Theses - Pure Mathematics and Mathematical Statistics

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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.
Open Access
Extreme values of non-Gaussian fields
Hofstetter, Michael
In recent years the extremal behaviour of log-correlated spatial Gaussian processes has drawn a lot of attention. Among many other results, it is known for the lattice discrete Gaussian free field (DGFF) in $d=2$ as well as for general log-correlated Gaussian fields, that the limiting law of the centred maximum is a randomly shifted Gumbel distribution. While for Gaussian fields the picture is rather complete, many difficulties arise when the field of interest in non-Gaussian. In this thesis, we study the extreme values of the non-Gaussian sine-Gordon field and $\textit{P}$($\phi$)$_{2}$ field on the unit torus in dimension $d=2$. Our analysis includes the $\phi_2^4$ field. To this end, we develop tools, which we subsequently use to establish results for their extreme values, which are analogous to the known results for the Gaussian free field in $d=2$. In particular, we prove that the centred global maximum of the non-Gaussian fields of interest converges in distribution to a randomly shifted Gumbel distribution, which confirms the conjectured behaviour of these fields. For the sine-Gordon field, we extend the scope of the extreme values to the local extremal process, which also includes information about the local extrema of the field. More precisely, we prove that this random measure converges to a certain Poisson point process with random intensity measure. For both the sine-Gordon field and the $\textit{P}$($\phi$)$_{2}$ field, the main tool is a coupling result between the well-studied Gaussian free field and the field of interest. This allows to represent the non-Gaussian field as a sum of the Gaussian free field and a difference field for which further probabilistic regularity estimates are established using renormalisation group and stochastic control techniques, in particular the Polchinski renormalisation group approach and the Bou$\acute e$-Dupuis variational formula.
Open Access
Polynomial Multi-Curve Models And Extensions In Mathematical Finance
Du Toit, Thomas
This thesis is organized into three chapters: In the first chapter, we introduce the changes that have occurred in the fixed-income market due to the credit crisis in 2007–2008. We then discuss the impact of this crisis on the pre-crisis relation between zero-coupon bonds and forward rate agreements written on the Libor/Euribor rates. This particularly, this includes describing the dynamics of spread rates in addition to interest rates, which to date had not been part of pre-crisis models. Following this, we introduce a multicurve model set-up in order to compute the non-arbitrage prices of a forward rate agreement. This is done by assuming that the underlying factor processes, which describe the dynamics of the interest and spread rates are given by a diffusion process such that bond prices can be expressed as polynomials and the forward Libor rates as rational functions. At the end of the first chapter, we consider a multi-curve model set-up in the discrete time setting. As in the continuous case, the bond prices can be expressed as polynomial and the forward Libor rates as rational functions. In this case we consider linear functions and give a calibration method. In the second chapter, we extend from diffusion processes to jump processes, allowing the factor process to have jumps. Our main result is the classification of such models for which the bond prices can be expressed as polynomials. These models are arbitrage-free in the sense that the discounted zero-coupon bond prices are local martingales. In the third chapter, we consider a specific type of volatility models, which all have the characteristic that the moments of the stock can be expressed as a polynomial. Further, we show that there is a relation between the k-th moment and the degree of the polynomial.
Open Access
Modular Temperley-Lieb Theory
Spencer, Robert; Spencer, Robert [0000-0003-2432-3392]
This dissertation considers the representation theory of Temperley-Lieb algebras, TLₙ , along with some related cellular algebras, over positive and mixed characteristic fields. The Temperley-Lieb algebras are planar diagrammatic algebras. They are generated by “(n, n)-diagrams”: planar pair matchings between n points. Multiplication is diagrammatic composition with closed loops resolving to linear factors of δ. Their representation theory depends greatly on the characteristic of the underlying ring, as well as the choice of δ. This thesis shows that the representation theories of the Temperley-Lieb algebras in the semi-simple case, characteristic zero, or “unmixed” positive characteristics are special cases of the mixed-characteristic theory. We do this purely diagrammatically, without recourse to any Schur-Weyl dualities. We then find the generating idempotent of the principle projective module. In the semi-simple case this is the famed Jones-Wenzl idempotent. However, this celebrated object is not well defined in arbitrary characteristic and we extend the work of Burrull, Libedinsky and Sentinelli in the unmixed positive characteristic case to the remaining two characteristic classes. Armed with these new elements, we consider truncations of TLₙ by certain tensors of such generalised idempotents. These algebras are still cellular and can be considered as endomorphism rings of tensor products of indecomposable Uq(𝔰𝔩₂)-modules in the given characteristic. We determine the cellular data for a number of classes of such truncations and give indications of a general, if computationally intensive, approach for the remaining classes. Finally, we examine how some of these results may extend to “webs”, generalisations of Temperley-Lieb diagrams, and prove a conjecture by Elias.
