Pure Mathematics
This is the collection for the University of Waterloo's Department of Pure Mathematics.
Recent Submissions

On Asynchronous Interference Channels
(University of Waterloo, 20161115)In the first part of the thesis, a decentralized wireless network of separate transmitterreceiver pairs is studied where there is no central controller to assign the resources to the users and users do not explicitly ... 
The Enigma History and Mathematics
(University of Waterloo, 20161025)In this thesis we look at the solution to the German code machine, the Enigma machine. This solution was originally found by Polish cryptologists. We look at the solution from a historical perspective, but most importantly, ... 
COMPATIBILITY OF EXTENSIONS OF A COMBINATORIAL GEOMETRY
(University of Waterloo, 20160929)Two extensions of a geometry are compatible with each other if they have a common extension. If the given extensions are elementary, their compatibility can be intrinsically described in terms of their corresponding linear ... 
Classical Field Theory in the BV Formalism
(University of Waterloo, 20160920)This document is a review of the perspective on classical eld theories presented in [2] and [3]. 
Strong Morita Equivalence and Imprimitivity Theorems
(University of Waterloo, 20160915)The purpose of this thesis is to give an exposition of two topics, mostly following the books \cite{R & W} and \cite{Wil}. First, we wish to investigate crossed product $C^*$algebras in its most general form. Crossed ... 
Graph C*algebras and the Abelian Core
(University of Waterloo, 20160826)We may associate to a C*algebra to the directed graph E by associating edges to partial isometries and vertices to pairwise orthogonal Hilbert spaces which satisfy some additional conditions. Such graph algebras were first ... 
Some results on binary forms and counting rational points on algebraic varieties
(University of Waterloo, 20160817)In this thesis we study several problems related to the representation of integers by binary forms and counting rational points on algebraic varieties. In particular, we establish an asymptotic formula for $R_F(Z)$, the ... 
NevanlinnaPick Spaces and Dilations
(University of Waterloo, 20160712)The majority of this thesis is devoted to the study of NevanlinnaPick spaces and their multiplier algebras. These spaces are Hilbert function spaces in which a version of the NevanlinnaPick interpolation theorem from ... 
Exotic Group C*algebras, Tensor Products, and Related Constructions
(University of Waterloo, 20160615)Recently there has been a rejuvenated interest in exotic group C*algebras, i.e., group C*algebras which are "intermediate" to the full and reduced group C*algebras. This resurgence began with the introduction of the ... 
The Algebraic KirchbergPhillips Conjecture for Leavitt Path Algebras
(University of Waterloo, 20150918)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 ... 
Elliptic Curves over Finite Fields and their lTorsion Galois Representations
(University of Waterloo, 20150908)Let $q$ and $\ell$ be distinct primes. Given an elliptic curve $E$ over $\mathbf{F}_q$, we study the behaviour of the 2dimensional Galois representation of $\mathrm{Gal}(\overline{\mathbf{F}_q}/\mathbf{F}_q) \cong ... 
Some additive results in F_q[t]
(University of Waterloo, 20150827)We collected several results in integers of additive number theory and translated to results in F_q[t]. The results we collected are related to slim exceptional sets and the asymptotic formula in Waring's problem, a ... 
Semistable rank 2 coHiggs bundles over Hirzebruch surfaces
(University of Waterloo, 20150807)It has been observed by S. Rayan that the complex projective surfaces that potentially admit nontrivial examples of semistable coHiggs bundles must be found at the lower end of the EnriquesKodaira classification. ... 
Ktheory for C*Algebras and for Topological Spaces
(University of Waterloo, 20150427)Ktheory is the study of a collection of abelian groups that are invariant to C*algebras or to locally compact Hausdorff spaces. These groups are useful for distinguishing C*algebras and topological spaces, and they are ... 
On the local positivity of line bundles on algebraic surfaces
(University of Waterloo, 20150109)In this thesis, we define a measure for how `positive' an effective line bundle is at a point on a variety, and prove that it is linear on the numerically effective subcone on a smooth, complete, toric surface. 
Interpolation Sets For Compact Abelian Groups
(University of Waterloo, 20140904)We will study various properties of I_0 and \epsilonKronecker sets. We show that most infinite sets in the discrete dual group contain infinite interpolation sets. 
Generalized Complex Structures on Kodaira Surfaces
(University of Waterloo, 20140805)In this thesis, we study generalized complex structures on Kodaira surfaces, which are nonK\"ahler surfaces that admit holomorphic symplectic structures. We show, in particular, that the moduli space of eventype generalized ... 
Families of Thue Inequalities with Transitive Automorphisms
(University of Waterloo, 20140617)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 ... 
Some aspects of Cantor sets
(University of Waterloo, 20140328)For every positive, decreasing, summable sequence $a=(a_i)$, we can construct a Cantor set $C_a$ associated with $a$. These Cantor sets are not necessarily selfsimilar. Their dimensional properties and measures have been ... 
Asymptotic Distributions for Block Statistics on Noncrossing Partitions
(University of Waterloo, 20140123)The set of noncrossing 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 ...