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We study parahoric Hitchin fibrations over complex smooth projective curves. These are analogues of the Grothendieck resolutions in the classical Springer theory. In particular, we describe the generic fibers of the parahoric Hitchin fibrations. When the parahoric subgroup corresponds to a parabolic subgroup of a semisimple complex linear algebraic group, we recover some results for type A and generalize them to groups of other types.

The horizontal mean curvature flow is an evolution of a hypersurface, which is interesting not only in a theoretical framework but also in applied science, for example in neurogeometry and computer science (e.g. [13, 14, 15]). The associated equation, roughly speaking, describes the motion of a hypersurface embedded in a sub-Riemannian geometry (e.g. the Heisenberg group or a Carnot group, see [4, 43]) in relation to its horizontal mean curvature.

This thesis is dedicated to the study of rational solutions to Diophantine equations, or equivalently rational points on varieties. In particular, we consider families of equations and ask how frequently rational solutions exist to equations in these families. We also study the frequency of examples of local-global principles, such as the Hasse principle and weak approximation, for equations in these families. We develop an asymptotic formula, in Chapter 3, for the number of biquadratic fields K of bounded discriminant for which the equation N_{K/Q}(x) = c fails to satisfy the Hasse principle for each c in Q*. In Chapter 4, we study the frequency of Hasse principle failures in a particular family of rational surfaces. Families of varieties whose base is a hypersurface of low degree is the subject of Chapter 5. In Chapter 6, we describe the frequency with which conics of the form aX² +bY² +cZ² = 0 have non-trivial rational points. Finally, in the last chapter we study weak approximation for certain quadric surface bundles over P².

This thesis is about a class of complex algebraic threefolds known as flops, which are an important part of the Minimal Model Program in birational geometry. Threefold flops are commonly studied via their enumerative invariants, and here we focus on one such type of invariant: refined Donaldson-Thomas invariants. We develop theoretical aspects of refined Donaldson-Thomas theory for threefold flops, which allow us to understand their stability conditions and cyclic A∞-deformation theory. With these new methods, we are able to sidestep common computational barriers in the field and fully determine the Donaldson-Thomas invariants for an infinite family of flops, which includes many new examples. Our results show that a refined version of the strong-rationality conjecture of Pandharipande-Thomas holds in this setting, and also that refined Donaldson-Thomas invariants are not sufficiently fine to determine flops. Where possible we work motivically, computing invariants in the Grothendieck ring of varieties, but we also produce Hodge theoretic realisations of the invariants.

The aim of this thesis is to study both holomorphic and algebraic flows on Shimura varieties. The first part of the thesis studies holomorphic flows, the main result is a hyperbolic analogue of the Bloch-Ochiai Theorem in the context of mixed Shimura varieties. This extends previous results of Ullmo and Yafaev for co-compact pure Shimura varieties. The proof follows the template set by the hyperbolic Ax-Linedmann-Weierstrass theorem of using the PilaWilkie counting theorem together with some volume inequalities to prove our result. The heart of the proof consists of two volume inequalities, first one for the intersection of a definable set with hyperbolic balls in Hermitian symmetric domains of non-compact type. The second for the intersection of a definable portion of a holomorphic curve with a fundamental domain for the action of a congruence group on a Hermitian symmetric domain of non-compact type. In the second part we study totally geodesic subvarieties of mixed Shimura varieties and algebraic flows. We show that contrary to the case of pure Shimura varieties, there is in general no inclusion either way between the concept of weakly special and totally geodesic subvariety in the mixed setting. Then we report an argument communicated by N. Mok which shows that unlike in the pure case there are totally geodesic submanifolds of a mixed Shimura variety that are not homogeneous. Finally we use these results on totally geodesic subvarieties to state and prove a generalisation of results of Ullmo and Yafaev on algebraic flows on pure Shimura varieties to the mixed case. The proof follows the pure case and uses a theorem of Ratner in arithmetic dynamics.

In this thesis we generalise the theory of exceptional covers of varieties to the context of difference algebraic geometry. We formulate the notion of difference exceptional covers and prove a difference exceptionality criterion analogous to the classical situation. We also obtain a Galois correspondence for difference ring extensions in the style of Borceux- Janelidze. Along the way we explore the background of exceptional varieties, Grothendieck's Galois categories and discuss a proof of the classical exceptionality criterion. An overview of difference algebraic geometry is given, motivated by the work of Tomaˇsi´c.

We construct shift operators on equivariant symplectic cohomology which generalise the shift operators on equivariant quantum cohomology in algebraic geometry. That is, given a Hamiltonian action of the torus $T$, we assign to a cocharacter of $T$ an endomorphism of $(S^1 \times T)$-equivariant Floer cohomology based on the equivariant Floer Seidel map. We prove the shift operator commutes with a connection. This connection is a multivariate version of Seidel's $q$-connection on $S^1$-equivariant Floer cohomology and generalises the Dubrovin connection on equivariant quantum cohomology. We prove that the connection is flat, which was conjectured by Seidel. As an application, we compute these algebraic structures for a few specific examples and provide a method to compute them for toric manifolds using the moment polytope.

