# Theses - Pure Mathematics and Mathematical Statistics

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Item Open Access Extremal Problems for Cycles, Paths and Set-SystemsMond, AdvaShow more This thesis consists of an introduction and five chapters, each devoted to a different combinatorial problem. What ties all problems considered in this thesis together is their extremal nature and the probabilistic point of view taken in their formulations or analysis. In the first three chapters we consider extremal questions regarding cycles. Cycles in graphs are one of the most natural and basic structures to study. In particular, in the context of extremal problems, questions regarding their appearance and their count has always been of a significant interest. In Chapter 2 we study the Turán number of long cycles in random and pseudo-random graphs. Denote by $ex(G(n,p),H)$ the random variable counting the number of edges in a largest subgraph of $G(n,p)$ without a copy of $H$. We determine the asymptotic value of $ex(G(n,p), C_t)$ where $C_t$ is a cycle of length $t$, for $p\geq \frac Cn$ and $A \log n \leq t \leq (1 - \varepsilon)n$, for constants $C$ and $A$. The size of $ex(G(n,p), C_t)$ depends substantially on the parity of $t$. In particular, our results match the classical result of Woodall on the Turán number of long cycles, and can be seen as its random version. This demonstrates a phenomenon known as the transference principle, introduced by Conlon and Gowers and by Schacht. In this context it can be interpreted as a random graph "inheriting'' its (relative) extremal properties from the classical deterministic case, i.e., the complete graph. In fact, our techniques apply in a more general sparse pseudo-random setting. We also prove a robustness-type result, showing the likely existence of cycles of prescribed lengths in a random subgraph of a graph with a nearly optimal density. Finally, we also present further applications of our main tool (the Key Lemma) for proving results on Ramsey-type problems about cycles in sparse random graphs. In Chapter 3 we count at least how many Hamilton cycles one can find in a hypergraph which is guaranteed to contain at least one. For $0\leq \ell 1/2$ has (asymptotically and up to a subexponential factor) at least as many Hamilton $\ell$-cycles as a typical random $k$-graph with edge-probability $\delta$. This significantly improves a result of Glock, Gould, Joos, Kühn and Osthus, and proves a conjecture of Ferber, Krivelevich and Sudakov for all values $0\leq \ellShow more Item Embargo Categorifications of Cluster VarietiesKrawchuk, ColinShow more In this thesis we study additive categorifications of algebraic varieties whose coordinate rings admit a cluster algebra structure. The presence of a cluster structure has important implications for the geometry of a variety including the existence of canonical linearly independent sets of regular functions. Additionally, methods for studying cluster algebras can be applied to obtain insights that are beyond the reach of conventional geometric tools. In order to associate a cluster algebra structure to a variety one needs to construct an initial seed of regular functions that generate the coordinate ring, and where each cluster variable generated from the initial seed is indeed a regular function. In the case of Grassmannians and their open positroid subvarieties, this initial seed is described by a strand diagram in the disk called a Postnikov diagram. Our first objective is to study the cluster structure on the homogeneous coordinate ring of the Grassmannian using the additive categorification introduced by Jensen King and Su. This abstract approach allows us to apply methods from representation theory, algebraic geometry and combinatorics. We use these tools to describe a class of (𝑘 − 1)-exact sequences present in the subcategory of Plücker modules. In the process, we generalise a construction of Jasso involving 𝑛-exact sequences formed by acyclic complexes. The second goal of this thesis is to develop a method for categorifying general cluster varieties. The mutation that defines these cluster structures is determined from the combinatorics of a frozen quiver with potential called a dimer quiver. In order to construct a categorification from those of simpler cluster varieties, we introduce a gluing operation for dimer quivers on surfaces. We use this operation to recover the boundary algebra obtained from a gluing when certain consistency conditions are met. As an application we determine the boundary algebra of a class of homogeneous strand diagrams on the annulus and we provide a method to determine the boundary algebra of the initial seed of a positroid variety through a decomposition into glued dimer quivers. Finally, we investigate a categorical interpretation of freezing cluster variables using reductions of stably 2-Calabi-Yau Frobenius extriangulated categories introduced by Faber, Marsh and Pressland. This work provides a first step towards constructing