In my thesis I investigate an algebraization problem. The simplest, but already nontrivial, problem in this direction is to find necessary and sufficient conditions for three graphs of smooth functions on a given interval to belong to an algebraic curve of degree three. The analogous problems were raised by Lie and Darboux in connection with the classification of surfaces of double translation; by Poincare and Mumford in connection with the Schottky problem; by Griffiths and Henkin in connection with a converse of Abel’s theorem; by Bol and Akivis in the connection with the algebraization problem in the theory of webs. Interestingly, the complex-analytic technique developed by Griftiths and Henkin for the holomorphic case failed to work in the real smooth setting. In the thesis I develop a technique of, what I call, complex moments. Together with a simple differentiation rule it provides a unified approach to all the algebraization problems considered so far (both complex-analytic and real smooth). As a result I prove two variants (’polynomial’ and ’rational’) of a converse of Abel’s theorem which significantly generalize results of Griffiths and Henkin. Already the ’polynomial’ case is nontrivial leading to a new relation between the algebraization problem in the theory of webs and the converse of Abel’s theorem. But, perhaps, the most interesting is the rational case as a new phenomenon occurs: there are forms with logarithmic singularities on special algebraic varieties that satisfy the converse of Abel’s theorem. In the thesis I give a complete description of such varieties and forms.

In this thesis we study the Ginzburg-Landau equations of superconductivity, which are among the basic nonlinear partial differential equations of Theoretical and Mathematical Physics. These equations also have geometric interest as equations for the section and connection of certain principal bundles and are related to Seiberg-Witten equations used extensively in Differential Geometry. In 1957, Abrokosov suggested that for sufficiently high magnetic fields there exist solutions for which all physical quantities have the periodicity of a lattice, with the magnetic field penetrating the superconductor at the vertices of the lattice (Abrikosov lattice solutions). The corresponding phenomenon was confirmed experimentally and is among the most interesting aspects of superconductivity and is discussed in every book on the subject. In 2003, Abrikosov was awarded the Nobel Prize in Physics for this discovery. Building on the previous results in the subject we prove the existence of such lattices in the case where each lattice cell contains a single quantum of magnetic flux, and in the general case reduce the problem to an n-dimensional problem, where n is the number of quanta of flux. We prove that for Type II superconductors, these solutions are stable, and in the case n = 1, we show that as the external magnetic field approaches the critical value at which superconductivity first appears, the lattice which minimizes the average free energy per lattice cell is the triangular lattice.

In my thesis I investigate an algebraization problem. The simplest, but already nontrivial, problem in this direction is to find necessary and sufficient conditions for three graphs of smooth functions on a given interval to belong to an algebraic curve of degree three. The analogous problems were raised by Lie and Darboux in connection with the classification of surfaces of double translation; by Poincare and Mumford in connection with the Schottky problem; by Griffiths and Henkin in connection with a converse of Abel’s theorem; by Bol and Akivis in the connection with the algebraization problem in the theory of webs. Interestingly, the complex-analytic technique developed by Griftiths and Henkin for the holomorphic case failed to work in the real smooth setting. In the thesis I develop a technique of, what I call, complex moments. Together with a simple differentiation rule it provides a unified approach to all the algebraization problems considered so far (both complex-analytic and real smooth). As a result I prove two variants (’polynomial’ and ’rational’) of a converse of Abel’s theorem which significantly generalize results of Griffiths and Henkin. Already the ’polynomial’ case is nontrivial leading to a new relation between the algebraization problem in the theory of webs and the converse of Abel’s theorem. But, perhaps, the most interesting is the rational case as a new phenomenon occurs: there are forms with logarithmic singularities on special algebraic varieties that satisfy the converse of Abel’s theorem. In the thesis I give a complete description of such varieties and forms.

