Pure Mathematics

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This is the collection for the University of Waterloo's Department of Pure Mathematics.

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Fourier Analysis of Local Fell Groups
(University of Waterloo, 2025-10-17) Vujičić, Aleksa
In 1972, Bagget showed that a separable locally compact group G is compact if and only if its dual space G^ is discrete. Curiously however, there are non-discrete groups whose duals are compact, and such a group was identified in the same paper. In a similar vein, one can define the “Fell group”, a semidirect product of the units of the p-adic integers 𝕆ₚ* acting via multiplication on the p-adic numbers ℚₚ, which Baggett shows is a noncompact group whose dual is not countable. This Fell group forms a basis of the novel work presented in this thesis. In Chapters 3 and 4, we look at the more general setting of the p-adic integers and numbers, known respectively as discrete valuation rings (DVRs) and local fields. We compile many known results about these objects, in order to generalise the theory of the Fell group to what we call the “local Fell groups”. While this is primarily background material from a variety of sources, there is additional work required to extend these results so that the theory is coherent and complete. We also briefly study finite-dimensional vector spaces over local fields. In Chapter 5, we analyse the Fourier and Fourier-Stieltjes algebras of these local Fell groups, which are of the form A ⋊ K for A abelian and K compact. These local Fell groups fall into a particular class of groups induced by actions for which the stabilisers are ‘minimal’, and we call such groups “cheap groups”. For groups of this form, we show that B(G) = B∞(G) ⊕ A(K) ∘ qK, where B∞(G) is the Fourier space generated by purely infinite representations. We also show that in group with countable open orbits (such as the local Fell groups) this simplifies further to B(G) = A(G) ⊕ A(K) ∘ qK. In an attempt to generalise this to higher-dimensional analogues, for which the above does not hold true, we examine the structure of B∞(G). In particular, we obtain a result for dimension two in terms of the projective space, and we show that this is in some sense the ‘best’ decomposition that can be made. Finally in Chapter 6, we study the amenability of the central Fourier algebra ZA(G) = A(G) ∩ ZL1(G) for G = 𝕆ₚ ⋊ 𝕆ₚ and its local field equivalents. We show that ZA(G) contains as a quotient the Fourier algebra of a hypergroup, which is induced by action of 𝕆ₚ* ↷ 𝕆ₚ. In general, if H is a hypergroup induced by an action K ↷ A, then there is a corresponding dual hypergroup H^ by the dual action. When this is the case, we show that these satisfy A(H) ≅ L1(H^), mimicking the classical result for groups. We also show that if H^ has orbits which ‘grow sufficiently large’, then via a result of Alaghmandan, the algebra L1(H^) is not amenable. In particular, this shows that ZA(G) is also not amenable, reaffirming a conjecture of Alaghmandan and Spronk.
The complexity of constraint system games with entanglement
(University of Waterloo, 2025-08-27) Mastel, Kieran
Constraint satisfaction problems (CSPs) are a natural class of decision problems where one must decide whether there is an assignment to variables that satisfies a given formula. Schaefer's dichotomy theorem and its extension to all alphabets due to Bulatov and Zhuk, shows that CSP languages are either efficiently decidable or NP-complete. It is possible to extend CSP languages to quantum assignments using the formalism of nonlocal games. The recent MIP*=RE theorem of Ji, Natarajan, Vidick, Wright, and Yuen shows that the complexity class MIP* of multiprover proof systems with entangled provers contains all recursively enumerable languages. As a consequence, general succinctly presented CSPs are RE-complete. We show that a wide range of NP-complete CSPs become RE-complete when the players are allowed entanglement, including all boolean CSPs, such as 3SAT and 3-colouring. This implies that these CSP languages remain undecidable even when not succinctly presented. Prior work of Grilo, Slofstra, and Yuen shows (via a technique called simulatable codes) that every language in MIP* has a perfect zero knowledge (PZK) MIP* protocol. The MIP*=RE theorem uses two-prover one-round proof systems. Hence, such systems are complete for MIP*. However, the construction in Grilo, Slofstra, and Yuen uses six provers, and there is no obvious way to get perfect zero knowledge with two provers via simulatable codes. This leads to a natural question: are there two-prover PZK-MIP* protocols for all of MIP*? In this work, we show that every language in MIP* has a two-prover one-round PZK-MIP* protocol, answering the question in the affirmative. For the proof, we use a new method based on a key consequence of the MIP*=RE theorem, which is that every MIP* protocol can be turned into a family of boolean constraint system (BCS) nonlocal games. This makes it possible to work with MIP* protocols as boolean constraint systems. In particular, it allows us to use a variant of a CSP due to Dwork, Feige, Kilian, Naor, and Safra that gives a classical MIP protocol for 3SAT with perfect zero knowledge. To prove our results, we develop a toolkit for analyzing the quantum soundness of reductions between constraint system (CS) games, which we expect to be useful more broadly. In this formalism, synchronous strategies for a nonlocal game correspond to tracial states on an algebra. We equip the algebra with a finitely supported weight that allows us to gauge the players' performance in the corresponding game using a weighted sum of squares. The soundness of our reductions hinges on guaranteeing that specific measurements for the players are close to commuting when their strategy performs well. To this end, we construct commutativity gadgets for all boolean CSPs and show that the commutativity gadget for graph 3-colouring due to Ji is sound against entangled provers. We define a broad class of CSPs that have simple commutativity gadgets. We show a variety of relations between the different ways of presenting CSPs as games. This toolkit also applies to commuting operator strategies, and our argument shows that every language with a commuting operator BCS protocol has a two prover PZK commuting operator protocol.
