Theses - Pure Mathematics and Mathematical Statistics

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Equivariant line bundles with connection on the Drinfeld upper half-space Ω⁽²⁾
Zhu, Yiyue
Ardakov and Wadsley developed a theory of $\mathscr{D}$-modules on rigid analytic spaces and established a Beilinson-Bernstein style localisation theorem for coadmissible modules over the locally analytic distribution algebra. Using this theory, they obtained admissible locally analytic representations of GL₂ by studying equivariant line bundles with connection on the Drinfeld half-plane Ω⁽¹⁾. In this thesis, we will follow the idea of Ardakov-Wadsley and extend their techniques to GL₃ by studying the Drinfeld upper half-space Ω⁽²⁾ of dimension 2.
Open Access
Results in Ramsey theory and extremal graph theory
Metrebian, Robert
In this thesis, we study several combinatorial problems in which we aim to find upper or lower bounds on a certain quantity relating to graphs. The first problem is in Ramsey theory, while the others are in extremal graph theory. In Chapter 2, which is joint work with Vojtěch Dvořák, we consider the Ramsey number $R(F_n)$ of the fan graph $F_n$, a graph consisting of $n$ triangles which all share a common vertex. Chen, Yu and Zhao showed that $\frac{9}{2}n-5 \leq R(F_n) \leq \frac{11}{2}n+6$. We build on the techniques that they used to prove the upper bound of $\frac{11}{2}n+6$, and adopt a more detailed approach to examining the structure of the graph. This allows us to improve the upper bound to $\frac{31}{6}n+15$. In Chapter 3, we work on a problem in graph colouring. Petruševski and Škrekovski recently introduced the concept of odd colouring, and the odd chromatic number of a graph, which is the smallest number of colours in an odd colouring of that graph. They showed that planar graphs have odd chromatic number at most $9$, and this bound was improved to $8$ by Petr and Portier. We consider the odd chromatic number of toroidal graphs, which are graphs that embed into a torus. By using the discharging method, along with detailed analysis of a remaining special case, we show that toroidal graphs have odd chromatic number at most $9$. In Chapter 4, which is joint work with Victor Souza, we consider a problem in the hypercube graph $Q_n$. Huang showed that every induced subgraph of the hypercube with $2^{n-1}+1$ vertices has maximum degree at least $\lceil\sqrt{n}\rceil$, which resolved a major open problem in computer science known as the Sensitivity Conjecture. Huang asked whether analogous results could be obtained for larger induced subgraphs. For induced subgraphs of $Q_n$ with $p2^n$ vertices, we find a simple lower bound that holds for all $p$, and substantially improve this bound in the range $\frac{1}{2} < p < \frac{2}{3}$ by analysing the local structure of the graph. We also find constructions of subgraphs achieving the simple lower bound asymptotically when $p = 1-\frac{1}{r}$.
Open Access
Semiparametric Methods for Two Problems in Causal Inference using Machine Learning
Klyne, Harvey
Scientific applications such as personalised (precision) medicine require statistical guarantees on causal mechanisms, however in many settings only observational data with complex underlying interactions are available. Recent advances in machine learning have made it possible to model such systems, but their inherent biases and black-box nature pose an inferential challenge. Semiparametric methods are able to nonetheless leverage these powerful nonparametric regression procedures to provide valid statistical analysis on interesting parametric components of the data generating process. This thesis consists of three chapters. The first chapter summarises the semiparametric and causal inference literatures, paying particular attention to doubly-robust methods and conditional independence testing. In the second chapter, we explore the doubly-robust estimation of the average partial effect — a generalisation of the linear coefficient in a (partially) linear model and a local measure of causal effect. This framework involves two plug-in nuisance function estimates, and trades their errors off against each other. The first nuisance function is the conditional expectation function, whose estimate is required to be differentiable. We propose convolving an arbitrary plug-in machine learning regression — which need not be differentiable — with a Gaussian kernel, and demonstrate that for a range of kernel bandwidths we can achieve the semiparametric efficiency bound at no asymptotic cost to the regression mean-squared error. The second nuisance function is the derivative of the log-density of the predictors, termed the score function. This score function does not depend on the conditional distribution of the response given the predictors. Score estimation is only well-studied in the univariate case. We propose using a location-scale model to reduce the problem of multivariate score estimation to conditional mean and variance estimation plus univariate score estimation. This enables the use of