▼ For K a compact subset of a Riemann surface, necessary and sufficient conditions are given for a function algebra containing A(K) to be all of C(K). Using these results, several conditions are given on a complex-valued function f so that the algebra generated by A(K) and f is all of C(K). In particular, the results are applied to a harmonic function f to give sufficient conditions for the algebra generated by A(K) and f to be all of C(K). Also, sufficient conditions are given for the algebra A(K) to be a maximal subalgebra of C(K). For X a compact subset of a Banach space, six properties that a compact subset K of the boundary of X can have in relation to the algebra A(X) are considered. These properties include the concepts of totally null set, zero set and null set. In the case of the finite and infinite dimensional polydiscs it is shown that five of the properties are equivalent, and a counterexample is given to show that the property of being a totally null set is weaker than the other five properties. This is in contrast to the unit ball of ℂn, where all six properties are known to be equivalent. The construction of the counterexample is then modified to give a condition on an algebra such that the property of being a totally null set is weaker than the property of being a null set. Finally, several conditions on an algebra are given, each of which implies that the property of being a totally null set is equivalent to the property of being a null set. Advisors/Committee Members: Izzo, Alexander J.

▼ This manuscript is to show the equivalency of the Axiom of Choice, Zorn's Lemma and Zermelo's Well-Ordering Principle. Starting with a brief history of the development of set history, we introduce the Axioms of Zermelo-Fraenkel, common applications of the axioms, and set theoretic descriptions of sets of numbers. The book, Introduction to Set Theory, by Karel Hrbacek and Thomas Jech is the primary resource with other sources providing additional background information. Advisors/Committee Members: McGovern, Warren Wm.

▼ In Chemistry, fullerenes are molecules composed entirely of carbon atoms, in the form of a hollow sphere, ellipsoid or tube, such that each atom is bonded with three other atoms and the atoms form pentagonal or hexagonal rings. The spherical fullerenes motivated the related mathematical concept: a fullerene graph is a trivalent plane graph such that all faces are pentagons and hexagons. The goal of this research is to prove the conjecture that there are exactly five l1-embeddable fullerenes. These are known to be the following fullerenes: F20(Ih),F26(D3h), F40(Td), F44(T), F80(Ih) (where the group of symmetry is given in parentheses for each fullerene). We proceed in proving this result by looking at the minimal distance between the pentagonal faces of the fullerene. In the cases when the minimal distance between pentagons is greater than two we obtain a contradiction, which leads us to conclude that in an l1-embeddable fullerene there must exist at least two pentagons that either are adjacent or have a common hexagonal neighbor. For the latter cases we show that the only possibilities are the five fullerenes listed above. Advisors/Committee Members: Shpectorov, Sergey.

▼ This paper will examine first the three spaces of constant curvature: Euclidean, spherical, and hyperbolic. Next, we consider the definitions and properties associated with Coxeter groups, reflection groups, and geometric reflection groups. This leads us to an interesting theorem about polytopes with angles that have are integral sub multiples of pi and how these polytopes tessellate these model spaces. These integers give a Coxeter matrix and corresponding Coxeter group. We list each of these possible polytopes in two dimensional Euclidean space and two dimensional spherical space. In hyperbolic space, there are infinitely many possibilities (based on the Gauss Bonnet theorem) and we specifically investigate the right angled case. From this theorem, we can also investigate regular polygons in hyperbolic 2-space and their tessellations. Lastly, we use the upper half plane model to construct a right angled hexagon centered at i. Advisors/Committee Members: Blok, Rieuwert.

▼ A fullerene is a carbon molecule where each carbon atom is chemically bonded to three other carbon atoms and the atoms form pentagonal and hexagonal rings and this molecule can be viewed as a finite connected trivalent plane graph, all of whose faces are pentagons and hexagons. Some research has focused on the graph theoretical properties of fullerenes and in some cases trying to determine a relationship between a graph theoretical property and chemical properties of the molecule. In this research we will focus on the graph theoretical property of a graph being l1-embeddable and discuss the face consistency of a particular class of fullerenes. This class is the hexagonal orange peel fullerenes with the symmetry group D6h and the pentagonal orange peel fullerenes with the symmetry groups D5h or I. Advisors/Committee Members: Shpectorov, Sergey.

▼ Phan’s theorem and the Curtis-Tits’ theorem are useful tools in the original proof of the Classification of Finite Simple Groups and the ongoing Gorenstein-Lyons-Solomon revision. Bennett, Gramlich, Hoffman and Shpectorov proved in a series of papers that Phan’s theorem and the Curtis-Tits’ theorem were results with very geometric proofs. They created a technique to prove these results which was generalized to produce what they called Curtis-Phan-Tits Theory. The present paper applies this technique to the orthogonal groups. A geometry is created on which a particular orthogonal group acts flag-transitively. The geometry is shown to be both connected and then simply connected when the dimension of the orthgonal group is at least five (except when the field is order three). After these facts are established Tits’ lemma is used to conclude that the orthogonal group is the universal completion of an interesting amalgam of subgroups that is associated with the geometry. This type of result is useful in the context of identifying a group when there is knowledge of the subgroup structure. Advisors/Committee Members: Hoffman, Corneliu.