In this thesis we obtain new estimates and regularity results for some fully nonlinear elliptic equations arising in conformal geometry. Our first set of new results concern local pointwise second derivative estimates for elliptic solutions to the σk-Yamabe equation. In Chapter 2 we obtain such estimates for W2,p-strong solutions on Euclidean domains, addressing both the so-called positive and negative cases. We explore two methods for obtaining these estimates: an integrability improvement argument coupled with Moser iteration, and a method using the Alexandrov-Bakelman-Pucci estimate. Our estimates are obtained in the more general context of augmented Hessian equations. In Chapter 3 we obtain similar estimates for smooth solutions on manifolds when k = 2. Our work here contributes to a growing literature on the regularity theory for the σk-Yamabe equation and, from a broader perspective, the regularity theory for fully nonlinear, non-uniformly elliptic equations. In Chapter 4 we study the existence of conformal metrics satisfying g-1 Agτ ∈ Γ2+, where Agτ is the trace-modified Schouten tensor. When τ = 1, this is an important question in the context of the σ2-Yamabe problem, and it is also of independent geometric and topological interest when τ ≤ 1. Our focus will be on three dimensions; we prove a new existence result when τ < 1, and obtain a new proof of a result of Ge, Lin & Wang [GLW10] when τ = 1, with an eye towards tackling some related problems. In Chapter 5 we obtain integral estimates for a fourth order perturbation of the (trace-modified) σ2-Yamabe equation in three dimensions, in the spirit of Chang, Gursky & Yang [CGY02b]. Our study of this equation is partly motivated by the existence problems considered in Chapter 4, but also from the analytic viewpoint of using fourth order regularisations to study non-uniformly elliptic PDEs of second order.

We present an order structure for tiling substitution systems of the plane. The order structure gives rise to a space filling curve which is defined over an iterative system akin to the given tiling substitution. We use this space filling curve to define a label set on the original tiles, inducing a new tiling with a factor map to the original. On the other hand, our new tiling also defines an almost one-to-one factor map to a one-dimensional tiling obtained from 'flattening' the space filling curve. We view this as a way of reducing dimension, giving new insights on 2-dimensional substitution tilings using symbolic dynamics.

Surface curvature plays a vital role in many biological processes. Examples include the organising of proteins in cell membranes, the tiling of cells in epithelial layers and the growth of virus capsids. A major technological benefit of micro- and nanoscale curvature is that it can guide colloidal self-assembly, a property which is of major importance in fields such as drug-delivery, biosensor fabrication, and the development of meta-materials. Recent advances in fabrication techniques, such as 3D-printing, have made a rich library of geometries available to experimentalists and engineers seeking to realise these complex and fascinating systems. Here, we use bespoke simulations and theoretical models to study the effects of surface curvature on two-dimensional systems of isotropically attractive colloids. We identify four important properties: the finite but boundary-free area of closed surfaces, the minimum perimeter of a patch with a given area, the difference between the Euclidean and geodesic separation of points on the surface, and the frustration of the hexagonal lattice in regions of non-zero Gaussian curvature. We also show that competition between these effects produces a range of novel behaviours. Starting from the simplest example of a sphere, we show that surface curvature has a strong effect on the gas-liquid nucleation profile and the size of the critical nucleus, as well as destroying the equivalence of the canonical and grand canonical ensembles. Then, focussing on surfaces with non-uniform curvature, we use tori to demonstrate that the different thermodynamic phases are localised to specific regions of the surface, and the transitions between them involve the translation of the colloidal assemble. Finally, we investigate the cone, where there is no Gaussian curvature but the mean curvature varies. We find that surface curvature can stabilise chiral and achiral crystals, and the ground states depend on both the range of the potential and whether it acts through Euclidean space or along geodesics.

Ever since Kronheimer's celebrated hyperkähler construction of gravitational instantons thirty years ago, many similar constructions have appeared in the mathematical literature, eventually producing a vast class of holomorphic symplectic manifolds nowadays called Conical Symplectic Resolutions (CSRs). So far, they have been considered in representation theory, algebraic and differential geometry, and also in theoretical physics as they arise from certain supersymmetric gauge theories. However, much of their symplectic geometry is unexplored. This thesis aims to make the first steps in this direction. There are two essentially different natural real symplectic structures on CSRs; for this reason, this thesis splits into two major parts. The first part views CSRs as Liouville real symplectic manifolds (the symplectic form is exact). We investigate the presence of smooth closed exact Lagrangian submanifolds. The main theorem is that for any CSR there is a non-empty collection of non-isotopic such Lagrangians which arise from contracting C�-actions that act on the holomorphic symplectic structure by weight one. Lagrangians obtained from this method will be called minimal components. In particular, we use these to obtain (non-zero) lower bounds for the rank of symplectic cohomology. We will investigate two large subfamilies of CSRs where we are able to count minimal components and describe them. The family of Nakajima Quiver Varieties, in particular, we study in detail those of Dynkin type A; and the family of resolutions of Slodowy varieties that arise from the representation theory of semisimple Lie algebras and involve Springer theory. In the latter, we also obtain some further families of Lagrangians using Springer theoretic methods and certain crystal operators. The second part studies CSRs with respect to non-exact symplectic structures arising from S1-invariant Kähler structures on CSRs. We construct a family of symplectic cohomologies for them, labelled by different contracting C*-actions. Although we prove that these vanish, this vanishing result allows us to obtain an action-dependent filtration by ideals on the ordinary cohomology of a CSR. Using Morse-Bott-Floer spectral sequences, we give many examples that explicitly describe these filtrations, in particular, we show an example where distinct filtrations arise from different C*-actions.

We aim to contribute to the problem of finding desingularisations of codimension-1 holomorphic foliations in arbitrary ambient dimension. To this end, we prove an alternate characterisation of simple singularities, which leads to an equivalent condition for the existence of a desingularisation in the non-dicritical case. We then give some results and conjectures to show when this condition is satisfied.