an arbitrary machine learning regression. Simulations confirm the desirable properties of our approaches, and code is made available in the R package drape (Doubly-Robust Average Partial Effects) available from https://github.com/harveyklyne/drape. In the third chapter, we consider testing for conditional independence of two discrete random variables X and Y given a third continuous variable Z. Conditional independence testing forms the basis for constraint-based causal structure learning, but it has been shown that any test which controls size for all null distributions has no power against any alternative. For this reason it is necessary to restrict the null space, and it is convenient to do so in terms of the performance of machine learning methods. Previous works have additionally made strong structural assumptions on both X and Y. A doubly-robust approach which does not make such assumptions is to compute a generalised covariance measure using an arbitrary machine learning method, reducing the test for conditional correlation to testing whether an asymptotically Gaussian vector has mean zero. This vector is often high-dimensional and naive tests suffer from a lack of power. We propose greedily merging the labels of the underlying discrete variables so as to maximise the observed conditional correlation. By doing so we uncover additional structure in an adaptive fashion. Our test is calibrated using a novel double bootstrap. We demonstrate an algorithm to perform this procedure in a computationally efficient manner. Simulations confirm that we are able to improve power in high-dimensional settings with low-dimensional structure, whilst maintaining the desired size control. Code is made available in the R package catci (CATegorical Conditional Independence) available from https://github.com/harveyklyne/catci.
Open Access
A Walk through the Forest: the Geometry and Topology of Random Systems
Halberstam, Noah
We prove several theorems on the geometry and topology of random walks and random forests, with analysis of the latter of these random systems often relying on analysis of the former and vice versa. The main models we consider are the static and dynamic random conductance models, the uniform spanning forest, the arboreal gas and countable Markov chains, and we will be interested in both the qualitative and quantitative behaviour of these systems over large scales. The quantitative properties of both the random system and its underlying medium are in this work and in general often encoded as a set of dimensions, or exponents, which govern how those properties scale asymptotically with distance or time. In addition to the analytical work above, we numerically investigate the relationships between the dimensions of fractal media and the random systems which sit upon them, and, in particular, provide evidence that universality should hold beyond the Euclidean setting. Material taken from a total of six papers is included. We also include an introduction explaining the background and context to these papers.
Open Access
Random conformally covariant metrics in the plane
Hughes, Liam
This thesis is in the broad area of random conformal geometry, combining tools from probability and complex analysis. We mainly consider *Liouville quantum gravity* (LQG), a model introduced in the physics literature in the 1980s by Polyakov in order to provide a canonical example of a random surface with conformal symmetries and formally given by the Riemannian metric tensor "$e^{\gamma h} (dx^2+dy^2)$'' where $h$ is a Gaussian free field (GFF) on a planar domain and $\gamma \in (0,2)$. Duplantier and Sheffield constructed the $\gamma$-LQG area and boundary length measures, which fall under the framework of Kahane's Gaussian multiplicative chaos. Later, a conformally covariant distance metric associated to $\gamma$-LQG was constructed for whole-plane and zero-boundary GFFs. In this thesis we describe the $\gamma$-LQG metric corresponding to a free-boundary GFF and derive basic properties and estimates for the boundary behaviour of the metric using GFF techniques. We use these to show that when one uses a conformal welding to glue together boundary segments of two appropriate independent LQG surfaces to get another LQG surface decorated by a *Schramm--Loewner evolution* (SLE) curve, the LQG metric on the resulting surface can be obtained as a natural metric space quotient of those on the two original surfaces. This generalizes results of Gwynne and Miller in the special case $\gamma = \sqrt{8/3}$ (for which the LQG metric can be explicitly described in terms of Brownian motion) to the entire subcritical range $\gamma \in (0,2)$. Moreover, we show that LQG metrics are infinite-dimensional (in the sense of Assouad) and thus that their embeddings into the plane cannot be quasisymmetric. We also consider chemical distance metrics associated to *conformal loop ensembles*, the loop version of SLE, using the imaginary geometry coupling to the GFF to bound the exponent governing the conformal symmetries of such a metric.