In this thesis we study the Ginzburg-Landau equations of superconductivity, which are among the basic nonlinear partial differential equations of Theoretical and Mathematical Physics. These equations also have geometric interest as equations for the section and connection of certain principal bundles and are related to Seiberg-Witten equations used extensively in Differential Geometry. In 1957, Abrokosov suggested that for sufficiently high magnetic fields there exist solutions for which all physical quantities have the periodicity of a lattice, with the magnetic field penetrating the superconductor at the vertices of the lattice (Abrikosov lattice solutions). The corresponding phenomenon was confirmed experimentally and is among the most interesting aspects of superconductivity and is discussed in every book on the subject. In 2003, Abrikosov was awarded the Nobel Prize in Physics for this discovery. Building on the previous results in the subject we prove the existence of such lattices in the case where each lattice cell contains a single quantum of magnetic flux, and in the general case reduce the problem to an n-dimensional problem, where n is the number of quanta of flux. We prove that for Type II superconductors, these solutions are stable, and in the case n = 1, we show that as the external magnetic field approaches the critical value at which superconductivity first appears, the lattice which minimizes the average free energy per lattice cell is the triangular lattice.

Doctor of Philosophy / Department of Mathematics / Zongzhu Lin / The purpose of this work is to study Hopf algebra analogs of constructions in the theory of p-adic representations of p-adic groups. We study Hopf algebras and comodules, whose underlying vector spaces are either Banach or compact inductive limits of such. This framework is unifying for the study of continuous and locally analytic representations of compact p-adic groups, affinoid and sigma-affinoid groups and their quantized analogs. We define the analog of Frechet-Stein structure for Hopf algebra (which play role of the function algebra), which we call CT-Stein structure. We prove that a compact type structure on a CT-Hopf algebra is CT-Stein if its dual is a nuclear Frechet-Stein structure on the dual NF-Hopf algebra. We show that for every compact p-adic group the algebra of locally analytic functions on that group is CT-Stein. We describe admissible representations in terms of comodules, which we call admissible comodules, and thus we prove that admissible locally analytic representations of compact p-adic groups are compact inductive limits of artinian locally analytic Banach space representations. We introduce quantized analogs of algebras Ur(sl2;K) from [7] thus giving an example of in fite-dimensional noncommutative and noncocommutative nonarchimedean Banach Hopf algebra. We prove that these algebras are Noetherian. We also introduce a quantum analog of U(sl2;K) and we prove that it is a (in fite-dimensional non-commutative and non-cocommutative) Frechet-Stein Hopf algebra. We study the cohomology theory of nonarchimedean comodules. In the case of modules and algebras this was done by Kohlhasse, following the framework of J.L. Taylor. We use an analog of the topological derived functor of Helemskii to develop a cohomology theory of non-archimedean comodules (this approach can be applied to modules too). The derived functor approach allows us to discuss a Grothendieck spectral sequence (GSS) in our context. We apply GSS theorem to prove generalized tensor identity and give an example, when this identity is nontrivial.

Doctor of Philosophy / Department of Mathematics / Charles N. Moore / The reader will recall that the classical $p$-Lebesgue spaces are those functions defined on a measure space $(X, \mu)$ whose modulus raised to the $p^{\rm th}$ power is integrable. This condition gives many quantitative measurements on the growth of the function, both locally and globally. Results and applications pertaining to such functions are ubiquitous. That said, the constancy of the exponent $p$ when computing $\int_X \abs{f}^p d\mu$ is limiting in the sense that it is intrinsically uniform in scope. Speaking loosely, there are instances in which one is concerned with the $p$ growth of a function in a region $A$ and its $q$ growth in another region $B$. As such, allowing the exponent to vary from region to region (or point to point) is a reasonable course of action. The task of developing such a theory was first taken up by Wladyslaw Orlicz in the 1930's. The theory he developed, of which variable Lebesgue spaces are a special case, was only intermittently studied and analyzed through the end of the century. However, at the turn of the millennium, several results and their applications sparked a focused and intense interest in variable $L^p$ spaces. It was found that with very few assumptions on the exponent function many of the classical structure and density theorems are valid in the variable-exponent case. Somewhat surprisingly, these results were largely proved using intuitive adaptations of well-established methods. In fact, this methodology set the tone for the first part of the decade, where a multitude of ``affirmative'' results emerged. While the successful adaptation of classical results persists to a large extent today, there are nontrivial situations in which one cannot hope to extend a result known for constant $L^p$. In this paper, we wish to explore both of the aforementioned directions of research. We will first establish the fundamentals for variable $L^p$. Afterwards, we will apply these fundamentals to some classical $L^p$ results that have been extended to the variable setting. We will conclude by shifting our attention to Littlewood-Paley theory, where we will furnish an example for which it is impossible to extend constant-exponent results to the variable case.

