Browsing Pure Mathematics by Title
Now showing items 143-162 of 165
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Some Problems in Multiplicative and Additive Number Theory
(University of Waterloo, 2020-07-27)In this thesis, we obtain several results in number theory. Let $k\geqslant 1$ be a natural number and $\omega_k(n)$ denote the number of distinct prime factors of a natural number $n$ with multiplicity $k$. We estimate ... -
Some results on binary forms and counting rational points on algebraic varieties
(University of Waterloo, 2016-08-17)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 ... -
Sparse Automatic Sets
(University of Waterloo, 2020-11-26)The theory of automatic sets and sequences arises naturally in many different areas of mathematics, notably in the study of algebraic power series in positive characteristic, due to work of Christol, and in Derksen's ... -
A spatial version of Wedderburn’s Principal Theorem
(Taylor & Francis, 2015)In this article we verify that ‘Wedderburn’s Principal Theorem’ has a particularly pleasant spatial implementation in the case of cleft subalgebras of the algebra of all linear transformations on a finite-dimensional vector ... -
Spectral Analysis of Laplacians on Certain Fractals
(University of Waterloo, 2007-12-06)Surprisingly, Fourier series on certain fractals can have better convergence properties than classical Fourier series. This is a result of the existence of gaps in the spectrum of the Laplacian. In this work we prove a ... -
Strong Morita Equivalence and Imprimitivity Theorems
(University of Waterloo, 2016-09-15)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 ... -
A survey of Roth's Theorem on progressions of length three
(University of Waterloo, 2011-12-19)For any finite set B and a subset A⊆B, we define the density of A in B to be the value α=|A|/|B|. Roth's famous theorem, proven in 1953, states that there is a constant C>0, such that if A⊆{1,...,N} for a positive integer ... -
A Survey of the Classification of Division Algebras
(University of Waterloo, 2008-09-29)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 ... -
Symmetrically-Normed Ideals and Characterizations of Absolutely Norming Operators
(University of Waterloo, 2018-08-23)The primary objective of this thesis is two fold: first, it is devoted to the study of absolutely norming operators (with respect to various arbitrary symmetric norms on $\mathcal{B}(\mathcal{H})$) with an eye towards the ... -
Techniques in operator algebras: classification, dilation and non-commutative boundary theory
(University of Waterloo, 2017-08-11)In this thesis we bring together several techniques in the theory of non-self-adjoint operator algebras and operator systems. We begin with classification of non-self-adjoint and self-adjoint operator algebras constructed ... -
Topics in G₂ geometry and geometric flows
(University of Waterloo, 2020-05-15)We study three different problems in this thesis, all related to G₂ structures and geometric flows. In the first problem we study hypersurfaces in a nearly G₂ manifold. We define various quantities associated to such a ... -
Tracial and ideal structure of crossed products and related constructions
(University of Waterloo, 2022-08-17)In this thesis, we concern ourselves with asking questions about the basic structure of group C*-algebras, crossed products, and groupoid C*-algebras. Specifically, we are concerned with two main topics. One is the simplicity ... -
Trisections of non-orientable 4-manifolds
(University of Waterloo, 2021-08-24)Broadly, this thesis is concerned with trying to understand 4-manifolds through 3-dimensional techniques. From the point of view of smooth manifolds, dimension four is quite unique; one striking illustration of this is the ... -
Tsirelson's problems and entanglement breaking rank
(University of Waterloo, 2018-08-13)There are two main themes of this thesis. In the first part, we study the various sets of correlations arising in the study of non-local games and Tsirelson's problems. In the second part we introduce the notion of ... -
Unitary Correlation Sets and their Applications
(University of Waterloo, 2019-07-04)We relate Connes' embedding problem in operator algebras to the Brown algebra Unc(n), which is defined as the universal unital C*-algebra generated by the entries of an n x n unitary matrix. In particular, we show that the ... -
Universal bounds for positive matrix semigroups
(Polish Academy of Sciences, 2016)We show that any compact semigroup of positive n×n matrices is similar (via a positive diagonal similarity) to a semigroup bounded by n√. We give examples to show this bound is best possible. We also consider the effect ... -
Upper Bounds for the Number of Integral Points on Quadratic Curves and Surfaces
(University of Waterloo, 2010-04-27)We are interested in investigating the number of integral points on quadrics. First, we consider non-degenerate plane conic curves defined over Z. In particular we look at two types of conic sections: hyperbolas with ... -
Use-Bounded Strong Reducibilities
(University of Waterloo, 2009-08-20)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 ... -
Vector Bundles on Non-Kähler Elliptic Surfaces
(University of Waterloo, 2023-04-20)This thesis studies two problems relating to moduli spaces of vector bundles on non-Kähler elliptic surfaces. The first project involves the holomorphic symplectic structure on smooth and compact moduli spaces of sheaves ... -
Von Neumann Algebras for Abstract Harmonic Analysis
(University of Waterloo, 2008-08-28)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 ...