▼ Let G be an affine group scheme defined over a field k, and denote by RepkG the category of finite dimensional representations of G over k. The principle of tannakian duality states that any neutral tannakian category is tensorially equivalent to RepkG for some affine group scheme G and field k, and conversely. Originally motivated by an attempt to find a first-order explanation for generic cohomology of algebraic groups, we study neutral tannakian categories as abstract first-order structures and, in particular, ultraproducts of them. One of the main theorems of this dissertation is that certain naturally definable subcategories of these ultraproducts are themselves neutral tannakian categories, hence tensorially equivalent to ComodA for some Hopf algebra A over a field k. We are able to give a fairly tidy description of the representing Hopf algebras of these categories, and explicitly compute them in several examples. The work done in this vein constitutes roughly half of this dissertation. The second half is much less abstract in nature, as we turn our attention to working out the representation theories of certain unipotent algebraic groups, namely the additive group Ga and the Heisenberg group H1. The results we obtain for these groups in characteristic zero are not at all new or surprising, but in positive characteristic they perhaps are. In both cases we obtain that, for a given dimension n, if p is large enough with respect to n, all n-dimensional modules for these groups in characteristic p are given by commuting products of representations, with the constituent factors resembling representations of the same group in characteristic zero. This has led us to define the ‘height-restricted ultraproduct’ of the categories RepkiG for a sequence of fields ki of increasing positive characteristic, and the above result can be summarized by saying that these height-restricted ultraproducts are tensorially equivalent to RepkGn, where Gn denotes a direct product of copies of G and k is a certain field of characteristic zero. We later use these results to extrapolate some generic cohomology results for these particular unipotent groups. Advisors/Committee Members: Hewitt, Dr. Paul R.

▼ Different forms of computational systems, such as binary computers and Turing Machines, are known to be able to efficiently simulate each other. This is the basis of the Church-Turing Thesis which stipulates that any sufficiently powerful model of computing is equivalent to any other – any algorithm in one model can be translated, in polynomial time, to an equivalent algorithm in another. In 1982 a new form of computation was introduced which is based on the effects of Quantum Mechanics. It is currently unknown if Quantum Computing can be efficiently simulated by a classical computer, and thus might be a more powerful system of computing. The aim of this dissertation is to study the complexity of quantum computations from the perspective of groups. Two groups will receive extensive study. Each Toffoli group is determined and a minimal generating set for each will be constructed, also the type of simple group created by the direct limit of the Toffoli groups will be determined. The frames of each Pauli group will be analyzed and lead to a different realization of the Clifford Group. Let h be the Hadamard Gate, a purely quantum operator, and let P2n be the collection of permutation matrices of alternating type. We will see how SO2n(ℤ[1/2]).〈H〉 = 〈(h⊗I)σ|σ∈P2n〉, and that this suggests an algorithm which decomposes a quantum operator into a sequence of basic operators which are purely quantum or purely classical. This metric, and another based on buildings, will be explored. Advisors/Committee Members: Hewitt, Paul.

▼ We study the tangent TG and cotangent bundles T*G of a Lie group G which are also Lie groups. Our main results are to show that on TG the canonical Jacobi ndomorphism field S is parallel with respect to the canonical Lie group connection Lie group and that dually on the cotangent bundle of G the canonical symplectic form is parallel with respect to the canonical connection. We next prove some theorems for Lie algebra extensions in which we can obtain a group representation for the extended algebra from the representation of the lower dimensional algebra. We also determine the Lie algebra of the automorphism group of three well known Lie algebras. Finally we study the Hamilton-Jacobi separability of conformally flat metrics and find a metric, Lagrangian and geodesics for the solvable codimension one nilradical six dimensional Lie Algebras where one exists. Advisors/Committee Members: Thompson, Gerard.

▼ We study stable capillary surfaces with planar or spherical boundary in the absence of gravity. If the boundary of the capillary surface is embedded in a plane, we prove that the only immersed stable capillary surface is the spherical cap. The second part of this dissertation treats the case when the capillary surface lies inside the unit ball in R3 with its boundary on the unit sphere. We construct a Killing vector field for the hyperbolic metric and use it to show that if the center of mass of the region bounded between the surface and the unit sphere is at the origin, the configuration cannot be stable. As a corollary of this approach we obtain a new proof of a theorem by Barbosa and do Carmo. We also provide a new proof of the stability of spherical caps on a plane or inside of the round ball, using exotic containers. Advisors/Committee Members: Wente, Henry.

