# Pure Mathematics

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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).

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Item Solitons with continuous symmetries(University of Waterloo, 2024-08-29) Lang, Christopher JamesIn this thesis, we develop a framework for classifying symmetric points on moduli spaces using representation theory. We utilize this framework in a few case studies, but it has applications well beyond these cases. As a demonstration of the power of this framework, we use it to study various symmetric solitons: instantons as well as hyperbolic, singular, and Euclidean monopoles. Examples of these objects are hard to come by due to non-linear constraints. However, by applying this framework, we introduce a linear constraint, whose solution greatly simplifies the non-linear constraints. This simplification not only allows us to easily find a plethora of novel examples of these solitons, it also provides a framework for classifying such symmetric objects. As an example, by applying this method, we prove that the basic instanton is essentially the only instanton with two particular kinds of conformal symmetry. Additionally, we study the symmetry breaking of monopoles, a part of their topological classification. We prove a straightforward method for determining the symmetry breaking of a monopole and explicitly identify the symmetry breaking for all Lie groups with classical, simply Lie algebras. We also identify methods for doing the same for the exceptional simple Lie groups.Item Perspectives on the moduli space of torsion-free G2-structures(University of Waterloo, 2024-08-27) Romshoo, FaisalThe moduli space of torsion-free G₂-structures for a compact 7-manifold forms a non-singular smooth manifold. This was originally proved by Joyce. In this thesis, we present the details of this proof, modifying some of the arguments using new techniques. Next, we consider the action of gauge transformations on the space of torsion-free G₂-structures. This gives us a new framework to study the moduli space. We show that the torsion-free condition under the action of gauge transformations almost exactly corresponds to a particular 3-form, which arises naturally from the G₂-structure and the gauge transformation, being harmonic when we add a "gauge-fixing" condition. This may be the first step in giving an alternate proof of the fact that the moduli space forms a manifold in our framework of gauge transformations.Item Tsirelson's Bound and Beyond: Verifiability and Complexity in Quantum Systems(University of Waterloo, 2024-08-23) Zhao, YumingThis thesis employs operator-algebraic and group-theoretical techniques to study verifiability and complexity in bipartite quantum systems. A bipartite Bell scenario consists of two non-interacting parties, each can make several quantum measurements. If the two parties share an entangled quantum state, their measurement outcomes can be correlated in surprising ways. In general, we do not directly observe the entangled state and measurement operators (which are referred to as a quantum model), only the resulting statistics (which are referred to as a "correlation") --- there are typically many different models achieving a given correlation. Hence it is remarkable that some correlation has a unique quantum model. A correlation with this property is called a self-test. In the first part of this thesis, we give a new definition of self-testing in terms of abstract states on C*-algebras. We show that this operator-algebraic definition of self-testing is equivalent to the standard one and naturally extends to the commuting operator framework for nonlocal correlations. We also propose an operator-algebraic formulation of robust self-testing. For many nonlocal games of interest, including synchronous games and XOR games, their optimal strategies correspond to tracial states on the associated game algebras. We show that for such nonlocal games, our operator-algebraic definition of robust self-testing is equivalent to the standard one. This, in turn, yields an implication from the uniqueness of tracial states on C*-algebras to robust self-testing for nonlocal games. To address how to compute the robustness function of a self-test explicitly, we provide an enhanced version of a well-known stability result due to Gowers and Hatami and show how it completes a common argument used in self-testing. Self-testing provides a powerful tool for verifying quantum computations. Given that reliable cloud quantum computers are becoming closer to reality, the concept of verifiability of delegated quantum computations is of central interest. Many models have been proposed, each with specific strengths and weaknesses. In the second part of this thesis, we put forth a new model where the client trusts only its classical processing, makes no computational assumptions, and interacts with a quantum server in a single round. In addition, during a set-up phase, the client specifies the size n of the computation and receives an untrusted, off-the-shelf (OTS) device that is used to report the outcome of a single measurement. We show how to delegate polynomial-time quantum computations in the OTS model. This also