Doctor of Philosophy / Department of Mathematics / Louis Crane / In this thesis we explore the possibilities of obtaining Topological Quantum Field Theories using cobordisms with corners to break further down in the structure of manifolds of a given dimension. The algebraic data obtained is described in the language of higher category theory, more precisely in its cubical approach which we explore here as well. Interesting connections are proposed to some important objects in Physics: the representations of the Poincaré group. Finally we will describe in great details the topological tools needed to describe the categories of cobordisms with corners and give some conjectures on their nature.

Doctor of Philosophy / Department of Mathematics / Diego Maldonado / Originally introduced in 1961 by Carl Gustav Axel Harnack [36] in the context of harmonic functions in R[superscript]2, the so-called Harnack inequality has since been established for solutions to a wide variety of different partial differential equations (PDEs) by mathematicians at different times of its historical development. Among them, Moser's iterative scheme [47-49] and Krylov-Safonov's probabilistic method [43, 44] stand out as pioneering theories, both in terms of their originality and their impact on the study of regularity of solutions to PDEs. Caffarelli's work [12] in 1989 greatly simplified Krylov-Safonov's theory and established Harnack's inequality in the context of fully non-linear elliptic PDEs. In this scenario, Caffarelli and Gutierrez's study of the linearized Monge-Ampere equation [15, 16] in 2002-2003 served as a motivation for axiomatizations of Krylov-Safonov-Caffarelli theory [3, 25, 57]. The main work in this dissertation is a new axiomatization of Krylov-Safonov-Caffarelli theory. Our axiomatic approach to Harnack's inequality in spaces of homogeneous type has some distinctive features. It sheds more light onto the role of the so-called critical density property, a property which is at the heart of the techniques developed by Krylov and Safonov. Our structural assumptions become more natural, and thus, our theory better suited, in the context of variational PDEs. We base our method on the theory of Muckenhoupt's A[subscript]p weights. The dissertation also gives an application of our axiomatic approach to Harnack's inequality in the context of infinite graphs. We provide an alternate proof of Harnack's inequality for harmonic functions on graphs originally proved in [21].

Master of Science / Department of Industrial and Manufacturing Systems Engineering / Jessica L. Heier Stamm / Gifts-in-kind (GIK) donations negatively affect the humanitarian supply chain at the point of receipt near the disaster site. In any disaster, as much as 50 percent of GIK donations are irrelevant to the relief efforts. This proves to be a significant issue to humanitarian organizations because the quantity and type of future GIK are uncertain, making it difficult to account for GIK donations at the strategic planning level. The result is GIK consuming critical warehouse space and manpower. Additionally, improper treatment of GIK can result in ill-favor of donors and loss of donations (both cash and GIK) and support for the humanitarian organization. This thesis proposes a robust facility location approach that mitigates the impact of GIK by providing storage space for GIK and pre-positions supplies to meet initial demand. The setting of the problem is strategic planning for hurricane relief along the Gulf and Atlantic Coasts of the United States. The approach uses a robust scenario-based method to account for uncertainty in both demand and GIK donations. The model determines the location and number of warehouses in the network, the amount of pre-positioned supplies to meet demand, and the amount of space in each warehouse to alleviate the impact of GIK. The basis of the model is a variant of the covering facility location model that must satisfy all demand and GIK space requirements. A computational study with multiple cost minimizing objective functions illustrates how the model performs with realistic data. The results show that strategic planning in the preparedness phases of the disaster management cycle will significantly mitigate the impact of GIK.