Open Access
Ranks of tensors and polynomials, with combinatorial applications
Karam, Thomas
This thesis will consist of three main chapters. It is a standard fact of linear algebra that every matrix with rank k contains a k ⨯ k submatrix with rank k. In Chapter 2 we generalise this fact asymptotically to a class of notions of rank for higher-order tensors, containing in particular the tensor rank, the slice rank and the partition rank. We show that for every integer d ⩾ 2 and every notion R in this class of notions of rank, there exist functions F$_{d,R}$ and G$_{d,R}$ such that if an order-d tensor has R-rank at least G$_{d,R}$(l) then we can restrict its entries to a product of sets X$_{1}$ ⨯ … ⨯ X$_{d}$ such that the restriction has R-rank at least l and the sets X$_{1}$,…,X$_{d}$ each have size at most F$_{d,R}$(l). Combining the proof methods that we use to prove this result with a few additional ideas then allows us to show that under a very natural condition we can furthermore require the sets X$_{1}$,…,X$_{d}$ to be pairwise disjoint. In Chapter 3 we extend to the case of restricted subsets a result of Green and Tao on the equidistribution of high-rank polynomials over finite prime fields. We show that for every fixed prime integer p, for every integer d ∈ [2, p-1], and for every non-empty subset S of F$_{p}$, it is true uniformly in n that if P: F$_{p}^{n}$ → F$_{p}$ is a polynomial with degree at most d such that P(x) is not approximately equidistributed on F$_{p}$ when x is chosen uniformly at random in S$^{n}$, then P coincides on S$^{n}$ with a polynomial which can be expressed as a function of a bounded number of polynomials of degree at most d-1. Our argument uses two results which are known by that point: the second main result of Chapter 2, and the fact that an order-d tensor over F$_{p}$ with high partition rank necessarily has high analytic rank. In Chapter 4 we prove approximation results for conditions on {0,1}$^{n}$ and similar sets when those conditions are defined using polynomials from F$_{p}^{n}$ to F$_{p}$ for some prime p. We show in particular that for every non-empty subset S of F$_{p}$, if for some linear forms φ$_{i}$ on F$_{p}^{n}$ and some subsets E$_{i}$ of F$_{p}$, the set U of all x ∈ S$^{n}$ satisfying all conditions φ$_{i}$(x) ∈ E$_{i}$ is dense inside S$^{n}$ then there exist a bounded number of pairs (θ$_{i}$, T$_{i}$), where the θ$_{i}$ are linear forms on F$_{p}^{n}$ and the T$_{i}$ are subsets of F$_{p}$, such that the set of x ∈ S$^{n}$ satisfying all conditions θ$_{i}$(x) ∈ T$_{i}$ is contained in U and has inside S$^{n}$ approximately the same density as U has inside S$^{n}$. As an application, we rule out a class of potential counterexamples to a first unsolved case of the polynomial density Hales-Jewett conjecture. We also generalise our approximation results in some other directions: in particular we deduce an approximation result (with a weaker formulation) for polynomials of small degree from the main result of Chapter 3.