Open Access
Hitchin Functionals, h-Principles and Spectral Invariants
Mayther, Laurence
This thesis investigates Hitchin functionals and $h$-principles for stable forms on oriented manifolds, with a special focus on $\mathrm{G}_2$ and $\widetilde{\mathrm{G}}_2$ 3- and 4-forms. Additionally, it introduces two new spectral invariants of torsion-free $\mathrm{G}_2$-structures. Part I begins by investigating an open problem posed by Bryant, $\textit{viz.}$ whether the Hitchin functional $\mathcal{H}_3$ on closed $\mathrm{G}_2$ 3-forms is unbounded above. Chapter 3 uses a scaling argument to obtain sufficient conditions for the functional $\mathcal{H}_3$ to be unbounded above and applies this result to prove the unboundedness above of $\mathcal{H}_3$ on two explicit examples of closed 7-manifolds with closed $\mathrm{G}_2$ 3-forms. Chapter 3 then proceeds to interpret this unboundedness geometrically, demonstrating an unexpected link between the functional $\mathcal{H}_3$ and fibrations, proving that the 'large volume limit' of $\mathcal{H}_3$ in each case corresponds to the adiabatic limit of a suitable fibration. The proof utilises a new, general collapsing result for singular fibrations between orbifolds, without assumptions on curvature, which is proved in Chapter 4. Chapter 5 broadens the focus of Part I to include the Hitchin functionals $\mathcal{H}_4$, $\widetilde{\mathcal{H}}_3$ and $\widetilde{\mathcal{H}}_4$ on closed $\mathrm{G}_2$ 4-forms, $\widetilde{\mathrm{G}}_2$ 3-forms and $\widetilde{\mathrm{G}}_2$ 4-forms respectively. In its main result, Chapter 5 proves that $\mathcal{H}_4,\widetilde{\mathcal{H}}_3,\widetilde{\mathcal{H}}_4$ are always unbounded above and below (whenever defined), and also that $\mathcal{H}_3$ is always unbounded below (whenever defined). As scholia, the critical points of the functionals $\mathcal{H}_4$, $\widetilde{\mathcal{H}}_3$ and $\widetilde{\mathcal{H}}_4$ are shown to be saddle points, and initial conditions of the Laplacian coflow which cannot lead to convergent solutions are shown to be dense. Part I ends with a short discussion of open questions, in Chapter 6. Part II investigates relative $h$-principles for closed, stable forms. After establishing some prerequisite algebraic results, Chapter 7 begins by proving that if a class of closed, stable forms satisfies the relative $h$-principle, then its corresponding Hitchin functional is automatically unbounded above. By utilising the technique of convex integration, Chapter 7 then obtains sufficient conditions for a class of closed, stable forms to satisfy the relative $h$-principle, a result which subsumes all previously established $h$-principles for closed stable forms. Until now, 12 of the 16 possible classes of closed stable forms have remained open questions with regard to the relative $h$-principle. In the main result of Part II, Chapters 7 and 8 prove the relative $h$-principle in 5 of these open cases. The remaining 7 cases are addressed in the final chapter of Part II, where it is conjectured that the relative $h$-principle holds in each case. Chapter 9 applies the $h$-principles established in this thesis to prove various results on the topological properties of closed $\widetilde{\mathrm{G}}_2$, $\mathrm{SL}(3;\mathbb{C})$ and $\mathrm{SL}(3;\mathbb{R})^2$ forms. Firstly, it characterises which oriented 7-manifolds admit closed $\widetilde{\mathrm{G}}_2$ forms, in the process introducing a new technique for proving the vanishing of natural cohomology classes on non-closed manifolds. Next, it introduces $\widetilde{\mathrm{G}}_2$-cobordisms of closed $\mathrm{SL}(3;\mathbb{C})$ and $\mathrm{SL}(3;\mathbb{R})^2$ 3-forms and proves that homotopic forms are $\widetilde{\mathrm{G}}_2$-cobordant. Additionally, Chapter 9 classifies $\mathrm{SL}(3;\mathbb{C})$ 3-forms up to homotopy and provides a partial classification result on homotopy classes of $\mathrm{SL}(3;\mathbb{R})^2$ 3-forms. Part II ends with a short discussion of open questions, in Chapter 10. Part III introduces and examines two new spectral invariants of torsion-free $\mathrm{G}_2$-structures. Although the notion of an invariant is a central theme in geometry and topology, currently, there is only one known invariant of torsion-free $\mathrm{G}_2$-structures: the $\overline{\nu}$-invariant of Crowley-Goette-Nordström. Part III defines two new invariants of torsion-free $\mathrm{G}_2$-structures, termed $\mu_3$- and $\mu_4$-invariants, by regularising the classical notion of Morse index for the Hitchin functionals $\mathcal{H}_3$ and $\mathcal{H}_4$ at their critical points. In general, there is no known way to compute $\overline{\nu}$ for $\mathrm{G}_2$-manifolds constructed via Joyce's `generalised Kummer construction'. Chapter 11 obtains closed formulae for $\mu_3$ and $\mu_4$ on the orbifolds used in Joyce's construction, leading to a conjectural discussion in Chapter 12 of how to compute $\mu_3$ and $\mu_4$ on Joyce's manifolds.