Shapes are nonlinear, multifarious objects and it is a nontrivial task to attach meaning to statements about their comparison. A central problem in the field of shape analysis is therefore to make sense of the notion of a metric on an abstract shape space. Through the lens of infinite-dimensional Riemannian geometry one can endow such spaces with different metrics stemming from nonrigid group actions to compute so-called geodesic matchings between shapes that share diffeomorphic or homeomorphic orbits. Owing to shape analysis being a relatively new field in the history of mathematics, several open questions require answering to reconcile theoretical development with numerical approximation. This project sets out to unite the framework of diffeomorphometry with linear functional analysis amenable to finite element methods. Shapes in this project will be images, curves or even measures, and we derive and compute convergent numerical approximations to the geodesic equations related to the motion on the aforementioned groups.

In this thesis we prove the birational non-rigidity of Picard rank 1 Fano 3-folds in codimension 4 having Fano index 1. This is done by explicitly constructing Sarkisov links for these varieties to other Mori fibre spaces. We also consider those Fano 3-folds in codimension 4 and Fano index 1 having Picard rank 2, and we identify a Mori fibre space in its birational equivalence class. In a final short chapter, we begin this program for Fano 3-folds in codimension 4 having Fano index 2 by demonstrating a construction of them as quotients of index 1 Fano 3-folds.

The classification of singular Fano 3-folds remains an open problem in algebraic geometry. The purpose of this thesis is to prove the existence of new families of singular Fano 3-folds, specifically those whose anticanonical embedding is in codimension 4. This is achieved using unprojections. The projection of a scheme Y with coordinate ring k[Y ] := C[x0; : : : ; xn]=IY is a scheme X defined by the coordinate ring k[X] := C[x0; : : : ; xm]=(IY \ C[x0; : : : ; xm]) where m < n. Unprojection is a method of adjoining variables and equations to the ring of X to recover Y . In Chapter 2, we define a new unprojection format allowing us to construct Gorenstein rings of codimension n + 2 from Gorenstein rings of codimension n. Following the naming conventions in the literature, these unprojections are type II and should be considered as type II1 unprojections. We focus on the case where n = 2. This constructs codimension 4 Gorenstein rings which we define explicitly in Chapter 3. In Chapter 4, we use codimension 4 Gorenstein rings to construct and prove the existence of 16 new families of codimension 4 Fano 3-folds. Demonstrably, each family corresponds to a distinct Hilbert series. By using type II1 unprojections as in [33], we construct a second topologically distinct family of codimension 4 Fano 3-folds for these Hilbert series. The Hilbert scheme of these Fano 3-folds, therefore, contains at least 2 components that parametrize distinct Fano 3-folds. In Chapter 5, we consider pre-existing families of codimension 4 Fano 3-folds which are described by [10] but also constructible using our methods.

Geometric Invariant Theory (GIT) is a powerful theory for constructing and studying the geometry of moduli spaces in algebraic geometry, as quotients of a parameter space by a linear group action. However GIT only applies to classification problems involving reductive group actions, and only produces a moduli space for the subclass of "stable" objects. Non-Reductive GIT was developed over the last ten years to overcome these limitations. The aim of this thesis is firstly to describe a systematic way of constructing and studying the geometry of quotients using both classical and Non-Reductive GIT, and secondly to show how these methods can be applied to the classification problem for Higgs bundles. In the first part we describe an algorithm which, given the linear action of a linear algebraic group with so-called "internally graded unipotent radical" on a projective variety, uses classical and Non-Reductive GIT to produce a non-empty open subset of the variety admitting a quasi-projective geometric quotient, together with an explicit projective completion of this quotient. Assuming the initial variety is smooth, we describe a procedure for calculating the Poincaré series of the resulting projective completion, which first requires showing that it can have at worst finite quotient singularities. We then obtain as a result a formula which can be viewed as the non-reductive analogue of the formula for the Poincaré series of the partial desingularisation of classical GIT quotients. In the second part we show how Non-Reductive GIT can be used to refine two instability stratifications of the stack of Higgs bundles: the Higgs Harder-Narasimhan and the Harder-Narasimhan stratifications. These refined stratifications satisfy the property that each stratum admits a quasi-projective coarse moduli space with an explicit projective completion. In the rank 2 case we provide a complete moduli-theoretic interpretation of these refined stratifications and study the geometry of the corresponding moduli spaces.

This thesis is about Arithmetic Geometry, a field of Mathematics in which techniques from Algebraic Geometry are applied to study Diophantine equations. More precisely, my research revolves around the theory of Shimura varieties, a special class of varieties including modular curves and, more generally, moduli spaces parametrising principally polarised abelian varieties of given dimension (possibly with additional prescribed structures). Originally introduced by Shimura in the ‘60s in his study of the theory of complex multiplication, Shimura varieties are complex analytic varieties of great arithmetic interest. For example, to an algebraic point of a Shimura variety there are naturally attached a Galois representation and a Hodge structure, two objects that, according to Grothendieck’s philosophy of motives, should be intimately related. The work presented here is largely motivated by the (recent progress towards the) Zilber–Pink conjecture, a far reaching conjecture generalising the André–Oort and Mordell–Lang conjectures. More precisely, we first prove a conjecture of Buium–Poonen which is an instance of the Zilber--Pink conjecture (for a product of a modular curve and an elliptic curve). We then present Galois-theoretical sufficient conditions for the existence of rational points on certain Shimura varieties: the moduli space of K3 surfaces and Hilbert modular varieties (the latter case is joint work with G. Grossi). The main idea underlying such works comes from Langlands programme: to a compatible system of Galois representations one can attach an analytic object (like a classi- cal/Hilbert modular form or a Hodge structure), which in turn determines a motive which eventually gives an algebraic point of a Shimura variety. We then prove a geometrical version of Serre’s Galois open image theorem for arbitrary Shimura varieties. We finally discuss representation-theoretical conditions for a variation of Hodge structures to admit an integral structure (joint work with E. Ullmo).