Open Access
Modern Methods for Variable Significance Testing
Lundborg, Anton Rask
This thesis concerns the ubiquitous statistical problem of variable significance testing. The first chapter contains an account of classical approaches to variable significance testing including different perspectives on how to formalise the notion of `variable significance'. The historical development is contrasted with more recent methods that are adapted to both the scale of modern datasets but also the power of advanced machine learning techniques. This chapter also includes a description of and motivation for the theoretical framework that permeates the rest of the thesis: providing theoretical guarantees that hold uniformly over large classes of distributions. The second chapter deals with testing the null that Y ⊥ X | Z where X and Y take values in separable Hilbert spaces with a focus on applications to functional data. The first main result of the chapter shows that for functional data it is impossible to construct a non-trivial test for conditional independence even when assuming that the data are jointly Gaussian. A novel regression-based test, called the Generalised Hilbertian Covariance Measure (GHCM), is presented and theoretical guarantees for uniform asymptotic Type I error control are provided with the key assumption requiring that the product of the mean squared errors of regressing Y on Z and X on Z converges faster than n$^{-1}$, where n is the sample size. A power analysis is conducted under the same assumptions to illustrate that the test has uniform power over local alternatives where the expected conditional covariance operator has a Hilbert--Schmidt norm going to 0 at a $\sqrt[n]{n}$-rate. The chapter also contains extensive empirical evidence in the form of simulations demonstrating the validity and power properties of the test. The usefulness of the test is demonstrated by using the GHCM to construct confidence intervals for the boundary point in a truncated functional linear model and to detect edges in a graphical model for an EEG dataset. The third and final chapter analyses the problem of nonparametric variable significance testing by testing for conditional mean independence, that is, testing the null that E(Y | X, Z) = E(Y | Z) for real-valued Y. A test, called the Projected Covariance Measure (PCM), is derived by considering a family of studentised test statistics and choosing a member of this family in a data-driven way that balances robustness and power properties of the resulting test. The test is regression-based and is computed by splitting a set of observations of (X, Y, Z) into two sets of equal size, where one half is used to learn a projection of Y onto X and Z (nonparametrically) and the second half is used to test for vanishing expected conditional correlation given Z between the projection and Y. The chapter contains general conditions that ensure uniform asymptotic Type I control of the resulting test by imposing conditions on the mean-squared error of the involved regressions. A modification of the PCM using additional sample splitting and employing spline regression is shown to achieve the minimax optimal separation rate between null and alternative under Hölder smoothness assumptions on the regression functions and the conditional density of X given Z=z. The chapter also shows through simulation studies that the test maintains the strong type I error control of methods like the Generalised Covariance Measure (GCM) but has power against a broader class of alternatives.
Embargo
On the Variational Theory of Yang-Mills-Higgs Energies and the Structure of the Singular Set of $\mathbb{Z}$$_{2}$-Harmonic Spinors
Parise, Davide
This dissertation investigates Yang-Mills-Higgs energies and $\mathbb{Z}$$_{2}$-harmonic spinors, two classes of objects at the interface between analysis and geometry. In the first part, we present joint work with Alessandro Pigati and Daniel Stern on the variational theory of the self-dual U(1)-Yang-Mills-Higgs functionals on a closed Riemannian manifold (M,g). This natural family of energies associated with sections u: M → L and metric connections ∇ of Hermitian line bundles has long been studied in differential geometry and theoretical physics. We show how its variational theory is related to the one of the (n − 2)-area functional by establishing a Γ-convergence result in the spirit of Modica and Mortola. With this in hand, we study the comparison between the corresponding min-max theories. Therefore, we relate the classical theory for C$^{1}$-unctionals to the min-max theory introduced by Almgren and Pitts in the setting of geometric measure theory. In particular, we prove that min-max values for the latter always provide a lower bound for the former. En route to proving this comparison, we introduce the gradient flow of the Yang-Mills-Higgs energies and establish a Huisken-type monotonicity result along the flow. We complement this by studying the long-time existence, uniqueness and continuous dependence on the initial data of the flow. In the second part of the dissertation, we focus on the notion of $\mathbb{Z}$$_{2}$-harmonic spinors. These objects were introduced in foundational work of Taubes when studying the compactification of moduli spaces of flat PSL(2, C)-connections over 3-manifolds. Their role is to abstract various limiting phenomena and they also appear in other contexts, for instance when dealing with the moduli space of solutions to the Kapustin-Witten equations, the Vafa-Witten equations, and the Seiberg-Witten equations with multiple spinors. In all of these cases, the role played by the zero loci of $\mathbb{Z}$$_{2}$-harmonic spinors is crucial. Based on the pioneering techniques of Simon in the setting of minimal submanifolds, we obtain structural results on the singular set of $\mathbb{Z}$$_{2}$-harmonic spinors, subject to the validity of frequency monotonicity (a condition implied by, for instance, an appropriate regularity assumption). More precisely, we prove uniqueness of the blow-ups for every point, excluding an exceptional set of zero 2-dimensional Hausdorff measure, hence answering a question left open in the work of Taubes. From this, we infer 2-rectifiability of the singular set and the branch set. We conclude by analysing the setting of lowest frequency value, in which case we have that, locally, the branch set is a C$^{1, \alpha}$ -submanifold.