Open Access
Absolutely Continuous Stationary Measures
Kittle, Samuel
This thesis studies the absolute continuity of stationary measures. Given a finite set of measurable maps $S_1, S_2, \dots, S_n$ on a measurable set $X$ and a probability vector $p_1, p_2, \dots, p_n$ we say that a probability measure $\nu$ on $X$ is stationary if $\begin{equation*} \nu = \sum_{i=1}^{n} p_i \nu \circ S_i^{-1}. \end{equation*}$ If $S_1, \dots, S_n$ are elements of *PSL*₂($\mathbb{R}$) acting on *X* = *P*¹($\mathbb{R}$), we get the notion of Furstenberg measures. If $S_1, \dots, S_n$ are similarities, affine maps, or conformal maps then $\nu$ is called a self-similar, self-affine, or self-conformal measure respectively. This thesis is concerned with Furstenberg measures and self-similar measures. Two fundamental questions about stationary measures are what are their dimensions and when are they absolutely continuous. This thesis deals with the second one of these. There are several classes of stationary measures which are known to be absolutely continuous for typical choices of parameters. For example Solomyak showed that for almost every $\lambda \in (1/2, 1)$ the Bernoulli convolution with parameter $\lambda$ is absolutely continuous. This was extended by Shmerkin who showed that the exceptional set has Hausdorff dimension zero. However, despite much effort, there are relatively few known explicit examples of stationary measures which are absolutely continuous. In this thesis we find sufficient conditions for self-similar measures and Furstenberg measures to be absolutely continuous. Using this we are able to give new examples. The techniques we use are largely inspired by the techniques of Hochman and Varj\'u though we introduce several new ingredients the most important of which is ``detail'' which is a quantitative way of measuring how smooth a measure is at a given scale.
Open Access
Homological stability of spaces of manifolds via E_k-algebras
Sierra, Ismael
In this thesis we study homological stability properties of different families of spaces using the technique of cellular *E_k*-algebras. Firstly, we will consider spin mapping class groups of surfaces, and their algebraic analogue —quadratic symplectic groups— using cellular *E₂*-algebras. We will obtain improvements in their stability results, which for the spin mapping class groups we will show to be optimal away from the prime 2. We will also prove that in both cases the $\mathbb{F}$₂-homology satisfies secondary homological stability. Finally, we will give full descriptions of the first homology groups of the spin mapping class groups and of the quadratic symplectic groups. Secondly, we will study the classifying spaces of the diffeomorphism groups of the manifolds *W*_*g*,1 ∶= *D*^2*n*#(*Sⁿ* x *Sⁿ*)^#*g*. We will get new improvements in the stability results, especially when working with rational coefficients. Moreover, we will prove a new type of stability result —quantised homological stability— which says that either the best integral stability result is a linear bound of slope 1/2 or the stability is at least as good as a line of slope 2/3.
Open Access
Topics in symplectic Gromov–Witten theory
Hirschi, Amanda
The main focus of this thesis is on the Gromov--Witten theory of general symplectic manifolds. Mohan Swaminathan and I construct a framework to define a virtual fundamental class for the moduli space of stable maps to a general closed symplectic manifold. Our construction, inspired by [AMS21], works for all genera and leads to a more straightfoward definition of symplectic Gromov--Witten invariants as was previously available. We prove a formula for the Gromov--Witten invariants of a product of two symplectic manifolds, conjectured in [KM94]. I generalise the product formula to a formula for the Gromov--Witten invariants of a suitable fibre product of symplectic manifolds. Our invariants satisfy the Kontsevich-Manin axioms and are extended to descendent Gromov--Witten invariants. I show that our definition of Gromov--Witten invariants agrees with the classical Gromov--Witten invariants defined by [RT97] for semipositive symplectic manifolds. Given a Hamiltonian group action on the target manifold, I construct equivariant Gromov--Witten invariants and prove a virtual Atiyah--Bott-type localisation formula, providing a tool for computations. Together with Soham Chanda and Luya Wang, I construct infinitely many exotic Lagrangian tori in complex projective spaces of complex dimension higher than $2$. We lift tori in $\mathbb{P}$², constructed by Vianna, and show that these lifts remain non-symplectomorphic, using an invariant derived from pseudoholomorphic disks. Noah Porcelli and I use Ljusternik-Schnirelmann theory, applied to moduli spaces of pseudoholomorphic curves, and homotopy theory to prove lower bounds on the number of intersection points of two (possibly non-transverse) Lagrangians in terms of the cuplength of the Lagrangian in generalised cohomology theories, improving previous lower bounds by Hofer.