The main objects of study in this thesis are schemes parametrizing morphisms from the projective line to projective varieties. The first part of this thesis is dedicated to understand properties obtained from the functor of points of this parameter space, such as the preservation of open an closed immersions of schemes. Furthermore, for any projective variety, there is a natural partition of these schemes on closed subschemes in terms of the degrees of the morphisms. In the second part of this thesis, we use the properties studied on the first part to find a natural partition of the scheme of morphisms from the projective line to the blow-up of projective spaces at finitely many points which refine the aforementioned partition. We fully characterize this refinement on the case of Del Pezzo surfaces obtained by blowing up the projective plane at up to eight points in general position. Moreover, we use this to characterize irreducible components parametrizing rational curves which are resolutions of singularities of plane curves via this blow-up and to compute their dimension.

In this thesis we study several problems related to the existence problem of invariant canonical metrics on Fano manifolds in the presence of an effective algebraic torus action. The first chapter gives an introduction. The second chapter reviews the existing theory of T-varieties and reviews various stability thresholds and K-stability constructions which we make use of to obtain new results. In the third chapter we discuss some joint work with my supervisor to find new Kähler-Ricci solitons on smooth Fano threefolds admitting a complexity one torus action. In the fourth chapter we present a new formula for the greatest lower bound on Ricci curvature, an invariant which is now known to coincide with Tian's delta invariant. In the fifth chapter we find new Kähler-Einstein metrics on some general arrangement varieties.

Recent developments of geometry in information theory gave rise to two contemporary branches of optimization theories: 'optimization over Riemannian manifolds' and the 'Information Geometric interpretation of stochastic optimization over Euclidean spaces'. Inspired by the geometrical insights of these two emerging fields of optimization theory, this thesis studies Stochastic Derivative-Free Optimization (SDFO) algorithms over Riemannian manifolds from a geometrical perspective. We begin our discussions by investigating the Information (statistical) Geometrical structure of probability densities over Riemannian manifolds. This establishes a geometrical framework for Riemannian SDFO algorithms incorporating both the statistical geometry of the decision space and the Riemannian geometry of the search space. Equipped with the geometrical framework, we return to optimization methods over Riemannian manifolds. Riemannian adaptations of SDFO in the literature use search information generated within the normal neighbourhoods around search iterates. While this is natural due to the local Euclidean property of Riemannian manifolds, the search information remains local. We address this restriction using only the intrinsic geometry of Riemannian manifolds. In particular, we construct an evolving sampling mixture distribution for generating non-local search populations on the manifold, which is done using the aforementioned geometrical framework. We propose a generalized framework for adapting SDFO algorithms on Euclidean spaces to Riemannian manifolds, which encompasses known methods such as Riemannian Covariance Matrix Adaptation Evolutionary Strategies (Riemannian CMA-ES). We formulate and propose a novel algorithm -- Extended RSDFO. The Extended RSDFO is compared to Riemannian Trust-Region method, Riemannian CMA-ES and Riemannian Particle Swarm Optimization in a set of multi-modal optimization problems over a variety of Riemannian manifolds.

We prove that there is a natural plectic weight filtration on the cohomology of Hilbert modular varieties in the spirit of Nekovář and Scholl. This is achieved with the help of Morel's work on weight t-structures and a detailed study of partial Frobenius. We prove in particular that the partial Frobenius extends to toroidal and minimal compactifications.

In this thesis we study two different, but related, dynamical systems on hypersurfaces of Euclidean space. First, we consider billiard dynamics in a real-analytic, strictly convex domain. In dimension 2, if a billiard trajectory begins with the velocity vector making a positive angle with the boundary, then this angle remains bounded away from zero at each collision due to the existence of a family of invariant curves called caustics. We show that in higher dimensions, trajectories drifting towards the boundary can exist. Namely, using the geometric techniques of Arnold diffusion, we show that in three or more dimensions, assuming the geodesic flow on the boundary of the domain has a hyperbolic periodic orbit and a transverse homoclinic, the existence of trajectories asymptotically approaching the billiard boundary is a generic phenomenon in the real- analytic topology. Secondly, we address the assumptions from the first problem. We show that real-analytic, strictly convex, closed surfaces in 3-dimensional Euclidean space generically have a hyperbolic closed geodesic with a transverse homoclinic. It is also shown that real-analytic, closed hypersurfaces with an elliptic closed geodesic generically have a hyperbolic closed geodesic and a transverse homoclinic geodesic, in arbitrary dimension, and without the assumption of strict convexity. These are among the first perturbation-theoretic results for real-analytic geodesic flows.

This thesis applies the techniques of non-archimedean geometry to the study of degenerations and compactifications of algebraic varieties. The central object we investigate is the so-called essential skeleton, a combinatorial object that lies embedded in non-archimedean spaces and encodes an important part of the geometry of the space. This originates in the work of Kontsevich and Soibelman on mirror symmetry, an important development in algebraic geometry that has its roots in mathematical physics. The interplay of the theory of Berkovich spaces, the ideas of mirror symmetry and the tools of birational geometry gives form and meaning to the study of the essential skeleton. Chapters 3 and 4 are built on the research paper The essential skeleton of a product of degenerations, in collaboration with Morgan Brown [BM19]. We establish the behaviour of the essential skeleton under some natural operations, and we merge the language of logarithmic geometry into the construction of Berkovich skeletons. As main application, we compute the essential skeleton of some degenerations of hyperkähler varieties. We consider Hilbert schemes of a semistable degeneration of K3 surfaces, and generalised Kummer constructions applied to a semistable degeneration of abelian surfaces. In both cases we find that the dual complex of the 2n-dimensional degeneration is homeomorphic to a point, n-simplex, or CPn, depending on the type of the degeneration and in accordance with the predictions of mirror symmetry. Chapters 5 to 7 are based on the joint work Essential skeletons of pairs and the geometric P=W conjecture with Mirko Mauri and Matthew Stevenson [MMS18]. We introduce and study an explicit formulation of the weight function, a key tool to define the essential skeleton, in the case of varieties defined over a non-archimedean trivially-valued field. As a result, we employ these techniques to compute the dual boundary complexes of certain character varieties: this provides the first evidence for the geometric P=W conjecture in the compact case, and the first application of Berkovich geometry in non-abelian Hodge theory.