cluster categories from a gluing decomposition of cluster varieties.Show more Item Open Access Combinatorial aspects of synchronisation, percolation, and arithmeticSeixas Souza, VictorShow more This thesis consists of an introduction and six chapters, each devoted to a different combinatorial problem. These problems cover several different directions of research in extremal, probabilistic and arithmetic combinatorics. In Chapter 2, we shall study the Kuramoto model, which is the prototypical model used for rigorous mathematical analysis in the field of synchronisation and nonlinear dynamics. A realisation of this model consists of a collection of identical oscillators with interactions given by the edges of a graph. We show that a graph with sufficient expansion must be globally synchronising, meaning that the Kuramoto model on such a graph will converge to the fully synchronised state, namely the one where all the oscillators have the same phase, for every initial state up to a set of measure zero. We show that if $\varepsilon > 0$ and $p \geq (1 + \varepsilon)(\log n)/n$, then the Kuramoto model on the Erdős-Rényi graph $G(n,p)$ is globally synchronising with probability tending to one as $n$ goes to infinity. This improves a previous result of Kassabov, Strogatz and Townsend, who have shown this with $p \gg (\log n)^2/n$, and so solve a conjecture of Ling, Xu and Bandeira. Our result is essentially best possible as $G(n,p)$ is not connected if $p = (1 - \varepsilon)(\log n)/n$. We also show that any $d$-regular Ramanujan graph is globally synchronising if $d \geq 600$. The same holds for a typical $d$-regular random graph under the same assumption on the degrees. In Chapter 3, we study a problem at the confluence of percolation theory and combinatorial game theory. The $(m,b)$ Maker-Breaker percolation game on $(\mathbb{Z}^2)_p$, introduced by Day and Falgas-Ravry, is played in the following way. Before the game starts, each edge of $\mathbb{Z}^2$ is removed independently with probability $1-p$. After that, Maker chooses a vertex $v_0$. Then, in each round, Maker and Breaker claim respectively $m$ and $b$ unclaimed edges of $G$. Breaker wins if after the removal of the edges claimed by him the component of $v_0$ becomes finite, and Maker wins if she can indefinitely prevent Breaker from winning. We show that for any $p < 1$, Breaker almost surely has a winning strategy for the $(1,1)$ game on $(\mathbb{Z}^2)_p$. This fully answers a question of Day and Falgas-Ravry, who showed that for $p = 1$ Maker has a winning strategy for the $(1,1)$ game. Further, we show that in the $(2,1)$ game on $(\mathbb{Z}^2)_p$ Maker almost surely has a winning strategy when $p > 0.9402$, while Breaker almost surely has a winning strategy when $p < 0.5278$. This shows that the threshold value of $p$ above which Maker has a winning strategy for the $(2,1)$ game on $\mathbb{Z}^2$ is non-trivial. We prove similar results in various settings, including other lattices and biases $(m,b)$. These results extend to the most general case where each edge is given to Maker with probability $\alpha$ and to Breaker with probability $\beta$ before the game starts. In Chapter 4, we are concerned with the Maclaurin inequalities for graphs, which are a broad generalisation of the classical theorems of Turán and Zykov. In a nutshell they provide an asymptotically sharp answer to the following question: what is the maximum number of cliques of size q in a $K_{r+1}$-free graph with a given number of cliques of size $s$? We prove an extensions of the graph Maclaurin inequalities with a weight function that captures the local structure of the graph. As a corollary, we settle a recent conjecture of Kirsch and Nir, which simultaneously generalise the previous localised results of Bradač; Malec and Tompkins; and the results of Kirsch and Nir. In Chapter 5, we explore multiplicative analogues of classical results in additive Ramsey theory. In particular, we are concerned with the following question: given an $r$-colouring of the interval $\{2, \dotsc, N \}$, what is the minimum number of monochromatic solutions of the equation $xy = z$? We give an asymptotically sharp answer for this question when $r = 2$. In this case, we also prove a stability version of this result. For general $r$, we provide polynomial upper and lower bounds on the minimum number of monochromatic solutions. For $r \leq 4$, these bounds match up to a polylogarithm factor. We also obtain results for more general multiplicative equations of the form $x_1^{a_1} \dotsb x_k^{a_k} = y$, where $a_1, \dotsc, a_k$ are fixed positive integers, at least one of which equals $1$. In Chapter 6 we are concerned with how efficiently one can encode the divisibility order. The dimension of a partially-ordered set $P$ is the smallest integer $d$ such that one can embed $P$ into a product of $d$ linear orders. We prove that the dimension of the divisibility order on the interval $\{1, \dotsc, n\}$ is bounded above by $C(\log n)^2 (\log \log