Open Access
Birational Invariance of Punctured Log Gromov-Witten Theory and Intrinsic Mirror Constructions
Johnston, Samuel
In this thesis, we investigate and resolve various problems related to log Gromov-Witten theory and their application to mirror symmetry. We first prove for log Calabi-Yau varieties satisfying a semi-positivity assumption that the Gross-Siebert logarithmic mirror construction encodes solutions to enumerative problems considered in the non-archimedean construction of Keel and Yu, and use this to show the two approaches agree in most cases when both can be constructed. We also prove a classical-quantum period correspondence for smooth Fano pairs, with the classical periods encoded in the Gross-Siebert mirror construction, and in particular give enumerative meaning to generating series of regularized quantum periods. The second main result of this thesis is a study of the behavior of punctured log Gromov-Witten theory under log étale modifications X ̃ → X, generalizing an investigation first carried out by Abramovich and Wise. We show that the moduli space of stable log maps to X ̃ can be described explicitly in terms of the moduli space of stable log maps to X, together with understanding of the change in tropical moduli spaces. We use this result to resolve various foundational questions in punctured log Gromov-Witten theory, as well as to show a certain form of log étale invariance of the intrinsic mirror algebra.
Open Access
Gross-Siebert Mirror Ring for Smooth log Calabi-Yau Pairs
Wang, Yu
In this paper, we exhibit a formula relating punctured Gromov-Witten invariants used by Gross and Siebert in [GS2] to 2-point relative/logarithmic Gromov-Witten invariants with one point-constraint for any smooth log Calabi-Yau pair (W, D). Denote by Na,b the number of rational curves in W meeting D in two points, one with contact order a and one with contact order b with a point constraint. (Such numbers are defined within relative or logarithmic Gromov-Witten theory). We then apply a modified version of deformation to the normal cone technique and the degeneration formula developed in [KLR] and [ACGS1] to give a full understanding of Ne−1,1 with D nef where e is the intersection number of D and a chosen curve class. Later, by means of punctured invariants as auxiliary invariants, we prove, for the projective plane with an elliptic curve (P2, D), that all standard 2-pointed, degree d, relative invariants with a point condition, for each d, can be determined by exactly one of these degree d invariants, namely N3d−1,1, plus those lower degree invariants. In the last section, we give full calculations of 2-pointed, degree 2, one-point-constrained relative Gromov-Witten invariants for (P2, D).
Open Access
Mirrors to Toric Degenerations via Intrinsic Mirror Symmetry
Goncharov, Evgeny
We explore the connection between two mirror constructions in Gross-Siebert mirror symmetry: toric degeneration mirror symmetry and intrinsic mirror symmetry. After briefly exploring the case of degenerations of elliptic curves, we show that the Gross-Siebert mirror construction for minimal relative log Calabi-Yau degenerations generalizes that for divisorial toric degenerations $\bar{\mathfrak{X}} \to \mathcal{S}$ of K3-s that have a smooth generic fibre. We achieve this by constructing a resolution of $\bar{\mathfrak{X}} \to \mathcal{S}$ to a relative minimal log Calabi-Yau degeneration $\mathfrak{X} \to \mathcal{S}$ and comparing the algorithmic scattering diagram $\bar{\mathfrak{D}}$ giving rise to the toric degeneration mirror $\check{\bar{\mathfrak{X}}}$ and the canonical scattering diagram $\mathfrak{D}$ giving rise to the intrinsic mirror $\check{\mathfrak{X}}$. Moreover, we vastly expand the construction and obtain a correspondence between the restriction of the intrinsic mirror to the (numerical) minimal relative Gross-Siebert locus and the universal toric degeneration mirror. We also discuss generalizing the results to higher dimensions. In particular, we construct log smooth resolutions for a natural family of toric degenerations of Calabi-Yau threefolds.