The essential dimension, ed_k(G), of an algebraic group G is an invariant that measures the complexity of G-torsors over fields. The correspondence between algebraic tori and finite subgroups of GL_n(Z) means one can establish bounds on ed_k(T) of an algebraic torus T by studying the action of the corresponding subgroup of GL_n(Z).

In this paper we prove structure results for the singular sets of stationary harmonic maps and mean curvature flows local to particular singularities. The original work is contained in Chapter 5 and Chapter 8. Chapters 1-5 are concerned with energy minimising maps and stationary harmonic maps. Chapters 6-8 are concerned with mean curvature flows and Brakke flows. In the case of stationary harmonic maps we consider a singularity at which the spine dimension is maximal, and such that the weak tangent map is homotopically non-trivial, and has minimal density amongst singularities of maximal spine dimen- sion. Local to such a singularity we show the singular set is a bi-Hölder continuous homeomorphism of the unit disk of dimension equal to the maximal spine dimension. A weak tangent map is translation invariant along a subspace, and invariant under dilations, so it completely defined by its values on a sphere. Such a map is said to be homotopically non-trivial if the mapping of a sphere into some target manifold cannot be deformed by a homotopy to a constant map. For an n-dimensional mean curvature flow we consider a singularity at which we can find a shrinking cylinder as a tangent flow, that collapses on an (n−1)-dimensional plane. Local to such a singularity we show that all singularities have such a cylindrical tangent, or else have lower Gaussian density than that of the shrinking cylinder. The subset of cylindrical singularities can be shown to be contained in a finite union of parabolic (n − 1)-dimensional Lipschitz submanifolds. In the case that the mean curvature flow arises from elliptic regularisation we can show that all singularities local to a cylindrical singularity with (n − 1)-dimensional spine are either cylindrical singularities with (n − 1)-dimensional spine, or contained in a parabolic Hausdorff (n − 2)-dimensional set.

All minimal models of a given variety are linked by special birational maps called flops, which are a type of codimension two surgery. A version of the Bondal-Orlov conjecture, proved by Bridgeland, states that if X and Y are smooth complex projective threefolds linked by a flop, then they are derived equivalent (i.e. their bounded derived categories of coherent sheaves are equivalent). Van den Bergh was able to give a new proof of Bridgeland's theorem using the notion of a noncommutative crepant resolution, which is in particular a ring A together with a derived equivalence between X and A. The ring A is constructed as an endomorphism ring of a decomposable module, and hence admits an idempotent e. Donovan and Wemyss define the contraction algebra to be the quotient of A by e; it is a finite-dimensional noncommutative algebra that is conjectured to completely recover the geometry of the base of the flop. They show that it represents the noncommutative deformation theory of the flopping curves, as well as controlling the flop-flop autoequivalence of the derived category of X (which, for the algebraic model A, is the mutation-mutation autoequivalence). In this thesis, I construct and prove properties of a new invariant, the derived contraction algebra, which I define to be Braun-Chuang-Lazarev's derived quotient of A by e. A priori, the derived contraction algebra - which is a dga, rather than just an algebra - is a finer invariant than the classical contraction algebra. I prove (using recent results of Hua and Keller) a derived version of the Donovan-Wemyss conjecture, a suitable phrasing of which is true in all dimensions. I prove that the derived quotient admits a deformation-theoretic interpretation; the proof is purely homotopical algebra and relies at heart on a Koszul duality result. I moreover prove that in an appropriate sense, the derived contraction algebra controls the mutation-mutation autoequivalence. These results both recover and extend Donovan-Wemyss's. I give concrete applications and computations in the case of partial resolutions of Kleinian singularities, where the classical contraction algebra becomes inadequate.

Homological projective duality describes the bounded derived category D(X) of coherent sheaves on X when the latter is a complete intersection of hypersurfaces of the same degree. We give a procedure for constructing new HP duals from old and move beyond HPD by studying complete intersections cut out by hypersurfaces of varying degrees. First, starting with a HP dual pair X, Y and smooth orthogonal linear sections X_{L^{\perp}} , Y_L, we prove that the blowup of X in X_{L^{\perp}} is naturally HP dual to Y_L. The result also holds true when Y is a noncommutative variety or just a category. We extend the result to the case where the base locus X_{L^{\perp}} is a multiple of a smooth variety and the universal hyperplane has rational singularities; here the HP dual is a weakly crepant categorical resolution of singularities of Y_L. Second, for a complete intersection X ⊂ P^n, we show that the nontrivial part of D(X) is a weakly crepant categorical resolutions of a certain fractional Calabi-Yau category. This turns out to give examples of the previous story where, starting with a noncommutative HP dual, the blowing up process nevertheless gives geometric HP duals. This is also a generalisation of Kuznetsov's story for nodal cubic fourfolds [32] and in particular we obtain new examples of Calabi-Yau categories that admit a geometric crepant resolution. Furthermore, we get a characterisation of complete intersections of Calabi-Yau Hodge type in terms of their derived category and infinitely many examples of rational varieties that are categorically representable in codimension 2.

