Browsing Mathematics (Faculty of) by Subject "Pure Mathematics"
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Generalisations of Roth's theorem on finite abelian groups
(University of Waterloo, 20121218)Roth's theorem, proved by Roth in 1953, states that when A is a subset of the integers [1,N] with A dense enough, A has a three term arithmetic progression (3AP). Since then the bound originally given by Roth has been ... 
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 ... 
Harmonic analysis of Rajchman algebras
(University of Waterloo, 20100831)Abstract harmonic analysis is mainly concerned with the study of locally compact groups, their unitary representations, and the function spaces associated with them. The Fourier and FourierStieltjes algebras are two of ... 
HigherDimensional Kloosterman Sums and the Greatest Prime Factor of Integers of the Form a_1a_2\cdots a_{k+1}+1
(University of Waterloo, 20070829)We consider the greatest prime factors of integers of certain form. 
Integral Moments of Quadratic Dirichlet Lfunctions: A Computational Perspective
(University of Waterloo, 20100427)In recent years, the moments of Lfunctions has been a topic of growing interest in the field of analytic number theory. New techniques, including applications of Random Matrix Theory and multiple Dirichlet series, have ... 
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. 
A kConjugacy Class Problem
(University of Waterloo, 20070907)In any group G, we may extend the definition of the conjugacy class of an element to the conjugacy class of a ktuple, for a positive integer k. When k = 2, we are forming the conjugacy classes of ordered pairs, when k ... 
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 ... 
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 ... 
The LangTrotter conjecture for Drinfeld modules
(University of Waterloo, 20110822)In 1986, Gupta and Murty proved the LangTrotter conjecture in the case of elliptic curves having complex multiplication, conditional on the generalized Riemann hypothesis. That is, given a nontorsion point P∈E(ℚ), they ... 
Lehmer Numbers with at Least 2 Primitive Divisors
(University of Waterloo, 20071024)In 1878, Lucas \cite{lucas} investigated the sequences $(\ell_n)_{n=0}^\infty$ where $$\ell_n=\frac{\alpha^n\beta^n}{\alpha\beta},$$ $\alpha \beta$ and $\alpha+\beta$ are coprime integers, and where $\beta/\alpha$ is ... 
Lower order terms of moments of Lfunctions
(University of Waterloo, 20110617)<p>Given a positive integer k, Conrey, Farmer, Keating, Rubinstein and Snaith conjectured a formula for the asymptotics of the kth moments of the central values of quadratic Dirichlet Lfunctions. The conjectured formula ... 
Maximal ideal space techniques in nonselfadjoint operator algebras
(University of Waterloo, 20130426)The following thesis is divided into two main parts. In the first part we study the problem of characterizing algebras of functions living on analytic varieties. Specifically, we consider the restrictions M_V of the ... 
Maximal Operators in R^2
(University of Waterloo, 20070813)A maximal operator over the bases $\mathcal{B}$ is defined as \[Mf(x) = \sup_{x \in B \in \mathcal{B}} \frac{1}{B}\int_B f(y)dy. \] The boundedness of this operator can be used in a number of applications including ... 
Mean Curvature Flow in Euclidean spaces, Lagrangian Mean Curvature Flow, and Conormal Bundles
(University of Waterloo, 20110810)I will present the mean curvature flow in Euclidean spaces and the Lagrangian mean curvature flow. We will first study the mean curvature evolution of submanifolds in Euclidean spaces, with an emphasis on the case of ... 
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 ... 
The MordellLang Theorem from the Zilber Dichotomy
(University of Waterloo, 20100430)We present a largely selfcontained exposition of Ehud Hrushovski's proof of the function field MordellLang conjecture beginning from the Zilber Dichotomy for differentially closed fields and separably closed fields. Our ... 
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 ... 
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 ... 
On a Question of Wintner Concerning the Sequence of Integers Composed of Primes from a Given Set
(University of Waterloo, 20070927)We answer to a Wintner's question concerning the sequence of integers composed of primes from a given set. The results generalize and develop the answer to Wintner’s question due to Tijdeman.