▼ HIV is a virus currently affecting approximately 33.3 million people worldwide. Since it's discovery in the early 1980s, researchers have strived to find treatment that helps the immune system eradicate the virus from the human body. A great deal of advances have been made in helping HIV infected individuals from advancing to AIDS, but no cure has yet been found. Researchers have found that the immune system is in a chronic state of activation during HIV infection and believe this could be a major contributor to the decline of immune system cell populations. Using analysis of systems of Ordinary Differential Equations, this paper serves to better understand the dynamics of the activated immune system during HIV infection. Both current and possible future therapies are considered. Advisors/Committee Members: Vayo, H. Westcott.

▼ In my dissertation I am looking at six dimensional Lie algebras which are solvable, indecomposable and that have a five-dimensional nilradical. Such algebras were classified by the Russian mathematician G. M. Mubarakzyanov in a paper published in 1963. Depending on the structure of the five dimensional nilradical his paper has nine paragraphs. The paper contains errors because calculations were done by hand and the list of algebras is incomplete. Also some of his Lie algebras are isomorphic between themselves. I made extensive use of MAPLE with some routines to help finesse Mubarakzyanov’s list. Precisely, I corrected all the misprints and completed his list of algebras: in Paragraph 6 he disregarded a case which gives two new Lie algebras. There is also one new Lie algebra in Paragraph 2, one in Paragraph 5 and two new Lie algebras in Paragraph 7. Advisors/Committee Members: Thompson, Gerard.

▼ Kashkarev has shown that the mod 2 Steenrod algebra is a prime ring. For any odd prime p, we prove that the mod p Steenrod algebra is also a prime ring. In sequel, for any prime p, we show that the mod p Steenrod algebra (a local ring with nil maximal ideal) has infinite little Krull dimension. This contrasts sharply with the case of a commutative (or noetherian) local ring with nil maximal ideal which must have little Krull dimension equal to 0. Also, we show that the Steenrod algebra has no Krull dimension, classical Krull dimension, or Gabriel dimension. Advisors/Committee Members: Odenthal, Charles.

▼ In this thesis we deal with the zero product problem and the commuting problem for Toeplitz operators on the Bergman space over the unit disk of the complex plane. For the zero product problem, we show that the zero product of two Toeplitz operators has only the trivial solution when one of the symbols has certain polar decomposition and the other is a general bounded symbol. As for the commuting problem, we show that if the Fourier series of the bounded function f is of the form f(reiθ) = ∑k=−∞N eikθ fk(r) where N is a positive integer, and Tf commutes with Tz+g̅, where g is a bounded analytic function on the open unit disk, then Tf is a nontrivial linear combination of Tz+g̅ and the identity operator I. Also, we describe all Toeplitz operators Tf that commutes with Tz+z̅, when the symbol f is integrable, with respect to the Lebesgue area measure, on the unit disk. Advisors/Committee Members: Nagisetty, Rao.

▼ First, we propose a semiparametric statistic for comparing two correlated ROC curves under a marginal density ratio model. When comparing the accuracies of two diagnostic tests, a paired design is often used, in which both diagnostic tests are administered on the same patient. As a result, the two ROC curves are correlated. A joint model makes assumption on the correlation structure which might be too strong and hard to be verified, thus risks misspecification bias when the model assumption is not correctly specified. The proposed marginal density ratio model relaxes this assumption on the correlation structure and is more robust. Second, we consider fitting logistic regression models to $2× 2$ contingency tables and derive explicit formulas for the maximum likelihood estimator, the asymptotic covariance matrix, the Wald statistic, the likelihood ratio statistic, and the score statistic in terms of the four cell frequency counts. We derive the asymptotic distributions of the Wald statistic, the likelihood ratio statistic, and the score statistic under local alternatives to the null hypothesis. We present some results on analysis of one real dataset. Finally, we conduct a simulation study on comparison of various estimating methodologies under the situation when some covariate variables have missing values. Missing data analysis is among the most popular methodology in modern statistics. The existing methodologies apply mostly when the missing values occur in the response variable. We examine their performance when the missing values occur in the covariate variables. Advisors/Committee Members: Zhang, Biao.