Department of Pure Mathematics and Mathematical Statistics (DPMMS)https://www.repository.cam.ac.uk/handle/1810/2137472023-12-07T09:55:55Z2023-12-07T09:55:55Z4471Hitchin Functionals, h-Principles and Spectral InvariantsMayther, Laurencehttps://www.repository.cam.ac.uk/handle/1810/3619152023-12-06T01:44:42Zdc.title: Hitchin Functionals, h-Principles and Spectral Invariants
dc.contributor.author: Mayther, Laurence
dc.description.abstract: 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.
Absolutely Continuous Stationary MeasuresKittle, Samuelhttps://www.repository.cam.ac.uk/handle/1810/3610442023-11-18T01:44:48Zdc.title: Absolutely Continuous Stationary Measures
dc.contributor.author: Kittle, Samuel
dc.description.abstract: 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*<sub>2</sub>($\mathbb{R}$) acting on *X* = *P*<sup>1</sup>($\mathbb{R}$), 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.
Homological stability of spaces of manifolds via E_k-algebrasSierra, Ismaelhttps://www.repository.cam.ac.uk/handle/1810/3610012023-11-17T01:41:17Zdc.title: Homological stability of spaces of manifolds via E_k-algebras
dc.contributor.author: Sierra, Ismael
dc.description.abstract: In this thesis we study homological stability properties of different families of spaces using the technique of cellular *E<sub>k</sub>*-algebras. Firstly, we will consider spin mapping class groups of surfaces, and their algebraic analogue —quadratic symplectic groups— using cellular *E<sub>2</sub>*-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 $\mathbb{F}$<sub>2</sub>-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*<sub>*g*,1</sub> ∶= *D*<sup>2*n*</sup>#(*S<sup>n</sup>* x *S<sup>n</sup>*)<sup>#*g*</sup>. 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.
Topics in symplectic Gromov–Witten theoryHirschi, Amandahttps://www.repository.cam.ac.uk/handle/1810/3610002023-11-17T01:41:24Zdc.title: Topics in symplectic Gromov–Witten theory
dc.contributor.author: Hirschi, Amanda
dc.description.abstract: 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 $\mathbb{P}$<sup>2</sup>, 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.
Birational Invariance of Punctured Log Gromov-Witten Theory and Intrinsic Mirror ConstructionsJohnston, Samuelhttps://www.repository.cam.ac.uk/handle/1810/3609232023-11-16T01:45:54Zdc.title: Birational Invariance of Punctured Log Gromov-Witten Theory and Intrinsic Mirror Constructions
dc.contributor.author: Johnston, Samuel
dc.description.abstract: 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.
Gross-Siebert Mirror Ring for Smooth log Calabi-Yau PairsWang, Yuhttps://www.repository.cam.ac.uk/handle/1810/3607602023-11-11T01:54:39Zdc.title: Gross-Siebert Mirror Ring for Smooth log Calabi-Yau Pairs
dc.contributor.author: Wang, Yu
dc.description.abstract: 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).
Mirrors to Toric Degenerations via Intrinsic Mirror SymmetryGoncharov, Evgenyhttps://www.repository.cam.ac.uk/handle/1810/3587932023-11-03T01:40:47Zdc.title: Mirrors to Toric Degenerations via Intrinsic Mirror Symmetry
dc.contributor.author: Goncharov, Evgeny
dc.description.abstract: 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.
Extremal results for graphs and hypergraphs and other combinatorial problemsJanzer, Barnabáshttps://www.repository.cam.ac.uk/handle/1810/3576722023-09-29T00:45:02Zdc.title: Extremal results for graphs and hypergraphs and other combinatorial problems
dc.contributor.author: Janzer, Barnabás
dc.description.abstract: 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 r<s, there are n-vertex graphs containing n^{r−o(1)} copies of K_s such that any K_r is contained in at most one K_s. This gives a natural generalisation of the Ruzsa–Szemerédi (6,3)-problem. We also show that there are properly edge-coloured n-vertex graphs with n^{r−1−o(1)} copies of K_r such that no K_r is rainbow, answering a question of Gerbner, Mészáros, Methuku and Palmer about generalised rainbow Turán numbers.
In Chapter 6, we continue the study of the generalised rainbow Turán problem: how many copies of H can a properly edge-coloured graph on n vertices contain if it has no rainbow copy of F? We determine the order of magnitude in essentially all cases when F is a cycle and H is a path or a cycle. In particular, we answer a question of Gerbner, Mészáros, Methuku and Palmer.
In Chapter 7, we consider the following problem. Let g(n,H) be the smallest k such that we can assign a k-edge-colouring f_v of K_n to each vertex v in K_n with the property that for any copy H_0 of H in K_n, there is some u∈V(H_0) such that H_0 is rainbow in f_u. This function was introduced by Alon and Ben-Eliezer, and we answer several of their questions. In particular, we determine all connected graphs H for which g(n,H)=n^{o(1)}, and show that for all ε>0 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<y_i for all i. Engel, Mitsis, Pelekis and Reiher showed that for any weak antichain A in Z^n, the sum of the sizes of its (n−1)-dimensional projections must be at least as large as its size |A|. They asked what the smallest possible value of the gap between these two quantities is in terms of |A|. We give an explicit weak antichain attaining the minimum for each possible value of |A|.
Finally, in Chapter 13, we study the following problem. Esperet, Gimbel and King introduced the orientation covering number of a graph G as the smallest k with the property that we can choose k orientations of G such that whenever x, y, z are vertices of G with xy,xz∈E(G), then there is a chosen orientation in which both xy and xz are oriented away from x. We prove that the orientation covering number of G is the same as that of K_{χ(G)}, answering a question of Esperet, Gimbel and King. We also determine the orientation covering numbers of complete graphs.
Cubical small-cancellation theory and large-dimensional hyperbolic groupsArenas, Macarenahttps://www.repository.cam.ac.uk/handle/1810/3575372023-09-27T00:46:20Zdc.title: Cubical small-cancellation theory and large-dimensional hyperbolic groups
dc.contributor.author: Arenas, Macarena
dc.description.abstract: 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.
Two-Dimensional Discrete Gaussian Model at High TemperaturePark, Jiwoonhttps://www.repository.cam.ac.uk/handle/1810/3547842023-08-23T00:43:25Zdc.title: Two-Dimensional Discrete Gaussian Model at High Temperature
dc.contributor.author: Park, Jiwoon
dc.description.abstract: 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.