yields an interactive proof system for all of QMA, which, furthermore, we show can be accomplished in statistical zero-knowledge. This provides the first relativistic (one-round), two-prover zero-knowledge proof systems for QMA. Mathematically, bipartite quantum measurement systems can be modeled by the tensor product of free *-algebras. The third part of this thesis studies the complexity of determining positivity of noncommutative polynomials in these algebras. An element of a *-algebra is said to be positive if it is non-negative in all *-representations. In many situations, we'd like to be able to decide whether an element is positive, and if it is, find a certificate of positivity. For noncommutative algebras, it is well known that an element of the free *-algebra is positive if and only if it is a sum of squares. This provides an effective way to determine if a given noncommutative *-polynomial is positive, by searching through sums of squares decompositions. We show that no such procedure exists for the tensor product of two free *-algebras: determining whether a *-polynomial of such an algebra is positive is coRE-hard. We also show that it is coRE-hard to determine whether a noncommutative *-polynomial is trace-positive. Our results hold if free *-algebras are replaced by other algebras that model quantum measurements, such as group algebras of free groups or free products of cyclic groups.Item Mathematical Aspects of Higgs & Coulomb Branches(University of Waterloo, 2024-08-21) Suter, AidenThis thesis contains results pertaining to different aspects of the 3d mirror symmetry between Higgs and Coulomb branches. In Chapter 2 we verify the 3d A-model Higgs branch conjecture formulated in [BF23] for SQED with n > 3 hypermultiplets. The conjecture claims that the associated variety of the boundary VOA for the 3d A-model is isomorphic to the Higgs branch of the physical theory. We demonstrate that the boundary VOA is L1(psl(n|n)) and show that its associated variety is the closure of the minimal nilpotent orbit, verifying the conjecture. In Chapter 3 we build on the work of [Web19a; Web22] by explicitly constructing a tilting generator for the derived category of coherent sheaves on T∗Gr(2, 4). This variety is the Coulomb branch for a quiver gauge theory and has functions described by a KRLW algebra. We achieve this result by constructing generators for modules over this diagrammatic algebra and identifying the coherent sheaves these correspond to.Item Topics in the Geometry of Special Riemannian Structures(University of Waterloo, 2024-07-26) Iliashenko, AntonThe thesis consists of two chapters. The first chapter is the paper named “Betti numbers of nearly G₂ and nearly Kähler 6-manifolds with Weyl curvature bounds” which is now in the journal Geometriae Dedicata. Here we use the Weitzenböck formulas to get information about the Betti numbers of compact nearly G₂ and compact nearly Kähler 6-manifolds. First, we establish estimates on two curvature-type self adjoint operators on particular spaces assuming bounds on the sectional curvature. Then using the Weitzenböck formulas on harmonic forms, we get results of the form: if certain lower bounds hold for these curvature operators then certain Betti numbers are zero. Finally, we combine both steps above to get sufficient conditions of vanishing of certain Betti numbers based on the bounds on the sectional curvature. The second chapter is the paper written with my supervisor Spiro Karigiannis named “A special class of k-harmonic maps inducing calibrated fibrations”, to appear in the journal Mathematical Research Letters. Here we consider two special classes of k-harmonic maps between Riemannian manifolds which are related to calibrated geometry, satisfying a first order fully nonlinear PDE. The first is a special type of weakly conformal map u:(Lᵏ,g)→(Mⁿ,h) where k≤n and α is a calibration k-form on M. Away from the critical set, the image is an α-calibrated submanifold of M. These were previously studied by Cheng–Karigiannis–Madnick when α was associated to a vector cross product, but we clarify that such a restriction is unnecessary. The second, which is new, is a special type of weakly horizontally conformal map u:(Mⁿ,h)→(Lᵏ,g) where n≥k and α is a calibration (n-k)-form on M. Away from the critical set, the fibres u⁻¹{u(x)} are α-calibrated submanifolds of M. We also review some previously established analytic results for the first class; we exhibit some explicit noncompact examples of the second class, where (M,h) are the Bryant–Salamon manifolds with exceptional holonomy; we remark on the relevance of this new PDE to the Strominger–Yau–Zaslow conjecture for mirror symmetry in terms of special Lagrangian fibrations and to the G₂ version by Gukov–Yau–Zaslow in terms of coassociative fibrations; and we present several open questions for future study.Item Horospherical geometry: combinatorial algebraic stacks and approximating rational points(University of Waterloo, 2024-07-17) Monahan, SeanThe purpose of this thesis is to explore and develop several aspects of the theory of horospherical geometry. Horospherical varieties are equipped with the action of a reductive algebraic group such that there is an open orbit whose points are