n)^{-2} \log \log \log n$ as $n$ goes to infinity. We also provide a corresponding lower bound of $c(\log n)^2 (\log \log n)^{-2}$, asymptotically. To obtain these bounds, we provide a refinement of a bound of Füredi and Kahn and exploit a connection between the dimension of the divisibility order and the maximum size of $r$-cover-free families. In Chapter 7 we discuss an improvement upon an inequality of Croot, Dobbs, Friedlander, Hetzel and Pappalardi on the number of Egyptian fractions. We show that for all $k \geq 2$, the number of integers $1 \leq a \leq n$ such that the equation $a/n = 1/m_1 + \dotsb + 1/m_k$ has a solution in integers $1 \leq m_1 \leq \dotsb \leq m_k$ is bounded above by $n^{1 - 1/2^{k-2} + o(1)}$ as $n$ goes to infinity.Show more Item Open Access Computations and lower bounds for scl and the relative Gromov seminormMarchand, Alexis; Marchand, Alexis [0000-0001-6594-4254]Show more The main purpose of this thesis is to give computations and estimates of stable commutator length, an invariant of groups that measures homological complexity and is connected to several notions of negative curvature in geometric group theory. A first aspect of this is to give exact computations, motivated by questions of rationality. A particular case of interest is that of surface groups. We introduce the relative Gromov seminorm, a new invariant that serves as an intermediate step in computations of scl. We show that several results about scl in free groups generalise to the relative Gromov seminorm in surface groups. We explain how bounded cohomology provides a dual to the relative Gromov seminorm, and apply those ideas to obtain new computations of scl. Another aspect of this thesis is to obtain lower bounds for stable commutator length in the presence of negative curvature. We do this by developing a new geometric method, where surfaces estimating scl are equipped with a combinatorial geometric structure called an angle structure, and for which there is a notion of curvature and a version of the Gauß–Bonnet formula leading to estimates of the Euler characteristic. We show in particular that our method yields a new proof of a theorem of Heuer on a sharp spectral gap for right-angled Artin groups.Show more Item Open Access Shock phenomena for the one-dimensional Boltzmann equation and related kinetic problemsWynter, Dominic LouisShow more In this thesis, we present a work on the derivation of fluid models from microscopic dynamics, inspired by Hilbert’s sixth problem, on the justification of continuum equations from atomistic interactions. In particular, we focus on approximations of two-dimensional incompressible flow by finitely many interacting point vortices. Additionally, we study approximations of compressible gas dynamics by the Boltzmann equation, which models gases using a statistical description of molecular interactions, in which particles interact by elastic collisions. Furthermore, we develop the stability theory of the Boltzmann equation in one dimension for general data. In Chapter 1, we present a summary of the works included in this thesis. The first part of this thesis addresses point-vortex approximations of the incompressible Navier-Stokes equations in two dimensions. In Chapter 2, we present a work deriving convergence of a stochastic vortex system with Brownian noise to solutions of the Navier- Stokes equations in the many particle mean-field limit, where vortices are assumed to be of mixed-sign. The main method of this chapter is to prove propagation of chaos of the stochastic differential equation by comparison to the mean-field equation and relative entropy methods. More precisely, we prove quantitative estimates on the growth in time of correlations between vortices, and we show that these vanish in the many particle limit, proving our result. The second part of this thesis concerns the dynamics of collisional gases depending on one spatial dimension. In Chapter 3, we prove a large-data global well-posedness theory for the Boltzmann equation in one spatial dimension, with periodic boundary conditions and on the line. We prove this result for truncated hard-sphere interactions, for which local well-posedness results and propagation of derivatives and moments are elementary. Global well-posedness then follows using a Lyapunov functional well-adapted to one-dimensional transport problems, whose derivative exhibits a gain of regularity which is sufficient to control the nonlinearity, by a nonlinear Gronwall-type estimate. This estimate also gives decay estimates for finite mass and finite energy initial data on the line. In Chapter 4, we construct shock profile solutions for the Boltzmann equation with long-range interactions, near Navier-Stokes viscous shocks. We prove uniqueness up to translation, and show by a Chapman-Enskog argument that small amplitude Boltzmann shocks are well approximated by the Navier-Stokes equation. Our construction follows by a linearization procedure in the Chapman-Enskog scheme and a fixed-point