Exotic constructions on covers branched over hyperplane arrangements
(University of Waterloo, 2025-08-27) Harris, Robert
The purpose of this thesis is to study and construct geometric structures on 4-manifolds from a starting point of combinatorial objects. In part as a consequence of embedded surfaces and codimension two submanifolds coinciding in dimension four, many of the tools that are used in higher dimensions fail or are underwhelming when applied to 4-manifolds. For this reason, the development and advancement of techniques that are applicable to 4-manifolds are of particular interest and importance to low dimensional topologists. The general techniques of interest are those that either construct a 4-manifold in a novel way or find interesting embedded submanifolds. Consequently, we shall contribute to the study of 4-manifolds by investigating ways to construct 4-manifolds with positive signature in addition to describing construction methods that can guarantee the existence of algebraically interesting embedded symplectic submanifolds. In this thesis we investigate how the combinatorial data of line arrangements and the algebraic data of their complements in rational surfaces can be utilized to construct symplectic 4-manifolds with arbitrarily large signatures through the method of branched coverings. In general, we not only show that these line arrangements can be used to provide asymptotic bounds for the existence of symplectic 4-manifolds but we also show that for any line arrangement, there exist symplectic branched covers with sufficiently nice geometric and topological properties. Namely, we show they contain embedded symplectic surfaces which carry their fundamental group.
Generic Absoluteness in Set Theory
(University of Waterloo, 2025-08-26) Slavitch, Noah
In this thesis, we describe the state of the field of generic absoluteness, that is, the study of which statements retain their truth value under any forced generic extension or series of generic extensions of $V$, along with what large cardinal assumptions are necessary or sufficient to cause formulas to become generically absolute. We detail some of the tools used in generic absoluteness, such as homogeneously Suslin trees, direct limits of models, extenders, and the incredibly powerful Stationary Tower used for stationary tower forcing, along with the effect specific large cardinal properties have when combined with the stationary tower. We describe various landmark generic absoluteness results including the following: Shoenfield's $\Sigma_2^1$-Absoluteness theorem, which establishes that all $\Sigma_2^1$ statements retain their truth across all transitive models of $ZF$ containing $\omega_1$. We also examine more advanced generic absoluteness results in the projective hierarchy, such as the equiconsistency projective absoluteness with the existence of $\om$-many strong cardinals, and that projective absoluteness follows from $\om$-many Woodin cardinals. Further up the hierarchy, we describe $\Sigma_1^2$-absoluteness conditioned on $CH$, which states that assuming a proper class of measurable Woodin cardinals and $CH$ holding in $V$, if $\phi$ is a $\Sigma_1^2$ statement and $G$ is a generic filter for $\p$ some partial order such that and $V[G] \models CH$, then $V \models \phi$ if and only if $V[G] \models \phi$. We also cover $\Sigma_2^2$-absoluteness with generic $\lozenge$, a generic form of Jensen's $\lozenge$ principle, and its relation to Neeman Games in $\Omega$-logic. Furthermore, we cover the generic absoluteness properties of universally Baire sets, $uB$, such as a proper class of Woodin cardinals implying for $A \in uB$ and and any generic extension $V[G]$ an elementary embedding $j:L(A, \R) \to L(A_{G}, \R_{G})$ such that $j(A) = A$ and $j\restriction_\R = id$, along with the more recent axiom of Sealing, which seeks to 'seal' the truth of the universally Baire sets, along with examinations of the $uB$-powerset of $uB$, $uBp$, such as Weak Sealing for $uBp$, a strengthening of Sealing. We examine particular inner models of interest, such as Chang-like models and their unique generic absoluteness properties. Finally, we give proofs of the Shoenfield $\bSigma_2^1$ absoluteness theorem, and the fact that $\bSigma^1_3$ absoluteness follows from a class of measurable cardinals.