stabilized by maximal unipotent subgroups. This includes the well-known classes of toric varieties and flag varieties. Using this orbit structure and representation-theoretic condition on the stabilizer, one can classify horospherical varieties using combinatorial objects called coloured fans. We give an overview of the main features of this classification through a new, accessible notational framework. There are two main research themes in this thesis. The first is the development of a combinatorial theory for horospherical stacks, vastly generalizing that for horospherical varieties. We classify horospherical stacks using combinatorial objects called stacky coloured fans, extending the theory of coloured fans. As part of this classification, we describe the morphisms of horospherical stacks in terms of maps between the stacky coloured fans, we completely describe the good moduli space of a horospherical stack, and we introduce a special, hands-on class of horospherical stacks called coloured fantastacks. The second major theme is using horospherical varieties to probe a conjecture in arithmetic geometry. In 2007, McKinnon conjectured that, for a given point on a projective variety, there is a sequence, lying on a curve, which best approximates this point. We verify a version of this conjecture for horospherical varieties, contingent on Vojta’s Main Conjecture, which says that there is a sequence, lying on a curve, which approximates the given point better than any Zariski dense sequence.Item Generalized GCDs as Applications of Vojta’s Conjecture(University of Waterloo, 2023-08-31) Pyott, NolanStarting 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 should hold. After some discussion on the general geometry and arithmetic needed to examine these questions, we take a quick look into how Vojta’s conjectures provide a generalization of our first result. In particular, we also note a case where this implies a similar equality on particular elliptic curves.Item Notions of Complexity Within Computable Structure Theory(University of Waterloo, 2023-08-28) MacLean, LukeThis 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 metatheorem of Antonio Montalb\'an's to simplify an otherwise difficult priority construction. We restrict our attention to linear orders, and ask if, given a computable linear order $\A$ with degree of categoricity $\boldsymbol{d}$, it is possible to construct a computable isomorphic copy of $\A$ such that the isomorphism achieves the degree of categoricity and furthermore, that we did not do this coding using a computable set of points chosen in advance. To ensure that there was no computable set of points that could be used to compute the isomorphism we are forced to diagonalize against all possible computable unary relations while we construct our isomorphic copy. This tension between trying to code information into the isomorphism and trying to avoid using computable coding locations, necessitates the use of a metatheorem. This work builds off of results obtained by Csima, Deveau, and Stevenson for the ordinals $\omega$ and $\omega^2$, and extends it to $\omega^\alpha$ for any computable successor ordinal $\alpha$. In chapter 3, which is joint work with Alvir and Csima, we study the Scott complexity of countable reduced Abelian $p$-groups. We provide Scott sentences for all such groups, and show some cases where this is an optimal upper bound on the Scott complexity. To show this optimality we obtain partial results towards characterizing the back-and-forth relations on these groups. In chapter 4, which is joint work with Csima and Rossegger, we study structures under enumeration reducibility when restricting oneself to only the positive information about a structure. We find that relations that can be relatively intrinsically enumerated from such information have a definability characterization using a new class of formulas. We then use these formulas to produce a structural jump within the enumeration degrees that admits jump inversion, and compare it to other notions of the structural jump. We finally show that interpretability of one structure in another using these formulas is equivalent to the existence of a positive enumerable functor between the classes of isomorphic copies of the structures.Item Mind the GAP: Amenability Constants and Arens Regularity of Fourier Algebras(University of Waterloo, 2023-08-28) Sawatzky, JohnThis 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 multiplication operations on the double dual $\cA^{**}$ introduced by Arens in 1971, and if these operations agree then the algebra $\cA$ is said to be Arens regular. We study Arens regularity of the closures of $A(G)$ in the multiplier and completely bounded multiplier norms, denoted $A_M(G)$ and $A_{cb}(G)$ respectively. We prove that if a nonzero closed ideal in $A_M(G)$ or $A_{cb}(G)$ is Arens regular then $G$ must be a discrete group. Amenable Banach algebras were first studied by B.E. Johnson in 1972. For an amenable Banach algebra $\cA$ we can consider its amenability constant $AM(\cA) \geq 1$. We are particularly interested in collections of amenable Banach algebras for which there exists a constant $\lambda > 1$ such that the values in the interval $(1,\lambda)$ cannot be attained as amenability constants. If $G$ is a compact group, then the central Fourier algebra is defined as $ZA(G) = ZL^1(G) \cap A(G)$ and endowed with the $A(G)$ norm. We study the amenability constant theory of $ZA(G)$ when $G$ is a finite group.Item Divisibility of Discriminants of Homogeneous Polynomials(University of Waterloo, 2023-08-28) Vukovic, AndrejWe 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 is divisible by $p^2$ for $p$ prime. We also prove several formulas about a certain polynomial $\Delta_d'$, first introduced in (Bhargava, Shankar, Wang, 2022), which behaves like an average over the partial derivatives of $\Delta_d$, the discriminant of degree $d$ polynomials. In particular, we prove that $\Delta_d'$ is irreducible when $d\geq 5$.Item Quantum superchannels on the space of quantum channels(University of Waterloo, 2023-06-20) Daly, Padraig ConorQuantum 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 is motivated by the study of quantum superchannels, which are maps whose input and output are quantum channels. Rather than taking the domain to be the space of all linear maps, as has been done in the past, we motivate and define superchannels by considering them as transformations on the operator system spanned by quantum channels. Extension theorems for completely positive maps allow us to apply the characterisation theorem for superchannels to this smaller set of maps. These extensions are non unique, showing two different superchannels act the same on all input quantum channels, and so this new definition on the smaller domain captures more precisely the action of superchannels as transformations between quantum channels. The non uniqueness can affect the auxilliary dimension needed for the characterisation as well as the tensor product of the superchannels.Item Finitary approximations of free probability, involving combinatorial representation theory(University of Waterloo, 2023-05-18) Campbell, JacobThis 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 random matrices and was initiated by Marcus, Spielman, and Srivastava in 2015. An alternate approach to some of their results for sums and products of randomly rotated matrices is presented, using techniques from combinatorial representation theory. Those techniques are then applied to the commutators of such matrices, uncovering the non-trivial but tractable combinatorics of immanants and Schur polynomials. The second is the connection between symmetric groups and random matrices, specifically the asymptotics of star-transpositions in the infinite symmetric group and the gaussian unitary ensemble (GUE). For a continuous family of factor representations of $S_{\infty}$, a central limit theorem for the star-transpositions $(1,n)$ is derived from the insight of Gohm-K\"{o}stler that they form an exchangeable sequence of noncommutative random variables. Then, the central limit law is described by a random matrix model which continuously deforms the well-known traceless GUE by taking its gaussian entries from noncommutative operator algebras with canonical commutation relations (CCR). This random matrix model generalizes results of K\"{o}stler and Nica from 2021, which in turn generalized a result of Biane from 1995.Item Vector Bundles on Non-Kähler Elliptic Surfaces(University of Waterloo, 2023-04-20) Boulter, EricThis 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 on Kodaira surfaces. We show that these moduli spaces are neither Kähler nor simply connected. Comparing to other known examples of compact holomorphic symplectic manifolds, this shows that if the moduli spaces are deformation equivalent to a known example, then they are Douady spaces of points on a Kodaira surface. The second problem deals with the interplay between singularities of moduli spaces of rank-2 vector bundles and existence of stable Vafa--Witten bundles on non-Kähler elliptic surfaces. By constructing a Vafa--Witten bundle in each filtrable Chern class of the elliptic surface when the base has genus g≥2, we show that such a moduli space is smooth as a ringed space if and only if every bundle in the moduli space is irreducible.Item Exactness and Noncommutative Convexity(University of Waterloo, 2022-09-02) Manor, NicholasThis 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 it relates to C*-exactness, i.e., the exactness of the reduced C*-algebra. It is known that these two properties coincide for discrete groups. The problem of whether this equivalence holds for general locally compact groups has recently been reduced by Cave and Zacharias to the case of totally disconnected unimodular groups. We prove that the equivalence does hold for the class of locally compact groups whose reduced C*-algebra admits a tracial state. In Chapter 2 we establish the dual equivalence of the category of generalized (i.e. potentially non-unital) operator systems and the category of pointed compact noncommutative (nc) convex sets, extending a result of Davidson and Kennedy. We then apply this dual equivalence to establish a number of results about generalized operator systems, some of which are new even in the unital setting. We develop a theory of quotients of generalized operator systems that extends the theory of