Doctor of Philosophy / Department of Mathematics / Charles N. Moore / The main purpose of this thesis is to derive the law of the iterated logarithm for tail sums in various contexts in analysis. The various contexts are sums of Rademacher functions, general dyadic martingales, independent random variables and lacunary trigonometric series. We name the law of the iterated logarithm for tail sums as tail law of the iterated logarithm. We first establish the tail law of the iterated logarithm for sums of Rademacher functions and obtain both upper and lower bound in it. Sum of Rademacher functions is a nicely behaved dyadic martingale. With the ideas from the Rademacher case, we then establish the tail law of the iterated logarithm for general dyadic martingales. We obtain both upper and lower bound in the case of martingales. A lower bound is obtained for the law of the iterated logarithm for tail sums of bounded symmetric independent random variables. Lacunary trigonometric series exhibit many of the properties of partial sums of independent random variables. So we finally obtain a lower bound for the tail law of the iterated logarithm for lacunary trigonometric series introduced by Salem and Zygmund.

Master of Science / Department of Industrial and Manufacturing Systems Engineering / Todd Easton / Integer Programs (IP) are a class of discrete optimization that have been used commercially to improve various systems. IPs are often used to reach an optimal financial objective with constraints based upon resources, operations and other restrictions. While incredibly beneficial, IPs have been shown to be NP-complete with many IPs remaining unsolvable. Traditionally, Branch and Bound (BB) has been used to solve IPs. BB is an iterative algorithm that enumerates all potential integer solutions for a given IP. BB can guarantee an optimal solution, if it exists, in finite time. However, BB can require an exponential number of nodes to be evaluated before terminating. As a result, the memory of a computer using BB can be exceeded or it can take an excessively long time to find the solution. This thesis introduces a modified BB scheme called the Octanary Branching Algorithm (OBA). OBA introduces eight children in each iteration to more effectively partition the feasible region of the linear relaxation of the IP. OBA also introduces equality constraints in four of the children in order to reduce the dimension of the remaining nodes. OBA can guarantee an optimal solution, if it exists, in finite time. In addition, OBA has been shown to have some theoretical improvements over traditional BB. During computational tests, OBA was able to find the first, second and third integer solution with 64.8%, 27.9% and 29.3% fewer nodes evaluated, respectively, than CPLEX. These integers were 44.9%, 54.7% and 58.2% closer to the optimal solution, respectively, when compared to CPLEX. It is recommended that commercial solvers incorporate OBA in the initialization and random diving phases of BB.

Doctor of Philosophy / Department of Mathematics / David Yetter / This dissertation begins the development of the deformation theorem of monoidal categories which accounts for the function that all arrow-valued operations, composition, the arrow part of the monoidal product, and structural natural transformation are deformed. The first chapter is review of algebra deformation theory. It includes the Hochschild complex of an algebra, Gerstenhaber's deformation theory of rings and algebras, Yetter's deformation theory of a monoidal category, Gerstenhaber and Schack's bialgebra deformation theory and Markl and Shnider's deformation theory for Drinfel'd algebras. The second chapter examines deformations of a small $k$-linear monoidal category. It examines deformations beginning with a naive computational approach to discover that as in Markl and Shnider's theory for Drinfel'd algebras, deformations of monoidal categories are governed by the cohomology of a multicomplex. The standard results concerning first order deformations are established. Obstructions are shown to be cocycles in the special case of strict monoidal categories when one of composition or tensor or the associator is left undeformed. At the end there is a brief conclusion with conjectures.

Doctor of Philosophy / Department of Mathematics / David R. Auckly / We study geometric variational problems for a class of models in quantum field theory known as Faddeev-Skyrme models. Mathematically one considers minimizing an energy functional on homotopy classes of maps from closed 3-manifolds into homogeneous spaces of compact Lie groups. The energy minimizers known as Hopfions describe stable configurations of subatomic particles such as protons and their strong interactions. The Hopfions exhibit distinct localized knot-like structure and received a lot of attention lately in both mathematical and physical literature. High non-linearity of the energy functional presents both analytical and algebraic difficulties for studying it. In particular we introduce novel Sobolev spaces suitable for our variational problem and develop the notion of homotopy type for maps in such spaces that generalizes homotopy for smooth and continuous maps. As the spaces in question are neither linear nor even convex we take advantage of the algebraic structure on homogeneous spaces to represent maps by gauge potentials that form a linear space and reformulate the problem in terms of these potentials. However this representation of maps introduces some gauge ambiguity into the picture and we work out 'gauge calculus' for the principal bundles involved to apply the gauge-fixing techniques that eliminate the ambiguity. These bundles arise as pullbacks of the structure bundles H[arrow pointing right with hook on tail]G[arrow pointing right]G/H of homogeneous spaces and we study their topology and geometry that are of independent interest. Our main results include proving existence of Hopfions as finite energy Sobolev maps in each (generalized) homotopy class when the target space is a symmetric space. For more general spaces we obtain a weaker result on existence of minimizers only in each 2-homotopy class.