Partial C*-dynamical systems and the ideal structure of partial reduced crossed products
(University of Waterloo, 2025-08-25) Kroell, Larissa
In this thesis, we study C*-algebras stemming from partial C*-dynamical systems. We develop equivariant injective envelopes associated to such systems, which allow us to obtain canonical connections to enveloping actions as well as results on the ideal structure of partial crossed products. We extend the theory of equivariant injective envelopes pioneered by Hamana in the 1980s to partial C*-dynamical systems. To do so, we introduce the category of generalized unital partial actions by allowing for partial *-automorphisms acting on families of special hereditary subalgebras. Utilizing properties of injective envelopes and the notion of an injective unitization of partial C*-dynamical systems, we argue that it suffices to consider unital objects in our category. This also allows us to connect our theory to Abadie’s notion of enveloping actions leading to a canonical relationship of their G-injective envelopes. Utilizing properties of injective envelopes, we introduce novel non-triviality conditions for partial *-automorphisms inspired by global C*-dynamics. We contrast this notion with existing conditions in the literature. Lastly, we study a non-commutative generalization of stabilizer subgroups for pseudo-Glimm ideals. In particular, we show that for Glimm ideals in the G-injective envelope, these stabilizer subgroups are in fact amenable — a result which is crucial for our main theorems regarding the ideal structure of partial reduced crossed products. Finally, our main application of the theory of G-injective envelopes is a characterization of the ideal intersection property for partial C*-dynamical systems subject to a cohomological condition as a generalization of the result for global group actions. To state this generalization, we utilize the dynamical conditions introduced previously and generalize the notion of equivariant pseudo-expectations to partial C*-dynamical systems. We also give a sufficient intrinsic condition in terms of non-commutative uniformly recurrent partial subsystems utilizing pseudo-Glimm ideals. As a consequence of our results, we obtain a full characterization of the ideal intersection property for partial actions on commutative C*-algebras in terms of freeness of the partial action on the spectrum of the G-injective envelope.
Quadratic Forms over Global Fields
(University of Waterloo, 2025-08-19) Totani, Yash
This thesis is structured in two parts. The first part explores certain binary quadratic forms over the polynomial ring $\mathbb{F}_q[T]$. We derive explicit formulas for the number of representations of a polynomial and estimate their moments in two asymptotic scenarios: the large finite field limit, where the field size $q$ grows with fixed polynomial degree $n$, and the large degree limit, where the degree $n$ increases while $q$ remains fixed. In the former, we employ a Dirichlet series framework to extract asymptotic behavior, while in the latter, we apply a refined partitioning of the space of polynomials of fixed degree to obtain sharp asymptotic estimates. The second part investigates the representation of integers as sums of an even number of triangular numbers. Using the Hardy–Littlewood circle method, we sum the associated singular series and establish its convergence to the Eisenstein component of the expressions obtained using the theory of modular forms, which are expressed in terms of generalized divisor functions.
On the distributions of prime divisor counting functions
(University of Waterloo, 2025-07-09) Das, Sourabhashis
Let k and n be natural numbers. Let ω(n) denote the number of distinct prime factors of n, Ω(n) denote the total number of prime factors of n counted with multiplicity, and ω_k(n) denote the number of distinct prime factors of n that occur with multiplicity exactly k. Let h ≥ 2 be a natural number. We say that n is h-free if every prime factor of n has multiplicity less than h, and h-full if all prime factors of n have multiplicity at least h. In 1917, Hardy and Ramanujan proved that both ω(n) and Ω(n) have normal order log log n over the natural numbers. In this thesis, using a new counting argument, we establish the first and second moments of all these arithmetic functions over the sets of h-free and h-full numbers. We show that the normal order of ω(n) is log log n for both h-free and h-full numbers. For Ω(n), the normal order is log log n over h-free numbers and h log log n over h-full numbers. We also show that ω_1(n) has normal order log log n over h-free numbers, and ω_h(n) has normal order log log n over h-full numbers. Moreover, we prove that the functions ω_k(n) with 1 < k < h do not have a normal order over h-free numbers, and that the functions ω_k(n) with k > h do not have a normal order over h-full numbers. In their seminal work, Erdős and Kac showed that ω(n) is normally distributed over the natural numbers. Later, Liu extended this result by proving a subset generalization of the Erdős–Kac theorem. In this thesis, we leverage Liu’s framework to establish the Erdős–Kac theorem for both h-free and h-full numbers. Additionally, we show that ω_1(n) satisfies the Erdős–Kac theorem over h-free numbers, while ω_h(n) satisfies it over h-full numbers.