argument. Translation invariance of the problem gives a linearized operator with a one-dimensional kernel, which we track using the stability theory of viscous shocks. Existence of solutions then follows by linearized energy estimates and by a Galerkin scheme, using stable/unstable dimensionality arguments. To overcome a loss of moment from the discretization procedure, we replace the collision operator Q with a regularized operator Q_κ, and take limits κ → 0 using functional-analytic continuity arguments.Show more Item Open Access The broken non-abelian X-ray transform and inverse problems for connections at high fixed frequencySt-Amant, SimonShow more This thesis is concerned with the recovery of a u(n) -valued connection A on a Hermitian vector bundle E of rank n over a smooth Riemannian or Lorentzian manifold with boundary (M, g) from different geometric inverse problems. The connection is to be recovered up to a gauge that is the identity on the boundary. The main inverse problem we study is the broken non-abelian X-ray transform. Given a connection A, its broken non-abelian X-ray transform is given by its parallel transports over a set of broken geodesics that start and end on the boundary. Here, a broken geodesic is a piecewise smooth curve formed by the composition of two geodesics that intersect transversally at a point. In Chapter 2, we consider the case where M = D\Ω is a causal diamond D in Minkowski space to which a small neighbourhood Ω of the observer's worldline has been removed, g is the induced Lorentzian metric, and the set of broken geodesics are the broken light rays that start and end in Ω. This transform was introduced by Chen-Lassas-Oksanen-Paternain (2022) to study inverse problems related to the Yang-Mills-Higgs equations. It is also known as the broken light ray transform. We show a stability estimate that takes the gauge into account, and we use it to show that, under a suitable fixing of the gauge, one can consistently recover a connection from noisy measurements of its broken non-abelian X-ray transform through Bayesian inversion. The broken non-abelian X-ray transform on an arbitrary Riemannian or Lorentzian manifold is studied in Chapter 3. We show that regardless of the geometry of M, the transform is injective up to gauge if we consider all possible broken geodesics with endpoints on the boundary. We also give a criterion to determine when the broken non-abelian X-ray transform is injective up to gauge over a fixed set of broken geodesics. In the subsequent chapters, we consider two inverse problems of Calderón type on a compact Riemannian manifold with boundary. The first asks whether one can recover a connection A from the Dirichlet-to-Neumann map DN_λ^A associated with the linear equation (Δ_A - λ^2)u = 0 as λ gets large. Here, Δ_A is the connection Laplacian. The second problem is analogous to the first but concerns the Dirichlet-to-Neumann map Λ_λ^A associated with the nonlinear equation (Δ_A - λ^2)u + |u|^2 u = 0 instead. We study both problems by using special quasimodes called Gaussian beams. Gaussian beams are approximate solutions of (Δ_A - λ^2)u = 0 that are supported on a small neighbourhood of a geodesic. We develop the theory of Gaussian beams on a vector bundle in Chapter 4. Our approach is novel because we view the solutions globally as sections of a jet bundle associated with the geodesic, we take care in prescribing their values on the boundary and our estimates are uniform with respect to certain families of geodesics. In Chapter 5, we show that given two connections A_1 and A_2, either the boundary values of their respective Gaussian beams agree to all order or there exists λ_0 depending on A_1 and A_2 such that DN_λ^{A_1} ≠ DN_λ^{A_2} for almost all λ > λ_0. Importantly, if the boundary values of the Gaussian beam agree to order 1, then the non-abelian X-ray transforms of the connections agree. Depending on the geometry of the underlying manifold, this can be enough to show that A_1 and A_2 are gauge equivalent. The proof relies on establishing a solvability result with vanishing boundary terms, computing lower-order terms in the asymptotics of an integral with a stationary phase and seeing the k-th power of the Laplacian as an inner product on the space of homogeneous polynomials of order k. In Chapter 6, we show that given A_1 and A_2 as well as a family of broken geodesics, either their broken non-abelian X-ray transforms agree on that family or there exists λ_0 such that Λ_λ^{A_1} ≠ Λ_λ ^{A_2} for almost all λ > λ_0. By the results of Chapter 3, if the family of broken geodesics is large enough, then the broken non-abelian X-ray transforms agree if and only if the connections are gauge equivalent. The proof relies on an identity derived from the third-order linearisation of Λ_λ^A. Finally, in Chapter 7, we relate the order of conjugacy of a manifold with boundary to the existence of pairs of nontrapped geodesics that intersect exactly once.Show more Item Open Access The logarithmic Quot