Open Access
Extremal results for graphs and hypergraphs and other combinatorial problems
Janzer, Barnabás; Janzer, Barnabás [0000-0002-9904-7188]
In this dissertation we present several combinatorial results, primarily concerning extremal problems for graphs and hypergraphs, but also covering some additional topics. In Chapter 2, we consider the following geometric problem of Croft. Let K be a convex body in R^d that contains a copy of another body S in every possible orientation. Is it always possible to continuously move any one copy of S into another, inside K? We prove that the answer is positive if S is a line segment, but, surprisingly, the answer is negative in dimensions at least four for general S. In Chapter 3, we study the extremal number of tight cycles. Sós and Verstraëte raised the problem of finding the maximum possible size of an n-vertex r-uniform tight-cycle-free hypergraph. When r=2 this is simply n−1, and it was unknown whether the answer is Θ(n^{r−1}) in general. We show that this is not the case for any r≥3 by constructing r-uniform hypergraphs with n vertices and Ω(n^{r−1}logn/loglogn)=ω(n^{r−1}) edges which contain no tight cycles. In Chapter 4, we study the following saturation question: how small can maximal k-wise intersecting set systems over [n] be? Balogh, Chen, Hendrey, Lund, Luo, Tompkins and Tran resolved this problem for k=3, and for general k showed that the answer is between c_k·2^{n/(k−1)} and d_k·2^{n/⌈k/2⌉}. We prove that their lower bound gives the correct order of magnitude for all k. In Chapter 5, we prove that for any r, s with r0 there exists r such that g(n,K_r) = Ω(n^{1−ε}). We also prove a family of special cases of a conjecture of Conlon, Fox, Lee and Sudakov about the so-called hypergraph Erdős–Gyárfás function. In Chapter 8, we study bootstrap percolation for hypergraphs. Consider the process in which, given a fixed r-uniform hypergraph H and starting with a given n-vertex r-uniform hypergraph G, at each step we add to G all edges that create a new copy of H. We are interested in maximising the number of steps that this process takes before it stabilises. For the case where H=K_s^{(r)} with s>r≥3, we show that the number of steps of this process can be Θ(n^r). This answers a recent question of Noel and Ranganathan. We also demonstrate that different and interesting maximal running times can occur for other choices of H. In Chapter 9, we study an extremal problem about permutations. How many random transpositions (meaning that we swap given pairs of elements with given probabilities) do we need to perform on a deck of cards to ‘shuffle’ it? We study several problems on this topic. Among other results, we show that at least 2n−3 such swaps are needed to uniformly shuffle the first two cards of the deck, proving a conjecture of Groenland, Johnston, Radcliffe and Scott. In Chapter 10, we study the following extremal problem on set systems introduced by Holzman and Körner. We say that a pair (a,b) of families of subsets of an n-element set is cancellative if whenever A,A′∈a and B∈b satisfy A∪B=A′∪B, then A=A′, and whenever A∈a and B,B′∈b satisfy A∪B=A∪B′, then B=B′. Tolhuizen showed that there exist cancellative pairs with |a||b| about 2.25^n, whereas Holzman and Körner proved an upper bound of 2.326^n. We improve the upper bound to about 2.268^n. This result also improved the then best known upper bound for a conjecture of Simonyi about ‘recovering pairs’ (the Boolean case of the ‘sandglass conjecture’), although the upper bound for Simonyi’s problem has since been further improved. In Chapter 11 we study a continuous version of Sperner’s theorem. Engel, Mitsis, Pelekis and Reiher showed that an antichain in the continuous cube [0,1]^n must have (n−1)-dimensional Hausdorff measure at most n, and they conjectured that this bound can be attained. This was already known for n=2, and we prove this conjecture for all n. Chapter 12 has similar motivations to the preceding chapter. A subset A of Z^n is called a weak antichain if it does not contain two elements x and y satisfying x_i
Open Access
Cubical small-cancellation theory and large-dimensional hyperbolic groups
Arenas, Macarena; Arenas, Macarena [0000-0003-4965-8121]
Given a finitely presented group Q and a compact special cube complex X with nonelementary hyperbolic fundamental group, we produce a non-elementary, torsion-free, cocompactly cubulated hyperbolic group Γ that surjects onto Q, with kernel isomorphic to a quotient of G = π_1X and such that max{cd(G),2} ≥ cd(Γ) ≥ cd(G)−1. Separately, we show that under suitable hypotheses, the second homotopy group of the coned-off space associated to a C(9) cubical presentation is trivial, and use this to provide classifying spaces for proper actions for the fundamental groups of many quotients of square complexes admitting such cubical presentations. When the cubical presentations satisfy a condition analogous to requiring that the relators in a group presentation are not proper powers, we conclude that the corresponding coned-off space is aspherical.