Characterizing cofree representations of SLn x SLm
(University of Waterloo, 2025-07-03) Kitt, Nicole
The study, and in particular classification, of cofree representations has been an interest of research for over 70 years. The Chevalley-Shepard-Todd Theorem provides a beautiful intrinsic characterization for cofree representations of finite groups. Specifically, this theorem says that a representation V of a finite group G is cofree if and only if G is generated by pseudoreflections. Up until the late 1900s, with the exception of finite groups, all of the existing classifications of cofree representations of a particular group consist of an explicit list, as opposed to an intrinsic group-theoretic characterization. However, in 2019, Edidin, Satriano, and Whitehead formulated a conjecture which intrinsically characterizes stable irreducible cofree representations of connected reductive groups and verified their conjecture for simple Lie groups. The conjecture states that for a stable irreducible representation V of a connected reductive group G, V is cofree if and only if V is pure. In comparison to the classifications comprised of a list of cofree representations, this conjecture can be viewed as an analogue of the Chevalley–Shepard–Todd Theorem for actions of connected reductive groups. The aim of this thesis is to further expand upon the techniques formulated by Edidin, Satriano, and Whitehead as a means to work towards the verification of the conjecture for all connected semisimple Lie groups. The main result of this thesis is the verification of the conjecture for stable irreducible representations V\otimes W of SLn x SLm satisfying dim V >= n^2 and dim W >= m^2. As the main group under study in this thesis is SLn x SLm, in Chapter 2 we provide a thorough analysis of the structure of irreducible representations of SLn from the view point of them being in one-to-one correspondence with irreducible representations of the Lie algebra Lie(SLn). The last section of Chapter 2 describes the general theory of irreducible representations of complex semisimple Lie algebras, with SLn as a toy example. In Chapter 3, we provide a brief introduction to Geometric Invariant Theory (GIT) and present the main results of the theory. We then discuss the history of GIT and the known characterization results for properties of representations that arise from GIT. In particular, we introduce cofree representations and the current classification results for cofree representations of certain classes of groups. We finish Chapter 3 by introducing pure representations and the conjecture formulated by Edidin, Satriano, and Whitehead. In Chapter 4, we verify that for all stable irreducible representations V\otimes W of SLn x SLm satisfying dim V >= n^2 and dim W >= m^2, V\otimes W is cofree if and only if V\otimes W if pure. This involves proving an upper bound on the dimension of pure representations of G_1 x G_2, with G_i connected reductive Lie groups. We also introduce two methods that can be used to show that a given representation is not pure. The last section in Chapter 4 discusses the difficulties and obstacles when trying to verify the conjecture for the remaining cases, namely when dim V < n^2 or dim W < m^2.
Synchronous and quantum games: Graphical and algebraic methods
(University of Waterloo, 2025-05-15) Goldberg, Adina
This is a mathematics thesis that contributes to an understanding of nonlocal games as formal objects. With that said, it does have connections to quantum physics and information theory. Nonlocal games are interactive protocols modelling two players attempting to win a game, by answering a pair of questions posed by the referee, who then checks whether their answers are correct. The players may have access to a shared quantum resource state and may use a pre-arranged strategy. Upon receiving their questions, they can measure this state, subject to some separation constraints, in order to select their answers. A famous example is the CHSH game of [Cla+69], where making use of shared quantum entanglement gives the players an advantage over using classical strategies. This thesis contributes to two separate questions arising in the study of synchronous nonlocal games: their algebraic properties, and their generalization to the quantum question-and-answer setting. Synchronous games are those in which players must respond with the same answer, given the same question. First, we study a synchronous version of the linear constraint game, where the players must attempt to convince the referee that they share a solution to a system of linear equations over a finite field. We give a correspondence between two different algebraic objects modelling perfect strategies for this game, showing one is isomorphic to a quotient of the other. These objects are the game algebra of [OP16] and the solution group algebra of [CLS17]. We also demonstrate an equivalence of these linear system games to graph isomorphism games on graphs parameterized by the linear system. Second, we extend nonlocal games to quantum games, in the sense that we allow the questions and answers to be quantum states of a bipartite system. We do this by quantizing the rule function, games, strategies, and correlations using a graphical calculus for symmetric monoidal categories applied to the category of finite dimensional Hilbert spaces. This approach follows the overall program of categorical quantum mechanics. To this generalized setting of quantum games, we extend definitions and results around synchronicity. We also introduce quantum versions of the classical graph homomorphism [MR16] and isomorphism [Ats+16] games, where the question and answer spaces are the algebras representing the “vertices” of quantum graphs, and we show that quantum tensor strategies realizing perfect correlations for these games correspond to morphisms between the underlying quantum graphs.
