(6, 2), 26·Sp(6, 2), 25·GL(5, 2), 210:(U5(2):2), 21+6 − :((31+2:8):2) and 22n·Sp(2n, 2) and 28·Sp(8, 2). In addition we give some general results on the non-split group 22n·Sp(2n, 2). / Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2012.

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The structure of connected graphs of given size and order that have minimal algebraic connectivity is investigated. It is shown that they must consist of a chain of cliques. Moreover, an upper bound for the number of maximal cliques of size 2 or larger is derived. (author's abstract) / Series: Research Report Series / Department of Statistics and Mathematics

This thesis consists of two parts, both of which involve the study of algebraic structures constructed via the multiplication of modules. In the first part we look at Hall algebras. We consider the Hall algebra of a cyclic quiver algebra with relations of length two and present a multiplication formula for the explicit calculation of products in this algebra. We then look at the case of a cyclic quiver with two vertices and describe the corresponding composition algebra as a quotient of the positive part of a quantised enveloping algebra of type Ã₁ We then look at quotients of Hall algebras of self-injective algebras. We give an abstract result describing the quotient of such a Hall algebra by the ideal generated by isomorphism classes of projective modules, and also a more explicit result describing quotients of Hall algebras of group algebras for cyclic 2-groups and some related polynomial algebras. The second part of the thesis deals with Green rings. We compare the Green rings of a group algebra and the corresponding Jennings algebra for certain p-groups. It is shown that these two Green rings are isomorphic in the case of a cyclic p-group. In the case of the Klein four group it is shown that the two Green rings are not isomorphic, but that there exist quotients of these rings which are isomorphic. It is conjectured that the corresponding quotients will also be isomorphic in the case of a dihedral 2-group. The properties of these quotients are studied, with the aim of producing evidence to support this conjecture. The work on Green rings also includes some results on the realisation of quotients of Green rings as group rings over ℤ.