This thesis is devoted to the study of the symplectic algebraic geometry of the moduli spaces of the nonabelian Hodge correspondence: namely, we consider character varieties of (open) Riemann surfaces and the moduli spaces of semistable Higgs bundles on a closed Riemann surface and aim at giving an answer to the following two questions: are those moduli spaces symplectic singularities? Do they admit symplectic resolutions? In the case of Higgs bundles on closed Riemann surfaces, we are able to completely solve the problem, using the work of Bellamy and Schedler, in combination with Simpson’s Isosingularity theorem. For what concerns the Betti moduli spaces, we take a different perspective and realise them as (singular open subsets of) multiplicative quiver varieties, and we study the aforementioned problem for the latter varieties. The present work represents a first step of a series of investigations on the symplectic algebraic geometry involved in the nonabelian Hodge correspondence.

This thesis studies crystalline Galois representations, which are certain constructions in p-adic Hodge theory whose most interesting property is that they arise as representations associated with modular forms. The main theorems proved in this thesis are partial results towards a conjecture about crystalline representations which dates back to computational observations by Buzzard and Gouvêa. Among the curiosities they noticed when computing the p-adic slopes of modular forms (i.e. the p-adic valuations of their pth coefficients) was that, for a certain class of so-called Γ_0(N)-regular primes when the level N is coprime to p, the p-adic slopes of eigenforms of level Γ_0(N) are always integers. Since, at least when p > 2, the reductions modulo p of the Galois representations associated with such modular forms are always reducible, this eventually pointed to a more general conjecture that the "locally reducible" eigenforms-i.e. those eigenforms of level coprime to p which have associated Galois representations whose reductions modulo p are reducible-always have integer slopes. In fact, while all representations associated with the aforementioned modular forms are crystalline, the class of all crystalline representations is larger and constructed entirely locally via p-adic Hodge theory (and therefore does not need any of the global structure coming from the world of modular forms). Thus the main conjecture tackled in this thesis is an entirely local statement saying that the slopes of locally reducible crystalline representations of even weight are always integers. The main result we prove in this thesis is that this conjecture is true when the slope is less than (p-1)/2. We additionally classify the reductions modulo p for a large class of crystalline representations.

Compressed manifold modes are solutions to an optimisation problem involving the $\ell_1$ norm and the orthogonality condition $X^TMX=I$. Such functions can be used in geometry processing as a basis for the function space of a mesh and are related to the Laplacian eigenfunctions. Compressed manifold modes and other alternatives to the Laplacian eigenfunctions are all special cases of generalised manifold harmonics, introduced here as solutions to a more general problem. An important property of the Laplacian eigenfunctions is that they commute with isometry. A definition for isometry between meshes is given and it is proved that compressed manifold modes also commute with isometry. The requirements for generalised manifold harmonics to commute with isometry are explored. A variety of alternative basis functions are tested for their ability to reconstruct specific functions -- it is observed that the function type has more impact than the basis type. The bases are also tested for their ability to reconstruct functions transformed by functional map -- it is observed that some bases work better for different shape collections. The Stiefel manifold is given by the set of matrices $X \in \mathbb{R}^{n \times k}$ such that $X^TMX = I$, with $M=I$. Properties and results are generalised for the $M \neq I$ case. A sequential algorithm for optimisation on the generalised Stiefel manifold is given and applied to the calculation of compressed manifold modes. This involves a smoothing of the $\ell_1$ norm. Laplacian eigenfunctions can be approximated by solving an eigenproblem restricted to a subspace. It is proved that these restricted eigenfunctions also commute with isometry. Finally, a method for the approximation of compressed manifold modes is given. This combines the method of fast approximation of Laplacian eigenfunctions with the ADMM solution to the compressed manifold mode problem. A significant improvement is made to the speed of calculation.

We prove two versions of the marked length-spectrum rigidity conjecture for a large class of non-positively curved spaces: CAT(0) cube complexes. Under weak assumptions, we show that proper cocompact actions of Gromov hyperbolic groups on CAT(0) cube complexes are fully determined by their combinatorial length functions. If the cube complexes are irreducible and have the geodesic extension property, the same result holds for non-proper, non-cocompact actions of arbitrary groups. While some of our arguments are well-known in negative curvature, until now no length-spectrum rigidity result had been obtained in a setting of non-positive curvature (except for some particular cases in dimension 2). Our spaces are allowed to have arbitrarily high dimension and can contain flats. Fixing a group Γ, our work sets the basis for the study of the space of all actions of Γ on CAT(0) cube complexes. As a first step in this direction, we use length functions to compactify large spaces of cubulations of Γ. This can be viewed as an analogue of Thurston's compactification of Teichmüller space. As a special case of our results, we construct a compactification of the Charney-Stambaugh-Vogtmann Outer Space for the group of untwisted outer automorphisms of an (irreducible) right-angled Artin group. This generalises the length function compactification of the classical Culler-Vogtmann Outer Space. Throughout the thesis, our main tool is a new notion of cross ratio on Roller boundaries of CAT(0) cube complexes. We develop a general framework reducing length-spectrum rigidity questions to the problem of extending cross-ratio preserving maps between (subsets of) Roller boundaries. We then study under what conditions such boundary maps extend to cubical isomorphisms. In the hyperbolic case, our results extend part of the work of Bourdon and Xie on Fuchsian buildings. Along the way, we also analyse the relationship between Roller boundaries, contracting boundaries and visual boundaries of CAT(0) cube complexes. For a Gromov hyperbolic group Γ, this allows us to embed the space of cubulations of Γ into the space of Γ-invariant cross ratios on the Gromov boundary of ∂∞Γ. This draws an unexpected similarity between cubulations and the many other geometric structures that can be encoded in terms of boundary cross ratios: for instance, points of Teichmüller space, Hitchin representations, geodesic currents.