Open Access
Two-Dimensional Discrete Gaussian Model at High Temperature
Park, Jiwoon; Park, Jiwoon [0000-0002-1159-2676]
The Discrete Gaussian model is a Gaussian free field on lattice restricted to take integer values. In dimension two, it was proved by the seminal work of Fröhlich-Spencer that the Discrete Gaussian model exhibits localisation-delocalisation phase transition. The phase transition is ubiquitous in two-dimensional statistical physics models, intriguing the need for a unified framework for studying these phenomena. The goal of this thesis is to apply rigorous renormalisation group method to study the two-dimensional discrete Gaussian model in the delocalised phase, thereby obtaining central limit theorems in long-distance limit—in physics literature, the renormalisation group is a standard apparatus used to study scaling phenomena, in particular computing critical exponents and proving scaling limits and universality. We study the central limit theorem in three different regimes, first on macroscopic scale, second on mesoscopic scale and the third on microscopic scale. The first two amount to studying the scaling limits of the spin model under different limit regimes, while the final one discusses both pointwise and limit results. The final results have in particular prolific by-products, producing analogues of a number of results proved for different interface models. The entire thesis is devoted to solving these problems, but the strategy of the proof we develop is expected to have general applicability. Indeed, we develop renormalisation technology in the first half (Chapter 2–4) that only has weak requirements on the model. Then in the rest of the thesis, we develop an analysis specific to our model to prove the main theorems.
Open Access
Faithfulness of highest-weight modules for Iwasawa algebras
Mann, Stephen
We prove faithfulness of infinite-dimensional generalised Verma modules for Iwasawa algebras corresponding to split simple Lie algebras with a Chevalley basis. We use this to prove faithfulness of all infinite-dimensional highest-weight modules in the case of type *A*. In this case we also show that all non-zero two-sided ideals of the corresponding Iwasawa algebras have finite codimension, and in particular all non-zero prime ideals are annihilators of finite-dimensional simple modules.
Open Access
Homology of Configuration Spaces of Surfaces as Mapping Class Group Representations
Stavrou, Andreas
In this thesis, we study the homology of configuration spaces of surfaces viewed as representations of the mapping class group of the surface, distinguishing between various flavours: ordered and unordered configurations, of closed surfaces and surfaces with boundary, and with different homology coefficients. In Chapter 2, we prove a version of the scanning isomorphism that is “untwisted” and equivariant with the mapping class group action. We further prove that scanning remembers a product arising from superposing configurations. We apply this equivariant scanning to compute the rational cohomology of unordered configurations of surfaces with boundary. In Chapter 3, we adapt certain cellular decompositions of compactified configuration spaces to obtain the kernel of the mapping class group action on the homology of unordered configurations of both kinds of surfaces and with any coeffiecients. Finally, in Chapter 4, we geometrically construct mapping classes deep in the Johnson filtration that act non-trivially on the homology of ordered configurations, in support of a conjecture by Bianchi, Miller and Wilson.
Open Access
p-arithmetic cohomology and p-adic automorphic forms
Tarrach Garcia, Guillem
The cohomology of an arithmetic group with coefficients in finite-dimensional representations can be described in terms of automorphic representations of the group. In this thesis, we prove similar results for the cohomology of an *S*-arithmetic groups (where *S* is a finite set of primes) with coefficients in different types of representations. For example, we show that the cohomology of (duals of) locally algebraic representations of the local groups at places in *S* can be described in terms of automorphic representations satisfying certain conditions determined by the locally algebraic representation. We show that the cohomology with coefficients in (duals of) locally analytic representations can be used to define *p*-adic automorphic forms and families of them (eigenvarieties). In particular, we are able to give constructions of these objects in many new cases, such as when the reductive group is not quasi-split at *p*. We also prove that these constructions are equivalent, in the cases where they are defined, to those obtained using overconvergent cohomology and to the Bernstein eigenvarieties constructed by Breuil-Ding.