spaceKennedy-Hunt, PatrickShow more We introduce the logarithmic Quot space which generalises the Quot schemes of a variety X to the geometry of a pair (X,D) with D a simple normal crossing divisor on X. Such logarithmic Quot spaces interact well with degeneration and gluing formulae. We prove that the logarithmic Quot space is a separated and universally closed logarithmic algebraic space, which in nice cases is bounded. A special case, called the logarithmic Hilbert space, parameterises families of proper monomorphisms, and in this way is exactly analogous to the classical Hilbert scheme. The new complexity in the logarithmic space can then be viewed as stemming from the complexity of proper monomorphisms in logarithmic geometry. Our construction generalises the logarithmic Donaldson--Thomas space studied by Maulik--Ranganathan to arbitrary rank and dimension, and the good degenerations of Quot schemes of Li--Wu to simple normal crossings geometries. The logarithmic Quot space admits a tropicalisation called the space of tropical supports, which is defined in terms of a natural moduli functor on the category of rational polyhedral cones. The space of tropical supports, a polyhedral analogue of the Hilbert scheme, is representable by ``piecewise linear spaces'' which are introduced here to generalise cone complexes to allow non--convex geometries. The key to studying geometry of logarithmic Quot spaces is the natural map from logarithmic Quot space to the space of tropical supports. A central example of a Quot scheme is a linear system. We characterise logarithmic linear systems as toric stacks and describe the associated fans in terms of the secondary polytopes of Galfand, Kapranov, and Zelevinksy. We use this characterisation to study logarithmic Pandharipande--Thomas spaces of toric surfaces. We calculate the Euler-Satake characteristics of logarithmic Pandharipande--Thomas spaces in a number of basic examples. Consider a surjection of sheaves O_X^n -> on a torus G_m^n. In establishing boundedness of the logarithmic Quot space, we give a tropical characterisation of the toric compactifications of G_m^n for which the closure of F is logarithmically flat. Our tropical characterisation gives an algebro--geometric interpretation to the Gr\"obner complex, refines a celebrated result of Tevelev, and illustrates how the structure of a piecewise linear space appears in nature.Show more Item Open Access Floer theory and spectral networksNho, Yoon JaeShow more This thesis consists of two papers. In the first paper, we study the relationship between Gaiotto-Moore-Neitzke's non-abelianization map and Floer theory. Given a complete GMN quadratic differential $\phi$ defined on a closed Riemann surface $C$, let $\tilde{C}$ be the complement of the poles of $\phi$. In the case where the spectral curve $\Sigma_{\phi}$ is exact with respect to the canonical Liouville form on $T^{\ast}\tilde{C}$, we show that an ``almost flat" $GL(1;\mathbb{C})$-local system $\mathcal{L}$ on $\Sigma_{\phi}$ defines a Floer cohomology local system $HF_{\epsilon}(\Sigma_{\phi},\mathcal{L};\mathbb{C})$ on $\tilde{C}$ for $0< \epsilon\leq 1$. Then we show that for small enough $\epsilon$, the non-abelianization of $\mathcal{L}$ is isomorphic to the family Floer cohomology local system $HF_{\epsilon}(\Sigma_{\phi},\mathcal{L};\mathbb{C})$. In the second paper, we extend Groman and Solomon's reverse isoperimetric inequality to pseudoholomorphic curves with punctures at the boundary and whose boundary components lie in a collection of Lagrangian submanifolds with intersections locally modelled on $\RR^n\cap (\RR^{k}\times \sqrt{-1}\RR^{n-k})$ inside $\CC^n$. Our construction closely follows the methods used by Duval and Abouzaid and corrects an error appearing in the latter approach.Show more Item Open Access Presentations of Some Finite GroupsSoicher, LeonardShow more Item Open Access Degenerating abelian varieties via log abelian varietiesZhao, HeerShow more This thesis is based on the author's paper [52]. It extends the construction of log Tate curves from [25, §1] to higher dimensions. The approach involves extensive applications of algebraic spaces and higher-dimensional convex geometry to create diverse models for any given split totally degenerate semi-stable abelian variety over a complete discrete valuation field. Such models are used to construct a log abelian variety over the corresponding discrete valuation ring (endowed with the canonical log structure), which extends the given abelian variety.Show more Item Open Access Games, Graphs, and GroupsVersteegen, LeoShow more This dissertation contains various combinatorial results about games, graphs and finite Abelian groups. A linear configuration is said to be *common* in an Abelian group $G$ if every 2-colouring of $G$ yields at least as many monochromatic instances of the configuration as a randomly chosen colouring. In Chapter 2, we show that every configuration containing a 4-term arithmetic progression is uncommon in $\mathbb{F}_p^n$ for primes $p\geq 5$ and large $n$ and in $\mathbb{Z}_p$ for large primes $p$. We call a graph $H$ *strongly common* if for every colouring $\phi$ of $K_n$ with two colours, the number of monochromatic copies of $H$ is at least the number of monochromatic copies of $H$ in a random colouring of $K_n$ with the same density of colour classes as $\phi$. In Chapter 3, we prove that if a graph has odd girth but is not a cycle, then it is not strongly common. We also discuss the commonness property for hypergraphs. A set $A\subset \mathbb{F}_p^n$ is *sum-free* if it contains no elements $x$, $y$, and $z$ such that $x+y=z$. If $p\equiv 2 \mod 3$, the maximal size of a sum-free in $\mathbb{F}_p^n$ is known to be $(p^n+p^{n-1})/3$. In Chapter 4, we show that if a sum-free subset of $\mathbb{F}_p^n$ is larger than $p^n/3-p^{n-1}/6+p^{n-2}$, then it is contained in $(p+1)/3$ cosets of a subspace of codimension 1. For $p=5$, we prove the stronger bound $1.2\cdot 5^{n-1}$. We say that a family $\mathcal{C}$ of graphs on $n$ vertices is a *linear graph code* if the symmetric difference of the edge sets of any two graphs in $\mathcal{C}$ is also the edge set of a graph in $\mathcal{C}$. In Chapter 5, we investigate the maximal size of a linear graph code that does not contain a copy of a fixed graph $H$. In particular, we show that for almost all graphs $H$ with an even number of edges, there exists $\varepsilon_H>0$ such that the size of a linear graph code without a copy of $H$ is at most $2^{\binom{n}{2}}/n^{\varepsilon_H}$. The game *cops and robbers* is played on a graph $G$ by two players, where one of them controls $k$ cop pieces and the other controls a single robber piece. The players take turns to move their pieces along the edges between vertices of $G$, and the cop player attempts to have his pieces *catch* the robber piece. In Chapter 6, we present a new algorithm that determines for any $G$ and $k$ whether the cops can catch the robbers with optimal play. We will also prove sufficient conditions for a cop-win in terms of the independence and domination numbers of $G$. In the *domination game*, two players called Dominator and Staller select vertices in a graph $G$ alternately. A vertex is said to be *dominated* if it has been selected or is adjacent to a selected vertex. Each selected vertex must dominate a new vertex, and the game ends once every vertex in $G$ is dominated. Dominator aims to keep the game as short as possible, while Staller tries to achieve the opposite. In Chapter 7, we prove that for any graph $G$ on $n$ vertices, Dominator has a strategy to end the game in at most $3n/5$ moves. In Chapter 8, we show that if $G$ has $n$ vertices and minimum degree 2, then Dominator has a strategy to end the game in at most $\lceil 10n/17 \rceil$ moves. Finally, in Chapter 9, we show that if we replace the notion of domination by *total domination*, then Dominator has a strategy to end the game in $3n/4$ moves.Show more Item Open Access Explicit moduli spaces for curves of genus 1 and 2Frengley, Samuel; Frengley, Samuel [0000-0002-8904-6253]Show more In this thesis we study pairs of $N$-congruent elliptic curves. That is, we study pairs of elliptic curves whose $N$-torsion subgroups are isomorphic as Galois modules. In particular, we study the moduli surfaces $Z_{N,r}$ which parametrise $N$-congruences of elliptic curves which raise the Weil pairing to a power of $r$. We first study the birational geometry of the surfaces $Z_{N,r}^{\text{sym}}$ which arise as quotients of the surfaces $Z_{N,r}$ by the involution swapping the roles of the $N$-congruent elliptic curves. Building on previous work of Kani--Schanz and Hermann we compute the geometric genus of $Z_{N,r}^{\text{sym}}$ and place them within the Enriques--Kodaira classification in many cases. As a corollary we deduce that the Humbert surface of discriminant $N^2$ is rational if and only if $N \leq 16$, or if $N = 18$, $20$, or $24$. We then study $N$-congruences between quadratic twists of elliptic curves. If $N$ has exactly two distinct prime factors we show that these are parametrised by double covers of certain modular curves. In many, but not all, cases the modular curves in question correspond to the normaliser of a Cartan subgroup of $\operatorname{GL}_2(\mathbb{Z}/N\mathbb{Z})$. By computing explicit models for these double covers we find all pairs, $(N,r)$, such that there exist infinitely many $j$-invariants of elliptic curves $E/\mathbb{Q}$ which are $N$-congruent with power $r$ to a quadratic twist of $E$. We also find an example of a $48$-congruence over $\mathbb{Q}$. We make a conjecture classifying nontrivial $(N,r)$-congruences between quadratic twists of elliptic curves over $\mathbb{Q}$. We give a more detailed analysis of the level 15 case and use elliptic Chabauty to determine the rational points on a modular curve of genus 2 and Jacobian of rank $2$ which arises as a double cover of the modular curve $X(\mathrm{ns}\, 3^+, \mathrm{ns}\, 5^+)$. As a consequence we obtain a