G-structures, where G is a Lie group, are a uniform characterisation of many differential geometric structures of interest in supersymmetric compactifications of string theories. Calabi-Yau n-folds for instance have torsion-free SU(n)-structure, while more general structures with non-zero torsion are required for heterotic flux compactifications. Exceptional geometries in dimensions 7 and 8 with G = G2 and Spin(7) also feature prominently in this thesis. We discuss multiple connections between such geometries and the world- sheet theory describing strings on them, especially regarding their chiral symmetry algebras, originally due to Odake and Shatashvili-Vafa in the cases where G is SU (n), G2 and Spin(7). In the first part of the thesis, we describe these connections within the formalism of operator algebras. We also realise the superconformal algebra for G2 by combining Odake and free algebras following closely the recent mathematical construction of twisted connected sum G2 manifolds. By considering automorphisms of this realisation, we speculate on mirror symmetry in this context. In the second part of the thesis, G-structures are studied semi-classically from the worldsheet point of view using (1, 0) supersymmetric non-linear σ- models whose target M has reduced structure. Non-trivial flux and instanton- like connections on vector bundles over M are also allowed in order to deal with general applications to superstring compactifications, in particular in the heterotic case. We introduce a generalisation of the so-called special holonomy W-symmetry of Howe and Papadopoulos to σ-models with Fermi and mass sectors. We also investigate potential anomalies and show that cohomologically non-trivial terms in the quantum effective action are invariant under a corrected version of this symmetry. Consistency with supergravity at first order in α' 0 is manifest and discussed. We finally relate marginal deformations of (1, 0) G 2 and Spin(7) σ-models with the cohomology of worldsheet BRST operators. We work at lowest order in α' 0 and study general heterotic backgrounds with bundle and non-vanishing torsion.

The purpose of this thesis is to provide model-theoretic structures to study Shimura varieties. Two main problems are studied. Transcendence Properties of j. This part is aimed at proving a general transcendence property for the j function in the spirit of Schanuel's conjecture. Such a result was proven for the exponential function by M. Bays, J. Kirby and A. Wilkie, and it relies heavily on the Ax-Schanuel theorem. An analogue of Ax-Schanuel for the j function was proven by J. Pila and J. Tsimerman, and this allowed to adapt the strategy used for the exponential to j. One concrete consequence of the main transcendence result is: Theorem. Let τ ϵ R be j-generic. Suppose z1,...,zn ϵ H+ and g ϵ GL2(Q(τ ))+ are such that z1,...,zn; gz1,...,gzn are in different GL2(Q)+-orbits (pairwise). Then: t.d.(j(z); j'(z); j"(z); j(gz); j'(gz); j"(gz)/τ) ≥ 3n: The term j-generic is analogous to that of being exponentially transcendental. Using o-minimality, we prove that all but countably many values for τ can be used. We also analyse a modular version of Schanuel's conjecture and prove that there are at most countably many essential counterexamples to this conjecture. Categoricity of Shimura Varieties. Let p : X+ → S(C) be a Shimura variety. We associate with it a countable two-sorted first-order language and analyse the complete first-order theory it determines. The question of interest in this part of the thesis of whether or not this theory is categorical. One immediate issue is that the theory cannot determine the size of the fibres of the map between the sorts, so we restrict our analysis to the models with the smallest possible fibres. Categoricity for Shimura curves was established by C. Daw and A. Harris. From their result, it is immediate that some form of arithmetic open image condition is needed for the categoricity of general Shimura varieties. Our goal is to postulate the correct open image condition and prove that it is equivalent to categoricity. This open image condition is not known in general, however, using known cases of the Mumford-Tate conjecture, we obtain unconditional categoricity of two well-known Shimura varieties: A2 and A3, the moduli spaces of principally polarised abelian varieties of dimension 2 and 3, respectively.

This thesis deals with several theoretical and computational problems in the theory of p-adic heights on elliptic curves. In Chapter 3 we provide a new algorithm for the computation of the Mazur--Tate p-adic height in a family of elliptic curves over the rationals. As an application, we study the conjectural non-degeneracy properties of the pairing induced by the height, as well as orders of vanishing and arithmetic interpretations of some of the coefficients of the regulator series. In Chapter 4 we use the decomposition of the p-adic height as a sum of local contributions to explore a number of problems related to the quadratic Chabauty method for determining integral points on hyperbolic curves. For instance, we remove the assumption of semistability in the description of the quadratic Chabauty sets X(Zp)2 containing the integral points X(Z) of an elliptic curve of rank at most 1. Motivated by a conjecture of Kim, we then investigate theoretically and computationally the set-theoretic difference X(Zp)2\X(Z) In Chapter 5 we describe an implementation of the Chabauty--Coleman method for genus 3 hyperelliptic curves over Q whose Jacobians have rank 1. This is used to compute the rational points on approximately 17,000 curves and to provide evidence for a conjecture of Stoll. In Chapter 6 we extend the explicit quadratic Chabauty techniques to compute integral points on elliptic curves and rational points on genus 2 bielliptic curves to general number fields. Finally, in Chapter 7 we prove that the first two coefficients in the series expansion around s=1 of the p-adic L-function of an elliptic curve over the rationals are related by a formula involving the conductor of the curve. The introductory Chapter 1 outlines the common themes of the thesis; Chapter 2 introduces some preliminary notions.

