• Generalized GCDs as Applications of Vojta’s Conjecture 

  Pyott, Nolan (University of Waterloo, 2023-08-31)

  Starting with an analysis of the result that for any coprime integers a and b, and some ϵ > 0, we have eventually that gcd(a^n − 1,b^n − 1) < a^ϵn holds for all n, we are motivated to look for geometric reasons why this ...

• Notions of Complexity Within Computable Structure Theory 

  MacLean, Luke (University of Waterloo, 2023-08-28)

  This thesis covers multiple areas within computable structure theory, analyzing the complexities of certain aspects of computable structures with respect to different notions of definability. In chapter 2 we use a new ...

• Mind the GAP: Amenability Constants and Arens Regularity of Fourier Algebras 

  Sawatzky, John (University of Waterloo, 2023-08-28)

  This thesis aims to investigate properties of algebras related to the Fourier algebra $A(G)$ and the Fourier-Stieltjes algebra $B(G)$, where $G$ is a locally compact group. For a Banach algebra $\cA$ there are two natural ...

• Divisibility of Discriminants of Homogeneous Polynomials 

  Vukovic, Andrej (University of Waterloo, 2023-08-28)

  We prove several square-divisibility results about the discriminant of homogeneous polynomials of arbitrary degree and number of variables, when certain coefficients vanish, and give characterizations for when the discriminant ...

• Quantum superchannels on the space of quantum channels 

  Daly, Padraig Conor (University of Waterloo, 2023-06-20)

  Quantum channels, defined as completely-positive and trace-preserving maps on matrix algebras, are an important object in quantum information theory. In this thesis we are concerned with the space of these channels. This ...

• Finitary approximations of free probability, involving combinatorial representation theory 

  Campbell, Jacob (University of Waterloo, 2023-05-18)

  This thesis contributes to two theories which approximate free probability by finitary combinatorial structures. The first is finite free probability, which is concerned with expected characteristic polynomials of various ...

• Vector Bundles on Non-Kähler Elliptic Surfaces 

  Boulter, Eric (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 ...

• Exactness and Noncommutative Convexity 

  Manor, Nicholas (University of Waterloo, 2022-09-02)

  This thesis studies two separate topics in connection to operator systems theory: the dynamics of locally compact groups, and noncommutative convex geometry. In Chapter 1 we study exactness of locally compact groups as ...

• Dilation methods in semigroup dynamics and noncommutative convexity 

  Humeniuk, Adam (University of Waterloo, 2022-08-25)

  Since seminal work of Stinespring, Arveson, and others, dilation theory has been an indispensable tool for understanding operator algebras. Dilations are fundamental to the representation theory of operator systems and ...

• Tracial and ideal structure of crossed products and related constructions 

  Ursu, Dan (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 ...

• On the Dynamical Wilf-Zeilberger Problem 

  Sun, Yuxuan (University of Waterloo, 2022-08-15)

  In this paper, we give an algorithmic solution to a dynamical analog of the problem of certifying combinatorial identities by Wilf-Zeilberger pairs. Given two sequences generated in a dynamical setting, we calculate an ...

• Integrality theorems for symmetric instantons 

  Whitehead, Spencer (University of Waterloo, 2022-08-05)

  Anti-self-dual (ASD) instantons on R4 are connections A on SU(N)-vector bundles with finite L2-norm and curvature satisfying the ASD equation. Solutions to this non-linear partial differential equation correspond to certain ...

• On Amenability Properties and Coideals of Quantum Groups 

  Anderson-Sackaney, Benjamin (University of Waterloo, 2022-07-27)

  We study amenability type properties of locally compact quantum groups and subobjects of quantum groups realized as submodules of their von Neumann algebras. An important class of such subobjects are the coideals, which ...

• Brands of cumulants in non-commutative probability, and relations between them 

  Perales, Daniel (University of Waterloo, 2022-07-06)

  The study of non-commutative probability revolves around the different notions of independeces, such as free, Boolean and monotone. To each type of independence one can associate a notion of cumulants that linearize the ...

• Dispersing representations of semi-simple subalgebras of complex matrices 

  Marcoux, Laurent W.; Radjavi, Heydar; Zhang, Yuanhang (Elsevier, 2022-06-01)

  In this paper we consider the problem of determining the maximum dimension of P?(A!B)P, where A and B are unital, semi-simple subalgebras of the set Mn of n⇥n complex matrices, and P 2 M2n is a projection of rank n. We ...

• OFF-DIAGONAL CORNERS OF SUBALGEBRAS OF L(Cn) 

  Marcoux, Laurent W.; Radjavi, Heydar; Zhang, Yuanhang (Elsevier, 2020-12-15)

  Let n ∈ N, and consider Cn equipped with the standard inner product. Let A ⊆ L(Cn) be a unital algebra and P ∈ L(Cn) be an orthogonal projection. The space L := P ⊥A|ran P is said to be an off-diagonal corner of A, and L ...

• MATRIX ALGEBRAS WITH A CERTAIN COMPRESSION PROPERTY I 

  Cramer, Zachary; Marcoux, Laurent W.; Radjavi, Heydar (Elsevier, 2021-07-15)

  An algebra A of n × n complex matrices is said to be projection compressible if P AP is an algebra for all orthogonal projections P ∈ Mn(C). Analogously, A is said to be idempotent compressible if EAE is an algebra for all ...

• ALGEBRAIC DEGREE IN SPATIAL MATRICIAL NUMERICAL RANGES OF LINEAR OPERATORS 

  Bernik, Janez; Livshits, Leo; MacDonald, Gordon W.; Marcoux, Laurent W.; Mastnak, Mitja; Radjavi, Heydar (American Mathematical Society, 2021-07-20)

  We study the maximal algebraic degree of principal ortho-compressions of linear operators that constitute spatial matricial numerical ranges of higher order. We demonstrate (amongst other things) that for a (possibly ...

• OPERATORS WHICH ARE POLYNOMIALLY ISOMETRIC TO A NORMAL OPERATOR 

  Marcoux, Laurent W.; Zhang, Yuanhang (American Mathematical Society, 2020-01-15)

  Let H be a complex, separable Hilbert space and B(H) denote the algebra of all bounded linear operators acting on H. Given a unitarily-invariant norm k · ku on B(H) and two linear operators A and B in B(H), we shall say ...

• Normal operators with highly incompatible off-diagonal corners 

  Marcoux, Laurent W.; Radjavi, Heydar; Zhang, Yuanhang (Polish Academy of Sciences, 2020-05-25)

  Let H be a complex, separable Hilbert space, and B(H) denote the set of all bounded linear operators on H. Given an orthogonal projection P∈B(H) and an operator D∈B(H), we may write D=[D1D3D2D4] relative to the decomposition ...

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