new proof of the class number 1 problem. We construct infinite families of pairs of (geometrically non-isogenous) $12$ and $14$-congruent elliptic curves defined over $\mathbb{Q}$. We also find two pairs of $15$-congruent elliptic curves over $\mathbb{Q}$. Our approach is to compute explicit birational models for the moduli surfaces $Z_{N,r}$ for each $N = 12$, $14$ and $15$. A key ingredient in the proof is to construct simple (algebraic) conditions for the $2$, $3$, or $4$-torsion subgroups of a pair of elliptic curves to be isomorphic as Galois modules. These conditions are given in terms of the $j$-invariants of the pair of elliptic curves. Finally, we study an application of congruences to visualising elements of the Tate--Shafarevich group of an abelian variety. In particular, we construct a number of genus 2 Jacobians $J/\mathbb{Q}$ with real multiplication by $\mathbb{Z}[\sqrt{2}]$ and whose Tate--Shafarevich groups contain an element of order $7$. Our approach is to find elliptic curves whose $7$-torsion subgroups are isomorphic as Galois modules to the $(3 \pm \sqrt{2})$-torsion subgroups of $J/\mathbb{Q}$.Show more Item Open Access Contributions to asymptotic theory in nonparametric statisticsDeo, AdityaShow more Nonparametric statistics, in which the parameter of interest is allowed to be infinite-dimensional, provides a theoretical lens through which to analyse the performance of modern data science and machine learning methods. The need for a rigorous understanding of these procedures has never been greater due to their widespread use throughout the sciences and beyond. This thesis studies a variety of nonparametric statistical models, and considers the problems of parameter estimation and uncertainty quantification. Chapter 2 studies the problem of density estimation on the d-dimensional torus, using Wasserstein distances as the loss function. We consider the question of constructing adaptive honest confidence sets, which have uniform coverage guarantees but also shrink at an optimal rate depending on the smoothness of the true parameter, measured on the Besov scale. For the classically studied L^p-losses, it is known that such adaptive confidence sets only exist for a narrow range of smoothnesses; in the case of L^∞-loss, no adaptation is possible at all. For the Wasserstein distances W_p, 1 ≤ p ≤ 2, we uncover a new phenomenon: that in dimensions d ≤ 4, one may construct adaptive confidence sets which can adapt to any smoothness. In higher dimensions, there is a limited range of smoothnesses which can be adapted to, but this interval is wider than for the L^p-distances. The non-existence results are based on an analysis of a related composite hypothesis testing problem, while the construction of our confidence sets follows a risk estimation approach. We prove similar results for the W_1-distance when the sampling domain is R^d. These are the first results in a statistical approach to adaptive uncertainty quantification with Wasserstein distances. Our analysis extends to other weak loss functions such as negative order Sobolev norms. In Chapter 3, we study a general non-linear inverse problem in which the parameter is assumed to have a compositional structure. Taking a Bayesian approach, we consider a deep Gaussian process (DGP) prior, which is designed to leverage the compositional structure of the parameter in order to achieve fast convergence rates. We show that the DGP prior does indeed consistently solve the inverse problem, proving a posterior contraction rate result; this contraction rate depends on the compositional structure of the true parameter, and the DGP prior is able to adapt to this unknown structure. We also show that Gaussian process priors are unable to exploit compositional structures in the same manner, by proving contraction rate lower bounds for a well-studied family of rescaled Whittle-Matérn process priors. When the dimension d is sufficiently large, these lower bounds are polynomially slower than the contraction rate achieved by the DGP prior. These results indicate that when the parameter has a complex structure other than straightforward Sobolev or Hölder smoothness, one must look beyond Gaussian priors to obtain good performance from the Bayesian method. Two examples of inverse problems encompassed by our general framework are the Darcy flow problem and the steady-state Schrödinger equation. Chapter 4 studies the empirical process arising from a multi-dimensional diffusion process with periodic drift field and diffusivity. In particular, we address the question of proving the Donsker property for Besov classes of functions. Donsker classes have been completely characterised in the case of scalar diffusions, but these arguments rely on the use of local time, which does not exist in higher dimensions. Instead, we utilise the smoothing properties of the generator of the diffusion process, an elliptic second order partial differential operator, to show that the diffusion empirical process is smoother than its