Co-Higgs Bundles and Poisson Geometry
(University of Waterloo, 2025-01-09) Ali Medina, Brady Miliwska; Moraru, Ruxandra
The relationship between co-Higgs bundles and holomorphic Poisson structures was first studied by Polishchuk, who established a one-to-one correspondence between rank 2 holomorphic co-Higgs fields $\Phi$ and Poisson structures on the total space of $\mathbb{P}(V)$. Matviichuk later extended this work by showing how to obtain co-Higgs fields from coisotropic Poisson structures $\sigma_{\Phi}$ on $\text{Tot}(V)$ and $\pi_{\Phi}$ on $\text{Tot}(\mathbb{P}(V))$ for rank greater than or equal to $3$. For diagonalizable holomorphic co-Higgs fields, Matviichuk introduced the notion of strong integrability as a necessary condition for the existence of these corresponding Poisson structures, focusing his analysis on open sets $\mathcal{U} \subset \mathbb{C}$ and $\mathbb{P}^1$. We extend these results in several directions. First, we prove that a Poisson structure on $\mathbb{P}(V)$ lifts to a quadratic Poisson structure on $V$ if and only if $p^* \omega_X \otimes p^*\det V^*$ admits a Poisson module structure, extending Matviichuk's result to the non-Calabi-Yau case. For co-Higgs fields over $\mathcal{U} \subset \mathbb{C}^n$, we classify all cases where $\sigma_\Phi$ is integrable, proving that $\Phi$ must be either a function multiple of a constant matrix or have only one nonzero column. We also show that for curves of genus $g \geq 1$, all co-Higgs fields are strongly integrable and characterize their explicit forms based on genus. Finally, we investigate the relationship between stability and Poisson geometry. We establish that $\Phi$-invariant subbundles correspond to Poisson subvarieties of $\mathbb{P}(V)$ and prove that the eigenvariety $E$ coincides with $\text{Zeroes}(\pi_\Phi)$. For rank 2 co-Higgs bundles over $\mathbb{P}^1$, we show the zero locus of $\pi$ decomposes into two components: a 2:1 cover corresponding to the spectral curve $S_\Phi$ and fibers over the zeros of $\Phi$. This decomposition provides geometric criteria for understanding stability. In particular, we prove that the spectral curve is irreducible and the zero locus consists only of the 2:1 cover component if and only if $(V,\Phi)$ is stable.
Solitons with continuous symmetries
(University of Waterloo, 2024-08-29) Lang, Christopher James; Charbonneau, Benoit
In 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.
Perspectives on the moduli space of torsion-free G2-structures
(University of Waterloo, 2024-08-27) Romshoo, Faisal; Karigiannis, Spiro
The 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.
Tsirelson's Bound and Beyond: Verifiability and Complexity in Quantum Systems
(University of Waterloo, 2024-08-23) Zhao, Yuming; Slofstra, William
This 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.
Mathematical Aspects of Higgs & Coulomb Branches
(University of Waterloo, 2024-08-21) Suter, Aiden; Webster, Ben
This 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.
Topics in the Geometry of Special Riemannian Structures
(University of Waterloo, 2024-07-26) Iliashenko, Anton; Karigiannis, Spiro
The 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.
Horospherical geometry: combinatorial algebraic stacks and approximating rational points
(University of Waterloo, 2024-07-17) Monahan, Sean; Satriano, Matthew
The 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.
Generalized GCDs as Applications of Vojta’s Conjecture
(University of Waterloo, 2023-08-31) Pyott, Nolan; McKinnon, David
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 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.
Notions of Complexity Within Computable Structure Theory
(University of Waterloo, 2023-08-28) MacLean, Luke; Csima, Barbara
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 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.
Mind the GAP: Amenability Constants and Arens Regularity of Fourier Algebras
(University of Waterloo, 2023-08-28) Sawatzky, John; Forrest, Brian; Wiersma, Matthew
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 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.
Divisibility of Discriminants of Homogeneous Polynomials
(University of Waterloo, 2023-08-28) Vukovic, Andrej; Wang, Jerry Xiaoheng; McKinnon, David
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 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 $d\geq 5$.

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