Browsing Pure Mathematics by Issue Date
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The Weil conjectures
(University of Waterloo, 20080124)In discussing the question of rational points on algebraic curves, we are usually concerned with ℚ. André Weil looked instead at curves over finite fields; assembling the counts into a function, he discovered that it ... 
The Normal Distribution of ω(φ(m)) in Function Fields
(University of Waterloo, 20080128)Let ω(m) be the number of distinct prime factors of m. A celebrated theorem of ErdösKac states that the quantity (ω(m)loglog m)/√(loglog m) distributes normally. Let φ(m) be Euler's φfunction. Erdös and Pomerance ... 
Koblitz's Conjecture for the Drinfeld Module
(University of Waterloo, 20080501)Let $E$ be an elliptic curve over the rationals without complex multiplication such that any elliptic curve $\mathbb{Q}$isogenous to $E$ has trivial $\mathbb{Q}$torsion. Koblitz conjectured that the number of primes less ... 
Quaternion Algebras and Quadratic Forms
(University of Waterloo, 20080512)The main goal of this Masters' thesis is to explore isomorphism types of quaternion algebras using the theory of quadratic forms, number theory and algebra. I would also present ways to characterize quaternion algebras, ... 
Operator Spaces and Ideals in Fourier Algebras
(University of Waterloo, 20080811)In this thesis we study ideals in the Fourier algebra, A(G), of a locally compact group G. For a locally compact abelian group G, necessary conditions for a closed ideal in A(G) to be weakly complemented are given, ... 
Artin's Primitive Root Conjecture and its Extension to Compositie Moduli
(University of Waterloo, 20080811)If we fix an integer a not equal to 1 and which is not a perfect square, we are interested in estimating the quantity N_{a}(x) representing the number of prime integers p up to x such that a is a generator of the cyclic ... 
Artin's Conjecture: Unconditional Approach and Elliptic Analogue
(University of Waterloo, 20080811)In this thesis, I have explored the different approaches towards proving Artin's `primitive root' conjecture unconditionally and the elliptic curve analogue of the same. This conjecture was posed by E. Artin in the year ... 
Branched Covering Constructions and the Symplectic Geography Problem
(University of Waterloo, 20080815)We apply branched covering techniques to construct minimal simplyconnected symplectic 4manifolds with small χ_h values. We also use these constructions to provide an alternate proof that for each s ≥ 0, there exists a ... 
Von Neumann Algebras for Abstract Harmonic Analysis
(University of Waterloo, 20080828)This thesis develops the theory of operator algebras from the perspective of abstract harmonic analysis, and in particular, the theory of von Neumann algebras. Results from operator algebras are applied to the study of ... 
A Survey of the Classification of Division Algebras
(University of Waterloo, 20080929)For a given field F we seek all division algebras over F up to isomorphism. This question was first investigated for division algebras of finite dimension over F by Richard Brauer. We discuss the construction of the Brauer ... 
Moment Polynomials for the Riemann Zeta Function
(University of Waterloo, 20090121)In this thesis we calculated the coefficients of moment polynomials of the Riemann zeta function for k= 4, 5, 6...13 using cubic acceleration, which is an improved method from quadratic acceleration. We then numerically ... 
Algebraic characterization of multivariable dynamics
(University of Waterloo, 20090326)Let X be a locally compact Hausdorﬀ space along with n proper continuous maps σ = (σ1 , · · · , σn ). Then the pair (X, σ) is called a dynamical system. To each system one can associate a universal operator algebra called ... 
Operator Theoretic Methods in NevanlinnaPick Interpolation
(University of Waterloo, 20090331)This Master's thesis will develops a modern approach to complex interpolation problems studied by Carath\'{e}odory, Nevanlinna, Pick, and Schur in the early $20^{th}$ century. The fundamental problem to solve is as follows: ... 
Equality of NumberTheoretic Functions over Consecutive Integers
(University of Waterloo, 20090430)This thesis will survey a group of problems related to certain numbertheoretic functions. In particular, for said functions, these problems take the form of when and how often they are equal over consecutive integers, n ... 
The O'NanScott Theorem for Finite Primitive Permutation Groups, and Finite Representability
(University of Waterloo, 20090806)The O'NanScott Theorem classifies finite primitive permutation groups into one of five isomorphism classes. This theorem is very useful for answering questions about finite permutation groups since four out of the five ... 
The Differential Geometry of Instantons
(University of Waterloo, 20090807)The instanton solutions to the YangMills equations have a vast range of practical applications in field theories including gravitation and electromagnetism. Solutions to Maxwell's equations, for example, are abelian gauge ... 
Degree Spectra of Unary relations on ω and ζ
(University of Waterloo, 20090813)Let X be a unary relation on the domain of (ω,<). The degree spectrum of X on (ω,<) is the set of Turing degrees of the image of X in all computable presentations of (ω,<). Many results are known about the types of degree ... 
On the Structure of Nonnegative Semigroups of Matrices
(University of Waterloo, 20090818)The results presented here are concerned with questions of decomposability of multiplicative semigroups of matrices with nonnegative entries. Chapter 1 covers some preliminary results which become useful in the remainder ... 
UseBounded Strong Reducibilities
(University of Waterloo, 20090820)We study the degree structures of the strong reducibilities $(\leq_{ibT})$ and $(\leq_{cl})$, as well as $(\leq_{rK})$ and $(\leq_{wtt})$. We show that any noncomputable c.e. set is part of a uniformly c.e. copy of ... 
Zigzags of Finite, Bounded Posets and Monotone NearUnanimity Functions and Jónsson Operations
(University of Waterloo, 20090925)We define the notion of monotone operations admitted by partially ordered sets, specifically monotone nearunanimity functions and Jónsson operations. We then prove a result of McKenzie's in [8] which states that if a ...