classical i.i.d. analogue. As an application, we deduce precise asymptotics for the Wasserstein-1 distance between the occupation measure and the invariant measure.Show more Item Open Access The Taylor-Wiles method for reductive groupsWhitmore, DmitriShow more We construct a local deformation problem for residual Galois representations ρ valued in an arbitrary reductive group Ĝ which we use to develop a variant of the Taylor–Wiles method. Our generalization allows Taylor–Wiles places for which the image of Frobenius is semisimple, a weakening of the regular semisimple constraint imposed previously in the literature. We introduce the notion of Ĝ-adequate subgroup, our corresponding ‘big image’ condition. When Ĝ → GLn is a faithful irreducible representation, we show that a subgroup is Ĝ-adequate if it is GLn-irreducible and the residue characteristic is sufficiently large. We apply our ideas to the case Ĝ = GSp4 and prove a modularity lifting theorem for abelian surfaces over a totally real field F which holds under weaker hypotheses than in the work of Boxer–Calegari–Gee–Pilloni. We deduce some modularity results for elliptic curves over quadratic extensions of F. We further apply our ideas to V. Lafforgue’s construction of the global Langlands correspondence in characteristic p for a semisimple group Ĝ, proving an automorphy lifting theorem under the assumption of Ĝ-adequate residual image, a weakening of the Ĝ-abundant condition appearing previously in the literature. We deduce potential automorphy of everywhere unramified Galois representations with Ĝ-adequate residual image. We also give results concerning existence of global cyclic base change and the possible finite images of automorphic Galois representations. The main technical difficulty is in proving an instance of local-global compatibility at certain level structures deeper than parahoric level, which we require for our implementation of the Taylor–Wiles method.Show more Item Open Access Phase transition for cutoff for random walks on random graphsŠarković, AndelaShow more In this thesis, we analyse the cutoff phenomenon on two different random graph models. First, we consider a variant of the configuration model with an embedded community structure and study the mixing properties of a simple random walk on it. Every vertex has a given number of internal, degint ≥ 3, and outgoing, degout, half-edges. Given a stochastic matrix Q, we pick a random perfect matching of the half-edges subject to the constraint that each vertex v has degint(v) neighbours inside its community and the proportion of outgoing half-edges from community i matched to a half-edge from community j is Q(i,j). Assuming the number of communities is constant and that they all have comparable sizes, we prove the following dichotomy: a simple random walk on the resulting graph exhibits cutoff if and only if the product of the Cheeger constant of Q and log n (where n is the number of vertices) diverges. In [5], Ben-Hamou established a dichotomy for cutoff for a non-backtracking random walk on a similar random graph model with 2 communities. We prove that the same characterisation of cutoff holds for a simple random walk. In the second part of the thesis, we analyse a graph G* obtained from a finite deterministic graph G = (V,E) by considering a random perfect matching of V and adding the corresponding edges to G with weight ε, while assigning weight 1 to the original edges of G. For various sequences of graphs Gn and corresponding weights εn, we establish if the (weighted) random walk on G*n has cutoff. In particular, we show a phase transition for two families of graphs, graphs with polynomial growth of balls, and graphs where the entropy of the simple random walk grows linearly up to the time of order log|Vn|. These include in particular tori, expander families and locally expanding families. We also show that this phase transition is sharp in the case of expander graphs and vertex transitive graphs with polynomial growth of balls.Show more Item Controlled Access Equivariant line bundles with connection on the Drinfeld upper half-space Ω⁽²⁾Zhu, YiyueShow more 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 GL_{2}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 GL_{3}by studying the Drinfeld upper half-space Ω⁽²⁾ of dimension 2.Show more Item Open Access Results in Ramsey theory and extremal graph theoryMetrebian, RobertShow more 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 $p = 1-\frac{1}{r}$.Show more Item Open Access Semiparametric Methods for Two Problems in Causal Inference using Machine LearningKlyne, HarveyShow more 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.Show more Item Open Access A Walk through the Forest: the Geometry and Topology of Random SystemsHalberstam, NoahShow more 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.Show more Item Open Access Random conformally covariant metrics in the planeHughes, LiamShow more 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.Show more