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In this paper we study the hierarchical structure of the 2-d polyominoes. We introduce a new infinite family of polyominoes which we prove tiles a strip. We discuss applications of algebra to tiling. We discuss the algorithmic decidability of tiling the infinite plane Z x Z given a finite set of polyominoes. We will then discuss tiling with rectangles. We will then get some new, and some analogous results concerning the possible hierarchical structure for the 3-d polycubes.

We consider Landau-Ginzburg models stemming from non-abelian groups comprised of non-diagonal symmetries, and we describe a rule for the mirror LG model. In particular, we present the non-abelian dual group G*, which serves as the appropriate choice of group for the mirror LG model. We also describe an explicit mirror map between the A-model and the B-model state spaces for two examples. Further, we prove that this mirror map is an isomorphism between the untwisted broad sectors and the narrow diagonal sectors in general.

The classical spherical and Euclidean geometries are easy to visualize and correspond to spaces with constant curvature 0 and +1 respectively. The geometry with constant curvature −1, hyperbolic geometry, is much more complex. A powerful theorem of Mostow and Prasad states that in all dimensions at least 3, the geometry of a finite-volume hyperbolic manifold (a space with local d-dimensional hyperbolic geometry) is determined by the manifold's fundamental group (a topological invariant of the manifold). A cusp is a part of a finite-volume hyperbolic manifold that is infinite but has finite volume (cf. the surface of revolution of a tractrix has finite area but is infinite). All non-compact hyperbolic manifolds have cusps, but only finitely many of them. In the fundamental group of such a manifold, each cusp corresponds to a cusp subgroup, and each cusp subgroup is associated to a point on the boundary of H^d, which can be identified with the (d − 1)-sphere. It is known that there are many one-cusped two- and three-dimensional hyperbolic manifolds. This thesis studies restrictions on the existence of 1-cusped hyperbolic d-dimensional manifolds for d ≥ 3. Congruence subgroups belong to a special class of hyperbolic manifolds called arithmetic manifolds. Much is known about arithmetic hyperbolic 3- manifolds, but less is known about arithmetic hyperbolic manifolds of higher dimensions. An important infinite class of arithmetic d-manifolds is obtained using SO(n, 1; Z), a subset of the integer matrices with determinant 1. This is known to produce 1-cusped examples for small d. Taking special congruence conditions modulo a fixed number, we obtain congruence subgroups of SO(n, 1; Z) which also have cusps but possibly more than one. We ask what congruence subgroups with one cusp exist in SO(n, 1; Z). We consider the prime congruence level case, then generalize to arbitrary levels. Covering space theory implies a relation between the number of cusps and the image of a cusp in the mod p reduced group SO(d+ 1, p), an analogue of the classical rotation Lie group. We use the sizes of maximal subgroups of groups SO(d + 1, p), and the maximal subgroups' geometric actions on finite vector spaces, to bound the number of cusps from below. Let Ω(d, 1; Z) be the index 2 subgroup in SO(d, 1; Z) that consists of all elements of SO(d, 1; Z) with spinor norm +1. We show that for d = 5 and d ≥ 7 and all q not a power of 2, there is no 1-cusped level-q congruence subgroup of Ω(d, 1; Z). For d = 4, 6 and all q not of the form 2^a3^b, there is no 1-cusped level-q congruence subgroup of Ω(d, 1; Z). / Mathematics

<p> This thesis demonstrates the effectiveness of matrix methods in algebraic number fields, the integral closure of pure cubic fields and the use of the Hessian and discriminant to determine integral closure. </p> / Thesis / Master of Science (MSc)

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