quotients of unital operator systems. In addition, we extend results of Kennedy and Shamovich relating to nc Choquet simplices. We show that a generalized operator system is a C*-algebra if and only if its nc quasistate space is an nc Bauer simplex with zero as an extreme point, and we show that a second countable locally compact group has Kazhdan's property (T) if and only if for every action of the group on a C*-algebra, the set of invariant quasistates is the quasistate space of a C*-algebra. In Chapter 3 we expand on recent work of C.K. Ng about duals of operator systems. Call a nonunital operator system S dualizable if its dual S* embeds into B(H) via a complete order embedding and complete norm equivalence. Through the categorical duality of nonunital operator systems to pointed noncommutative convex sets discussed in Chapter 3, we characterize dualizability of S using geometric conditions on the nc quasistate space K in two ways. Firstly, in terms of an nc affine embedding of K into the nc unit ball of a Hilbert space satisfying a bounded positive extension property for nc affine functions. Secondly, we show that Ng's characterization is dual to a normality condition between K and its real nc cone. As applications, we obtain some permanence properties for dualizability and duality of mapping cones in quantum information, and give a new nc convex-geometric proof of Choi's Theorem.Item Dilation methods in semigroup dynamics and noncommutative convexity(University of Waterloo, 2022-08-25) Humeniuk, AdamSince 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 (non-selfadjoint) operator algebras. This thesis is a compilation of three research papers in operator algebras and noncommutative convexity linked by their use of dilations and operator systems. A semicrossed product is a non-selfadjoint operator algebra encoding the action of a semigroup on an operator or C*-algebra. In Chapter 2, we describe the C*-envelopes of a large class of semicrossed products. We prove that, when the positive cone of a discrete lattice ordered abelian group acts on a C*-algebra, the C*-envelope of the associated semicrossed product is a full corner of a crossed product by the whole group. After dilating the semigroup action to an automorphic action of the whole group using a direct product construction, we explicitly compute the Shilov ideal and therefore compute the C*-envelope. This generalizes a result of Davidson, Fuller, and Kakariadis from Z_+^n to the class of all discrete lattice ordered abelian groups. Chapters 3 and 4 present results in noncommutative (or ``matrix") convexity. By the noncommutative Kadison duality of Webster-Winkler and Davidson-Kennedy in the unital setting, and Kennedy-Kim-Manor in the nonunital setting, the category of compact noncommutative (nc) convex sets is dual to the category of operator systems. Thus nc convexity allows a new avenue to study operator systems geometrically. In Chapter 3, we prove a noncommutative generalization of the classical Jensen's Inequality for multivariable nc functions which are convex in each variable separately. The proof involves a sequence of dilations resembling a noncommutative analogue of Fubini's Theorem. This extends a single-variable nc version of Jensen's Inequality of Davidson and Kennedy. We demonstrate an application of the multivariable separate nc Jensen's Inequality to free semicircular systems in free probability. In Chapter 4, we discuss duals of operator systems. Recently, C.K. Ng obtained a nice duality theory for operator systems. Call a (possibly nonunital) operator system S dualizable if its dual S* embeds into B(H) via a complete order embedding and complete norm equivalence. Through the nonunital noncommutative Kadison duality of Kennedy, Kim, and Manor, we characterize dualizability of S using geometric conditions on its associated nc convex quasistate space K in two ways. Firstly, in terms of an nc affine embedding of K into the nc unit ball of a Hilbert space satisfying a certain extension property. Secondly, we show that Ng's characterization is dual to a normality condition between K and the nc cone R_+ K. As applications, we obtain some permanence properties for dualizability, and give a new nc convex-geometric proof of Choi's Theorem.Item Tracial and ideal structure of crossed products and related constructions(University of Waterloo, 2022-08-17) Ursu, DanIn 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 of these algebras, and we either extend work that was already done in the case of group C*-algebras and crossed products, or characterize simplicity altogether in the case of groupoid C*-algebras. The other is the structure of traces on these algebras, in particular in the case of crossed products. In the third chapter, we give complete descriptions of the tracial states on both the universal and reduced crossed products of a C*-dynamical system consisting of a unital C*-algebra A and a discrete group G. In particular, we also answer the question of when the tracial states on the crossed products are in canonical bijection with the G-invariant tracial states on A. This generalizes the unique trace property for discrete groups. The analysis simplifies greatly in various cases, for example when the