Pure Mathematics
This is the collection for the University of Waterloo's Department of Pure Mathematics .
Research outputs are organized by type (eg. Master Thesis, Article, Conference Paper).
Waterloo faculty, students, and staff can contact us or visit the UWSpace guide to learn more about depositing their research.
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Differential Operators on Manifolds with G2Structure
(University of Waterloo, 20201216)In this thesis, we study differential operators on manifolds with torsionfree G2structure. In particular, we use an identification of the spinor bundle S of such a manifold M with the bundle R ⊕ T*M to reframe statements ... 
Sparse Automatic Sets
(University of Waterloo, 20201126)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 ... 
Coefficient spaces arising from locally compact groups
(University of Waterloo, 20200817)This thesis studies two disjoint topics involving coefficient spaces and algebras associated to locally compact groups. First, Chapter 3 investigates the connection between amenability and compactness conditions on locally ... 
Recurrence in Algebraic Dynamics
(University of Waterloo, 20200728)The Dynamical MordellLang Conjecture states that if a polynomial orbit has infinite intersection with a closed set in an algebraic variety, then the intersection must occur periodically. Although this problem is unsolved ... 
Some Problems in Multiplicative and Additive Number Theory
(University of Waterloo, 20200727)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 ... 
Applications of Operator Systems in Dynamics, Correlation Sets, and Quantum Graphs
(University of Waterloo, 20200724)The recent works of KalantarKennedy, KatsoulisRamsey, Ozawa, and DykemaPaulsen have demonstrated that many problems in the theory of operator algebras and quantum information can be approached by looking at various ... 
Topics in G₂ geometry and geometric flows
(University of Waterloo, 20200515)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 ... 
On Hopf Ore Extensions and Zariski Cancellation Problems
(University of Waterloo, 20200429)In this thesis, we investigate Ore extensions of Hopf algebras and the Zariski Cancellation problem for noncommutative rings. In particular, we improve upon the existing conditions for when $T=R[x; \sigma, \delta]$ is a ... 
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 finitedimensional vector ... 
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 ... 
Reducibility of operator semigroups and values of vector states
(Springer, 20170801)Let S be a multiplicative semigroup of bounded linear operators on a complex Hilbert space H, and let Ω be the range of a vector state on S so that Ω = {⟨Sξ, ξ⟩ : S ∈ S} for some fixed unit vector ξ ∈ H. We study the ... 
Compact ideals and rigidity of representations for amenable operator algebras
(Polish Academy of Sciences, 2019)We examine rigidity phenomena for representations of amenable operator algebras which have an ideal of compact operators. We establish that a generalized version of Kadison’s conjecture on completely bounded homomorphisms ... 
Residual finite dimensionality and representations of amenable operator algebras
(Elsevier, 20190415)We consider a version of a famous open problem formulated by Kadison, asking whether bounded representations of operator algebras are automatically completely bounded. We investigate this question in the context of amenable ... 
Abelian, amenable operator algebras are similar to C∗ algebras
(Duke University Press, 201612)Suppose that H is a complex Hilbert space and that ℬ(H) denotes the bounded linear operators on H. We show that every abelian, amenable operator algebra is similar to a C∗algebra. We do this by showing that if 𝒜⊆ℬ(H) is ... 
Ranges of vector states on irreducible operator semigroups
(Springer, 2016)Let 𝜑 be a linear functional of rank one acting on an irreducible semigroup S of operators on a finite or infinitedimensional Hilbert space. It is a wellknown and simple fact that the range of 𝜑 cannot be a singleton. ... 
On selfadjoint extensions of semigroups of partial isometries
(American Mathematical Society, 2016)Let S be a semigroup of partial isometries acting on a complex, infinite dimensional, separable Hilbert space. In this paper we seek criteria which will guarantee that the selfadjoint semigroup T generated by S consists ... 
Linear preservers of polynomial numerical hulls of matrices
(Elsevier, 20190815)Let Mn be the algebra of all n × n complex matrices, 1 ≤ k ≤ n − 1 be an integer, and φ : Mn −→ Mn be a linear operator. In this paper, it is shown that φ preserves the polynomial numerical hull of order k if and only if ... 
Compressible Matrix Algebras and the Distance from Projections to Nilpotents
(University of Waterloo, 20191115)In this thesis we address two problems from the fields of operator algebras and operator theory. In our first problem, we seek to obtain a description of the unital subalgebras $\mathcal{A}$ of $\mathbb{M}_n(\mathbb{C})$ ... 
Algebraic Approaches to State Complexity of Regular Operations
(University of Waterloo, 20191015)The state complexity of operations on regular languages is an active area of research in theoretical computer science. Through connections with algebra, particularly the theory of semigroups and monoids, many problems ... 
Rational approximations on smooth rational surfaces
(University of Waterloo, 20190809)In this thesis, we study a conjecture made by D. McKinnon about rational approximations to rational points in algebraic varieties. The conjecture states that if a rational point P on a variety X lies on a rational curve, ...