▼ Convex optimization is an important branch of operations research. It generalizes linear programming and offers powerful tools for modelling problems and discovering optimal solutions to real world problems. Mathematically it is an interesting topic because it ties together many branches: linear algebra, multivariable calculus, and numerical analysis, to name a few. Modelling a problem as a convex optimization problem can be challenging but offers many benefits. Algorithm design is critically important to ensure precision of solutions that solve with minimal computation power. From an engineering perspective it is also incredibly useful, since many more situations can be modeled than with linear programming alone. Exponentials, distances, and many other functions arise in engineering problems all the time, and require convex optimization to optimally design. To fully appreciate convex optimization, I believe both a solid understanding of the theory and exploring an in-depth applied problem are necessary. After studying the theory in the fall, I became interested in the facility layout problem. I read about the nature of the problem, current algorithmic approaches to solving it, and how convex optimization could be used to formulate the problem. Finally, I created a mock scenario to solve: designing a summer camp in the best way possible. This scenario was little more than a colorful motivation to solve the facility layout problem, yet provided a sense of realism with which to create the data needed for the problem. I took a convex formulation of the problem from my textbook, changed some facets of how it was created, and expanded upon the ideas they presented. I was successful in solving my instance of the problem to optimality. This paper will be laid out in four broad sections, building our knowledge base from the ground up and mirroring my progression through the topic. In the first, I will explain convexity of sets and functions. The material in this section is based on that presented in the textbook Convex Optimization. I will define the vocabulary to discuss optimization problems, and explain what convex optimization is and why it is a useful tool. In the second section, I will overview the facility layout problem and describe commercially available software and the benefits and drawbacks to each. In the third section, I will present my model, the contributions I have made, and define my data. Finally, in the last section I will examine my results graphically and numerically and reflect on future research directions. Advisors/Committee Members: Bosch, Bob.

▼ The main focus of this paper will be on two very different areas in which topology is relevant to the study of infinite graphs. The first is the mechanics of compactness proofs, which use a particular group of lemmas to extend results about finite subgraphs to apply to an entire infinite graph. We will explore these results by using them to prove a result of de Bruijn and Erdos, that an infinite graph is k-colorable if its finite subgraphs are k-colorable, in several different ways. The second area is a relatively new area of study pioneered by Diestel which redefines certain concepts of graph theory in terms of a topology on a graph. Specifically, we find that certain basic features of the cycle space cannot be extended verbatim to infinite graphs. But if we define the cycle space in terms of homeomorphic images of the circle S1 in a compactified topology on the graph, we can find extensions. This will be motivated in more detail and some of the consequences explored. Advisors/Committee Members: Woods, Kevin.

▼ Infinite sets are a fundamental object of modern mathematics. Surprisingly, the existence of infinite sets cannot be proven within mathematics. Their existence, or even the consistency of their possible existence, must be justified extra-mathematically or taken as an article of faith. We describe here several varieties of large infinite set that have a similar status in mathematics to that of infinite sets, i.e. their existence cannot be proven, but they seem both reasonable and useful. These large sets are known as large cardinals. We focus on two types of large cardinal: inaccessible cardinals and measurable cardinals. Assuming the existence of a measurable cardinal allows us to disprove a questionable statement known as the Axiom of Constructibility (V=L). Advisors/Committee Members: Wilmer, Elizabeth.

▼ Toric algebra is a field of study that lies at the intersection of algebra, geometry, and combinatorics. Thus, the algebraic properties of the toric ideal IA defined by the vector configuration A are often characterizable via the geometric and combinatorial properties of its corresponding toric variety and A, respectively. Here, we focus on the property of normality of A. A normal vector configuration defines the toric ideal of a normal toric variety. However, the definition of normality of A is based entirely on the algebraic structures associated with A without regard to any of its combinatorial properties. In this paper, we discuss two attempts to provide a combinatorial characterization of normality of A. Particularly, we show that the properties "the convex hull of A possesses a unimodular covering" and "A is a Δ-normal configuration" are both sufficient conditions for normality of A, but neither is necessary. This suggests that another combinatorial property is required to provide the desired characterization of normality of A. Advisors/Committee Members: Woods, Kevin.