conjugacy classes of the original group G are all finite, and in other cases gives previously known results, for example when the original C*-algebra A is commutative. We also obtain results and examples in the case of abelian groups that contradict existing results in the literature of Bédos and Thomsen. Specifically, we give a finite-dimensional counterexample, and provide a correction to the result of Thomsen. The fourth chapter is a short note on results in the von Neumann crossed product case that were never submitted for publication, and the author suspects might potentially be folklore, but cannot actually find anywhere. We extend the results on C*-crossed products from the third chapter to the case of von Neumann crossed products. In particular, we obtain results that characterize when a G-invariant normal tracial state on M has a unique normal tracial extension to the crossed product. As a consequence, we also characterize when such crossed products are finite factors. In the fifth chapter, we consider the notion of a plump subgroup that was recently introduced by Amrutam. This is a relativized version of Powers' averaging property, and it is known that Powers' averaging property for G is equivalent to C*-simplicity. With this in mind, we introduce a relativized notion of C*-simplicity, and show that for normal subgroups it is equivalent to plumpness, along with several other characterizations. For the sixth chapter, we prove a generalized version of Powers' averaging property that characterizes simplicity of reduced crossed products of a commutative unital C*-algebra C(X) and a discrete group G. As a consequence, we generalize results of Hartman and Kalantar on unique stationarity to the state space of the crossed product and to Kawabe's generalized space of amenable subgroups. This further lets us generalize a result of the first coauthor of the original publication of this chapter and Kalantar on simplicity of intermediate C*-algebras. For the seventh chapter, we characterise, in several complementary ways, étale groupoids with locally compact Hausdorff space of units whose essential groupoid C*-algebra has the ideal intersection property, assuming that the groupoid is either Hausdorff or sigma-compact. This leads directly to a characterisation of the simplicity of this C*-algebra which, for Hausdorff groupoids, agrees with the reduced groupoid C*-algebra. Specifically, we prove that the ideal intersection property is equivalent to the absence of essentially confined amenable sections of isotropy groups. For groupoids with compact space of units we moreover show that this is equivalent to the uniqueness of equivariant pseudo-expectations. A key technical idea underlying our results is a new notion of groupoid action on C*-algebras including the essential groupoid C*-algebra itself. For minimal groupoids, we further obtain a relative version of Powers averaging property. Examples arise from suitable group representations into simple groupoid C*-algebras. This is illustrated by the example of the quasi-regular representation of Thompson's group T with respect to Thompson's group F, which satisfies the relative Powers averaging property in the Cuntz algebra O2.Item On the Dynamical Wilf-Zeilberger Problem(University of Waterloo, 2022-08-15) Sun, YuxuanIn 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 upper bound N ≥ 1 such that whenever the first N terms of the two sequences agree pairwise, the two sequences agree term-by-term. Then, we give an algorithm that can be used to check whether two such sequences agree term-by-term. Our methods are mainly based on the theory of Chow rings of algebraic varieties.Item Integrality theorems for symmetric instantons(University of Waterloo, 2022-08-05) Whitehead, SpencerAnti-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 algebraic data via the celebrated ADHM correspondence. While much is known about the space of instantons, it is still difficult to give explicit examples of them, aside from classes of solutions provided by certain ansatze. The perspective in this thesis is that of symmetry: by introducing a suitable notion of a nice group action on an instanton, one expects that the condition of 'equivariance with respect to the symmetry group' to reduce the number of parameters present in the ADHM equations, thus allowing for the creation of solutions not visible to existing ansatze. Through this method of symmetry, a theory of symmetric instantons is developed and applied it in particular to the case of finite-energy ASD solutions on R4 with symmetry a compact subgroup of Spin(4). This theory acts as a framework in which previous work on symmetric instantons may be realized, and in particular allows for a number of '(algebraic) integrality' results for solutions to the symmetric instanton equations. Using the equivariant index theorem the 'SU(2) restriction' ansatz used in previous work is proved to give the only non-trivial class of solutions to the symmetric instanton equations for certain symmetry subgroups of SU(2). Additionally, a question of Allen and Sutcliffe on the existence of a non-trivial instanton with symmetries of the 600-cell occurring at a charge lower than that of the JNR bound of 119 is resolved in