Browsing Mathematics (Faculty of) by Subject "Pure Mathematics"
Now showing items 1-20 of 66
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Algebraic characterization of multivariable dynamics
(University of Waterloo, 2009-03-26)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 ... -
The Algebraic Kirchberg--Phillips Conjecture for Leavitt Path Algebras
(University of Waterloo, 2015-09-18)This essay is meant to be an exposition of the theory of Leavitt path algebras and graph C*-algebras, with an aim to discuss some current classification questions. These two classes of algebras sit on opposite sides of a ... -
Amenability for the Fourier Algebra
(University of Waterloo, 2007-08-29)The Fourier algebra A(G) can be viewed as a dual object for the group G and, in turn, for the group algebra L1(G). It is a commutative Banach algebra constructed using the representation theory of the group, and from which ... -
Artin's Conjecture: Unconditional Approach and Elliptic Analogue
(University of Waterloo, 2008-08-11)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 ... -
Artin's Primitive Root Conjecture and its Extension to Compositie Moduli
(University of Waterloo, 2008-08-11)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 ... -
Asymptotic Distributions for Block Statistics on Non-crossing Partitions
(University of Waterloo, 2014-01-23)The set of non-crossing partitions was first studied by Kreweras in 1972 and was known to play an important role in combinatorics, geometric group theory, and free probability. In particular, it has a natural embedding ... -
Asymptotic Estimates for Rational Spaces on Hypersurfaces in Function Fields
(University of Waterloo, 2010-06-29)The ring of polynomials over a finite field has many arithmetic properties similar to those of the ring of rational integers. In this thesis, we apply the Hardy-Littlewood circle method to investigate the density of rational ... -
Branched Covering Constructions and the Symplectic Geography Problem
(University of Waterloo, 2008-08-15)We apply branched covering techniques to construct minimal simply-connected symplectic 4-manifolds with small χ_h values. We also use these constructions to provide an alternate proof that for each s ≥ 0, there exists a ... -
The Cohomology Ring of a Finite Abelian Group
(University of Waterloo, 2013-01-25)The cohomology ring of a finite cyclic group was explicitly computed by Cartan and Eilenberg in their 1956 book on Homological Algebra. It is surprising that the cohomology ring for the next simplest example, that of a ... -
Complexity of Classes of Structures
(University of Waterloo, 2013-08-28)The main theme of this thesis is studying classes of structures with respect to various measurements of complexity. We will briefly discuss the notion of computable dimension, while the breadth of the paper will focus on ... -
Contributions to the model theory of partial differential fields
(University of Waterloo, 2013-08-28)In this thesis three topics on the model theory of partial differential fields are considered: the generalized Galois theory for partial differential fields, geometric axioms for the theory of partial differentially closed ... -
Degree Spectra of Unary relations on ω and ζ
(University of Waterloo, 2009-08-13)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 ... -
The Differential Geometry of Instantons
(University of Waterloo, 2009-08-07)The instanton solutions to the Yang-Mills equations have a vast range of practical applications in field theories including gravitation and electro-magnetism. Solutions to Maxwell's equations, for example, are abelian gauge ... -
Digraph Algebras over Discrete Pre-ordered Groups
(University of Waterloo, 2013-01-25)This thesis consists of studies in the separate fields of operator algebras and non-associative algebras. Two natural operator algebra structures, A ⊗_max B and A ⊗_min B, exist on the tensor product of two given unital ... -
The Eberlein Compactification of Locally Compact Groups
(University of Waterloo, 2013-01-24)A compact semigroup is, roughly, a semigroup compactification of a locally compact group if it contains a dense homomorphic image of the group. The theory of semigroup compactifications has been developed in connection ... -
Elliptic Curves over Finite Fields and their l-Torsion Galois Representations
(University of Waterloo, 2015-09-08)Let $q$ and $\ell$ be distinct primes. Given an elliptic curve $E$ over $\mathbf{F}_q$, we study the behaviour of the 2-dimensional Galois representation of $\mathrm{Gal}(\overline{\mathbf{F}_q}/\mathbf{F}_q) \cong ... -
Equality of Number-Theoretic Functions over Consecutive Integers
(University of Waterloo, 2009-04-30)This thesis will survey a group of problems related to certain number-theoretic functions. In particular, for said functions, these problems take the form of when and how often they are equal over consecutive integers, n ... -
Families of Thue Inequalities with Transitive Automorphisms
(University of Waterloo, 2014-06-17)A family of parameterized Thue equations is defined as F_{t,s,...}(X, Y ) = m, m ∈ Z where F_{t,s,...}(X,Y) is a form in X and Y with degree greater than or equal to 3 and integer coefficients that are parameterized by ... -
The Fourier algebra of a locally trivial groupoid
(University of Waterloo, 2012-01-10)The goal of this thesis is to define and study the Fourier algebra A(G) of a locally compact groupoid G. If G is a locally compact group, its Fourier-Stieltjes algebra B(G) and its Fourier algebra A(G) were defined by ... -
Free semigroup algebras and the structure of an isometric tuple
(University of Waterloo, 2011-06-28)An n-tuple of operators V=(V_1,…,V_n) acting on a Hilbert space H is said to be isometric if the corresponding row operator is an isometry. A free semigroup algebra is the weakly closed algebra generated by an isometric ...