Doctor of Philosophy / Department of Mathematics / Gerald H. Hoehn / In this work, we study the complex N-Spin bordism groups of semifree circle actions and elliptic genera of level N. The notion of complex N-Spin manifolds (or simply N-manifolds) was introduced by Hoehn in [Hoh91]. Let the bordism ring of such manifolds be denoted by U;N and the ideal in U;N Q generated by bordism classes of connected complex N-Spin manifolds admitting an e ffective circle action of type t be denoted by IN;t. Also, let the elliptic genus of level n be denoted by 'n. It is conjectured in [Hoh91] that IN;t = \ njN n - tker('n): Our work gives a complete bordism analysis of rational bordism groups of semifree circle actions on complex N-Spin manifolds via traditional geometric techniques. We use this analysis to give a determination of the ideal IN;t for several N and t, and thereby verify the above conjectural equation for those values of N and t. More precisely, we verify that the conjecture holds true for all values of t with N 9, except for case (N; t) = (6; 3) which remains undecided. Moreover, the machinery developed in this work furnishes a mechanism with which to explore the ideal INt for any given values of N and t.

Doctor of Philosophy / Department of Mathematics / Zongzhu Lin / The Weyl algebra is the algebra of differential operators on a commutative ring of polynomials in finitely many variables. In Hayashi1990, Hayashi defines an algebra which he refers to as the quantized n-th Weyl algebra given by a deformation of the classical Weyl algebra. In luntsdifferential, Lunts and Rosenberg define [beta] and quantum differential operators for localization of quantum groups by deforming the relations that algebras of differential operators satisfy. In Iyer2007, Iyer and Mccune compute the quantum differential operators on the polynomial algebra with n variables. One naturally wonders ``What is the relationship between the quantized Weyl algebra and the quantum differential operators on the polynomial algebra with n variables?" In this thesis we answer this question by comparing the natural representations of U[subscript]q(sl[subscript]2) emerging from each algebra. Additionally, we connect the differential operators on the big cell of the flag variety of U[subscript]q(sl[subscript]n) with our deformed algebras. We also show the relationship between these algebras of differential operators and those appearing in the quantum Beilinson-Bernstein equivalence. Next we discuss analogous results in the case of [beta]-differential operators, as introduced in luntsdifferential. We consider both deformations on the underlying coordinate rings, and of the algebra of differential operators. We relate these results to the gluing problem for differential operators on noncommutative coordinate rings. We collect some of the different deformations of the usual Weyl algebra, and compare them based on a common bicharacter [beta]. Finally, we show a geometric result need in order to be able to glue deformed spaces and have their algebras of deformed differential operators cohere.

Master of Science / Department of Mathematics / Marianne Korten / The goal for this paper is to present material from Gilbarg and Trudinger’s Elliptic Partial Differential Equations of Second Order chapter 7 on Sobolev spaces, in a manner easily accessible to a beginning graduate student. The properties of weak derivatives and there relationship to conventional concepts from calculus are the main focus, that is when do weak and strong derivatives coincide. To enable the progression into the primary focus, the process of mollification is presented and is widely used in estimations. Imbedding theorems and compactness results are briefly covered in the final sections. Finally, we add some exercises at the end to illustrate the use of the ideas presented throughout the paper.

Doctor of Philosophy / Department of Mathematics / Christopher Pinner / Exponential and character sums occur frequently in number theory. In most applications one is only interested in estimating such sums. Explicit evaluations of such sums are rare. In this thesis we succeed in evaluating three types of sums when p is a prime and m is sufficiently large. The twisted monomial sum, the binomial character sum, and the generalized Jacobi sum. We additionally show that these are all sums which can be expressed in terms of classical Gauss sums.