the negative. Finally, ADHM data for two new instantons symmetric under the binary icosahedral group occurring at charges 13 and 23 are presented, as well as the software package used to generate them.Item On Amenability Properties and Coideals of Quantum Groups(University of Waterloo, 2022-07-27) Anderson-Sackaney, BenjaminWe 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 offer a way of defining a “quasisubgroup” for locally compact quantum groups. Chapters 3, 4, and 5 are based on [3], [2], and [4] respectively. In Chapter 3, we establish the notion of a non-commutative hull of a left ideal of L1(Gb) for a discrete quantum group G. Non-commutative spectral synthesis is defined too, and is related to a certain Ditkin’s property at infinity, allowing for a description of the closed left ideals of L1(Gb) for many known compact quantum groups Gb from the literature. We apply this work to study weak∗ closed in ideals in the quantum measure algebra of coamenable compact quantum groups and certain closed ideals in L1(Gb) which admit bounded right approximate identities in relation to coamenability of Gb (Theorem 3.3.14). In Chapter 4, we study relative amenability and amenability of coideals of a discrete quantum group, and coamenability of coideals of a compact quantum group. Making progress towards answering a coideal version of a question of [65], we prove a duality result that generalizes Tomatsu’s theorem [122] (lemmas 4.4.14 and 4.1.9). Consequently, we characterize the reduced central idempotent states of a compact quantum group (Corollary 4.1.2). In Chapter 5, we study tracial and G-invariant states of discrete quantum groups. A key result here is that tracial idempotent states are equivalently G-invariant idempotent states (Proposition 5.3.12). A consequence is the resolution of an open problem in [96, 22] in the discrete case, namely that amenability of G is equivalent to nuclearity of and the existence of a tracial state on Cr(Gb) (Corollary 5.3.14). We also obtain that simplicity of Cr(Gb) implies no G-invariant states exist (Corollary 5.3.15). Finally, we prove existence and uniqueness results of traces in terms of the cokernel, HF , of the Furstenberg boundary and the canonical Kac quotient of Gb. In Chapter 6, we develop a notion of operator amenability and operator biflatness of the action of a completely contractive Banach algebra on another completely contractive Banach algebra. We study these concept on various actions defined for locally compact quantum groups and their quantum subgroups, and relate them to usual operator amenability and other related properties, including amenability, coamenability, and compactness.Item Brands of cumulants in non-commutative probability, and relations between them(University of Waterloo, 2022-07-06) Perales, DanielThe 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 addition of independent random variables. These notions of cumulants are a clear analogue of classic cumulants that linearize the addition of independent random variables. The family of set partitions P plays a key role in the combinatorial study of probability because several formulas relating moments to a brand of cumulants can be expressed as a sum indexed by set partitions. An intriguing fact observed in the recent research literature was that non-commutative cumulants sometimes have applications to other areas of non-commutative probability than the one they were designed for. Thus one wonders if there are nice combinatorial formulas to directly transition from one brand of cumulants to another. This thesis is concerned with the study of interrelations between different brands of cumulants associated to classic, Boolean, free and monotone independences. My development is naturally divided in four topics. For the first topic I focus on free cumulants, and I use them in the study of the distribution of the anti-commutator ab + ba of two free random variables a and b. This follows up on some questions raised a while ago by Nica and Speicher in the 1990’s. I next consider the notion of convolution in the framework of a family of lattices, which goes back to work of Rota and collaborators in the 1970’s. I focus on the lattices of non-crossing partitions NC and put into evidence a certain group of semi-multiplicative functions, which encapsulate the moment-cumulant formulas and the inter-cumulant transition formulas for several known brands of cumulants in non-commutative probability. I next extend my considerations to the framework of P, which allows us to include the classical cumulants in the picture. Moreover, I do this in a more structured fashion by introducing a notion of iterative family of partitions S in P. This unifies the considerations related to NC and P, and also provides a whole gallery of new examples. Finally, it is important to make the observation that the group of semi-multiplicative functions which arise in connection to an iterative family is in fact a dual structure. Namely, it appears as the group of characters for a certain Hopf algebra over the partitions in S. In particular, the antipode promises to serve as a universal inversion tool for moment-cumulant formulas and for transition formulas between different brands of cumulants.