Open Access
Poset saturation and other combinatorial results
Ivan, Maria; Ivan, Maria [0000-0003-0817-3777]
In this dissertation we discuss a number of combinatorial results. These results fall into four broad areas: poset saturation, Ramsey theory, pursuit and evasion, and union-closed families. Chapter 2 is dedicated to the area of poset saturation. Given a finite poset P, we call a family F of subsets of [n] P-saturated if F does not contain an induced copy of P, but adding any other set to F creates an induced copy of P. The size of the smallest P-saturated family with ground set [n] is called the induced saturated number of P, which is denoted by sat∗(n,P). In this chapter we look at four posets: the butterfly, the diamond, the antichain and the poset N . We establish a linear lower bound for the butterfly, a lower bound of(2√2 − o(1))√n for the diamond, a lower bound of √n for the poset N , and the exact saturation number for the 5-antichain and the 6-antichain. Chapter 3 is dedicated to two different Ramsey theory questions. In Section 3.1 we establish a Ramsey characterisation of eventually periodic words. More precisely, for a finite colouring of X∗ (the set of finite words on alphabet X) we say that a factorisation x = u1u2 · · · of an infinite word x is ‘super-monochromatic’ if each word uk1 uk2 · · · ukn, where k1 < · · · < kn, is the same colour. We show that a word x is eventually periodic if and only if for every finite colouring of X∗ there is a suffix of x having a super-monochromatic factorisation. This has been a conjecture for quite some time. In Section 3.2 we investigate the question of whether or not, given a finite colouring of the rationals or the reals, we can find an infinite subset with the property that the set of all its finite sums and products is monochromatic. The main result of this section is the existence of a finite colouring of the rationals with the property that no infinite set whose denominators contain only finitely many primes has the set of all of its finite sums and products monochromatic. In Chapter 4 we explore the game of cops and robbers on infinite graphs. The main question is: for which graphs can one guarantee that the cop has a winning strategy? In the finite case these graphs are precisely the ‘constructible’ graphs, but the infinite case is not well understood. For example, we exhibit a graph that is cop-win but not constructible. This is the first known such example. On the other hand, every constructible graph is a weak cop win (meaning that the cop can eventually force the robber out of any finite set). We also investigate how this notion relates to the notion of ‘locally constructible’ (every finite graph is contained in a finite constructible subgraph). The main result of this chapter is the construction of a locally constructible graph that is not a weak cop win. Surprisingly, this graph may even be chosen to be locally finite. Finally, in Chapter 5 we discuss the union-closed conjecture which asserts that for any union-closed family of sets, there exists an element of the ground set contained in at least half of the sets of the family. Our attention is on the small sets of union-closed families. More precisely, we construct a class of union-closed families of sets such that the frequency of the elements of the minimal sets is o(1) – so that these elements are not generally in half of the sets of union-closed families.
Open Access
Effective integrality results in arithmetic dynamics
Young, Marley
Given a rational function f defined over a number field K, S. Ih conjectured the finiteness of f-preperiodic points which are S-integral relative to a given non-preperiodic point β. This conjecture remains open, but certain special cases have been proved. We formulate a generalisation of Ih's conjecture, considering a semigroup $\mathcal{G}$ generated by rational functions (along with an appropriate notion of preperiodic points) defined over K, instead of a single map, and prove some of the known cases in this context. We moreover make our results effective. Given an arbitrary, finitely generated rational semigroup $\mathcal{G}$, we prove our generalisation of Ih's conjecture under certain local conditions on the non-preperiodic point β, generalising a result of Petsche. As an application, we obtain bounds on the number of S-units in certain doubly-indexed dynamical sequences. In the case of a single, unicritical polynomial f_c(z)=z^d+c, with β set to be the critical point 0, for parameters c outside a small region, we give an explicit bound which depends only on the number of places of bad reduction for f_c. As part of the proof, we obtain novel lower bounds for the v-adically smallest preperiodic point of f_c for each place v of K. Finally, when $\mathcal{G}$ is a finitely generated semigroup of monomial maps, we prove the conjecture without any assumptions on β, and moreover give a bound which is uniform as β varies over number fields of bounded degree. This generalises results of Baker, Ih and Rumely, which were made uniform by Yap.