Doctor of Philosophy / Department of Mathematics / Alexander G. Ramm / A new method, the Dynamical Systems Method (DSM), justified recently, is applied to solving ill-conditioned linear algebraic system (ICLAS). The DSM gives a new approach to solving a wide class of ill-posed problems. In Chapter 1 a new iterative scheme for solving ICLAS is proposed. This iterative scheme is based on the DSM solution. An a posteriori stopping rules for the proposed method is justified. We also gives an a posteriori stopping rule for a modified iterative scheme developed in A.G.Ramm, JMAA,330 (2007),1338-1346, and proves convergence of the solution obtained by the iterative scheme. In Chapter 2 we give a convergence analysis of the following iterative scheme: u[subscript]n[superscript]delta=q u[subscript](n-1)[superscript]delta+(1-q)T[subscript](a[subscript]n)[superscript](-1) K[superscript]*f[subscript]delta, u[subscript]0[superscript]delta=0, where T:=K[superscript]* K, T[subscript]a :=T+aI, q in the interval (0,1),\quad a[subscript]n := alpha[subscript]0 q[superscript]n, alpha_0>0, with finite-dimensional approximations of T and K[superscript]* for solving stably Fredholm integral equations of the first kind with noisy data. In Chapter 3 a new method for inverting the Laplace transform from the real axis is formulated. This method is based on a quadrature formula. We assume that the unknown function f(t) is continuous with (known) compact support. An adaptive iterative method and an adaptive stopping rule, which yield the convergence of the approximate solution to f(t), are proposed in this chapter.

Master of Science / Department of Mathematics / David Auckly / This report introduces the fiber bundles. It includes the definitions of fiber bundles such as vector bundles and principal bundles, with some interesting examples. Reduction of the structure groups, and covering homotopy theorem and some specific computation using obstruction classes, Cech cohomology, Stiefel-Whitney classes, and first Chern classes are included.

Master of Science / Department of Electrical and Computer Engineering / Caterina M. Scoglio / In network theory, a complex network represents a system whose evolving structure and dynamic behavior contribute to its robustness. The study of complex networks, though young, spans diverse domains including engineering, science, biology, sociology, psychology, and business, to name a few. Regardless of the field of interest, robustness defines a network’s survivability in the advent of classical component failures and at the onset of cryptic malicious attacks. With increasingly ambitious initiatives such as GENI and FIND that seek to design future internets, it becomes imperative to define the characteristics of robust topologies, and to build future networks optimized for robustness. This thesis investigates the characteristics of network topologies that maintain a high level of throughput in spite of multiple attacks. To this end, we select network topologies belonging to the main network models and some real world networks. We consider three types of attacks: removal of random nodes, high degree nodes, and high betweenness nodes. We use elasticity as our robustness measure and, through our analysis, illustrate that different topologies can have different degrees of robustness. In particular, elasticity can fall as low as 0.8% of the upper bound based on the attack employed. This result substantiates the need for optimized network topology design. Furthermore, we implement a trade off function that combines elasticity under the three attack strategies and considers the cost of the network. Our extensive simulations show that, for a given network density, regular and semi-regular topologies can have higher degrees of robustness than heterogeneous topologies, and that link redundancy is a sufficient but not necessary condition for robustness.