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1 
Modules and comodules over nonarchimedean Hopf algebrasLyubinin, Anton January 1900 (has links)
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 padic representations of padic 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 padic groups, affinoid and sigmaaffinoid
groups and their quantized analogs. We define the analog of FrechetStein structure for
Hopf algebra (which play role of the function algebra), which we call CTStein structure.
We prove that a compact type structure on a CTHopf algebra is CTStein if its dual is a nuclear FrechetStein structure on the dual NFHopf algebra. We show that for every compact padic group the algebra of locally analytic functions on that group is CTStein. We describe admissible representations in terms of comodules, which we call admissible comodules, and
thus we prove that admissible locally analytic representations of compact padic 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 fitedimensional 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 fitedimensional noncommutative and
noncocommutative) FrechetStein 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 nonarchimedean 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.

2 
On variable Lebesgue spacesNguyen, Peter Quoc Hiep January 1900 (has links)
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 variableexponent case. Somewhat surprisingly, these results were largely proved using intuitive adaptations of wellestablished 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 LittlewoodPaley theory, where we will furnish an example for which it is impossible to extend constantexponent results to the variable case.

3 
Cobordism theory of semifree circle actions on complex nspin manifolds.Ahmad, Muhammad Naeem January 1900 (has links)
Doctor of Philosophy / Department of Mathematics / Gerald H. Hoehn / In this work, we study the complex NSpin bordism groups of semifree circle actions and
elliptic genera of level N.
The notion of complex NSpin manifolds (or simply Nmanifolds) 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 NSpin 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 NSpin 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.

4 
Deformations of differential operatorsBischof, Bryan E. January 1900 (has links)
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 nth 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 BeilinsonBernstein 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.

5 
Sobolev spacesClemens, Jason January 1900 (has links)
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.

6 
Prime power exponential and character sums with explicit evaluationsPigno, Vincent January 1900 (has links)
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.

7 
Fundamental concepts on Fourier Analysis (with exercises and applications)Dixit, Akriti January 1900 (has links)
Master of Science / Department of Mathematics / Diego M. Maldonado / In this work we present the main concepts of Fourier Analysis (such as Fourier series,
Fourier transforms, Parseval and Plancherel identities, correlation, and convolution) and
illustrate them by means of examples and applications. Most of the concepts presented
here can be found in the book "A First Course in Fourier Analysis" by David W.Kammler.
Similarly, the examples correspond to over 15 problems posed in the same book which have
been completely worked out in this report. As applications, we include Fourier's original
approach to the heat flow using Fourier series and an application to filtering onedimensional
signals.

8 
Renormalizations of the Kontsevich integral and their behavior under band sum moves.Gauthier, Renaud January 1900 (has links)
Doctor of Philosophy / Department of Mathematics / David Yetter / We generalize the definition of the framed Kontsevich integral initially presented in [LM1]. We study the behavior of the renormalized framed Kontsevich integral Z[hat]_f under band sum moves and show that it can be further renormalized into some invariant Z[widetilde]_f that is wellbehaved under moves for which link components of interest are locally put on top of each other. Originally, Le, Murakami and Ohtsuki ([LM5], [LM6]) showed that another choice of normalization is better suited for moves for which link components involved in the band sum move are put side by side. We show the choice of renormalization leads to essentially the same invariant and that the use of one renormalization or the other is just a matter of preference depending on whether one decides to have a horizontal or a vertical band sum. Much of the work on Z[widetilde]_f relies on using the tangle chord diagrams version of Z[hat]_f ([ChDu]). This leads us to introducing a matrix representation of tangle chord diagrams, where each chord is represented by a matrix, and tangle chord diagrams of degree $m$ are represented by stacks of m matrices, one for each chord making up the diagram. We show matrix congruences for some appropriately chosen matrices implement on the modified Kontsevich integral Z[widetilde]_f the band sum move on links. We show how Z[widetilde]_f in matrix notation behaves under the Reidemeister moves and under orientation changes. We show that for a link L in plat position, Z_f(L) in book notation is enough to recover its expression in terms of chord diagrams. We elucidate the relation between Z[check]_f and Z[widetilde]_f and show the quotienting procedure to produce 3manifold invariants from those as introduced in [LM5] is blind to the choice of normalization, and thus any choice of normalization leads to a 3manifold invariant.

9 
A law of the iterated logarithm for general lacunary seriesZhang, Xiaojing January 1900 (has links)
Doctor of Philosophy / Department of Mathematics / Charles N. Moore

10 
KTheory in categorical geometryBunch, Eric January 1900 (has links)
Doctor of Philosophy / Department of Mathematics / Zongzhu Lin / In the endeavor to study noncommutative algebraic geometry, Alex Rosenberg defined
in [13] the spectrum of an Abelian category. This spectrum generalizes the prime spectrum
of a commutative ring in the sense that the spectrum of the Abelian category R − mod is
homeomorphic to the prime spectrum of R. This spectrum can be seen as the beginning of
“categorical geometry”, and was used in [15] to study noncommutative algebriac geometry.
In this thesis, we are concerned with geometries extending beyond traditional algebraic
geometry coming from the algebraic structure of rings. We consider monoids in a monoidal
category as the appropriate generalization of rings–rings being monoids in the monoidal
category of Abelian groups. Drawing inspiration from the definition of the spectrum of
an Abelian category in [13], and the exploration of it in [15], we define the spectrum of
a monoidal category, which we will call the monoidal spectrum. We prove a descent condition which is the mathematical formalization of the statment “R − mod is the category
of quasicoherent sheaves on the monoidal spectrum of R − mod”. In addition, we prove
a functoriality condidition for the spectrum, and show that for a commutative Noetherian
ring, the monoidal spectrum of R − mod is homeomorphic to the prime spectrum of the ring
R.
In [1], Paul Balmer defined the prime tensor ideal spectrum of a tensor triangulated cat
gory; this can be thought of as the beginning of “tensor triangulated categorical geometry”.
This definition is very transparent and digestible, and is the inspiration for the definition in
this thesis of the prime tensor ideal spectrum of an monoidal Abelian category. It it shown
that for a polynomial identity ring R such that the catgory R − mod is monoidal Abelian,
the prime tensor ideal spectrum is homeomorphic to the prime ideal spectrum.

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