The notion of a symplectic groupoid first arose as an apparatus to solve the quantization problem for Poisson manifolds. It is well known that not every Poisson manifold can be globally realised by a symplectic groupoid. A key motivation for this thesis was Lu and Weinstein's construction of a (global) symplectic double groupoid for an arbitrary Poisson Lie group. We develop an extensive exposition of their results, and analyse some of the possible extensions of their construction. In particular, we produce a symplectic double groupoid for any pair of dual Poisson groupoids where the underlying Lie groupoid structures are of trivial type. Alongside these ideas, we also study the actions of double Lie structures. A detailed account of the actions of double Lie groupoids is given, and notions for the actions of LA-groupoids are defined. As an application of these double actions, we consider an alternative approach to Xu's study of Poisson reduced spaces for actions of a symplectic groupoid. This approach is then extended to consider the Poisson reduced spaces for more general actions of Poisson groupoids.

The space of stability conditions is a complex manifold, constructed from the data of a triangulated category. The notion of a stability condition is motivated by ideas arising in string theory, and arrives in algebraic geometry via mirror symmetry. This motivation suggests that in some cases, the geometry of the space of stability conditions should be not just a complex manifold, but that it should additionally carry the structure of a Frobenius manifold. The main focus of this work is the construction of the spaces of stability conditions associated to the Calabi–Yau-n triangulated categories of the Kronecker quiver, and the exploration of the connection between these spaces and the quantum cohomology of the projective line, which is a complex manifold with a known and well studied Frobenius structure.

This thesis develops new stereo vision methods for the reconstruction and analysis of complex surfaces, such as riverbeds. Depth and surface orientation estimates are crucial to the understanding of 3D scene geometry, from calibrated stereo images. In this work, we propose new visibility and disparity magnitude constraints, for slanted patches in the scene. These constraints can be used to associate geometrically feasible planes with each point, in a 3D disparity space representation. The new constraints are validated in the PatchMatch Stereo framework of Bleyer et al. (BMVC, 2011). In order to estimate the plane parameters in this algorithm, we modify the original spatial propagation procedure, and introduce a gradient-free non-linear optimiser. These improvements allow us to achieve accurate disparity maps, with sub-pixel precision, according to the Middlebury stereo benchmark. In addition to surface orientation, curvature information is needed for a full understanding of the local surface structure. The PatchMatch surface model is planar, and does not directly estimate the local curvature. We propose a local quadric surface model, which uses both the spatial position and the estimated surface normals, in the disparity space. We also propose principal curvature and principal direction constraints, which ensure that the local quadric model is geometrically feasible. Finally, we describe the design and capture of a new photogrammetric dataset, which can be used to study topographic changes in riverbed morphology, over time. The dataset is challenging for conventional stereo matching algorithms, because the visible surface consists of sand, which lacks large-scale image features. This laboratory dataset comprises thirty-nine calibrated stereo pairs, plus fifteen ground-truth depth maps, obtained by a laser scanner. We used this dataset to validate the stereo vision methods developed in the thesis, in relation to potential geomorphological applications.

C∞-schemes are a generalisation of manifolds that have nice properties such as the existence of fibre products. C∞-schemes have been used as a model for synthetic differential geometry, as in Dubuc, Kock, and Moerdijk and Reyes, and for defining derived differential geometry as in Lurie, and Spivak. Manifolds with corners are a generalisation of manifolds locally modelled on [0,∞)k × Rn-k, and their smooth maps behave well with respect to the corners as in Melrose. In particular, Joyce describes a corner functor from the category of manifolds with corners to the category of 'interior' manifolds with corners with mixed dimension. C∞-algebraic geometry with corners is the study of C∞-rings and C∞-schemes with corners, which we define in this thesis. We define (local/interior/firm) C∞-rings with corners, and study categorical properties such as the existence of limits and colimits using various adjoint functors. We describe a spectrum functor from C∞-rings with corners to local C∞-ringed spaces with corners, and show this a right adjoint to a global sections functor. We define C∞-schemes with corners using this spectrum functor. We show there is a full and faithful embedding of the category of manifolds with corners into the category of firm C∞-schemes with corners, and that fibre products of firm $C^\infty$-schemes with corners exist. We show that manifolds with corners are affine under geometric conditions. We define (b-)cotangent sheaves of C∞-schemes with corners and show they correspond to the (b-)cotangent bundles of manifolds with corners of Joyce. We describe the categories of interior local C∞-ringed spaces with corners and interior firm C∞-schemes with corners. We construct corner functors for both of these categories, which are right adjoint to the inclusion of these interior spaces/schemes into the non-interior ones. We show that these corner functors correspond to the corner functor for manifolds with corners. We expect applications of this work in defining derived spaces with corners in derived differential geometry, and we explore the connections of this work to log geometry and the positive log differentiable spaces of Gillam and Molcho.

In this thesis we aim to persuade the readers to reconsider homogeneous model theory as a framework in which to study Algebraic Stability Theory. We argue that the tight link between homogeneous model theory and categorical amalgamation constructions gives it a flexibility beyond what the ordinary first-order context can offer while simultaneously it allows the development of a stability-theoretic machinery very close to classical first-order stability. Thus we first introduce homogeneous model theory and categorical amalgamation theory separately (Chapters 1 and 2), laying out our own contributions alongside the known theory, before we utilise their interactions to prove related results pertinent to several different questions in algebraic stability theory. In Chapter 3 we give a very general result on omitting (possibly isolated) types from homogeneous models, generalising an analogous theorem by Hrushovski and Itai [18] regarding differential fields. Our method makes it possible to omit any family of types of trivial geometry from a superstable (and simple) homogeneous model without impacting the dividing structure and thus the stability class of the model. In Chapter 4 we apply homogeneous model theory to the study of Hrushovski constructions, enabling us to give a general treatment of the fundamental stabilitytheoretic properties beyond the combinatorial case to which such general accounts are usually confined. We obtain a general characterisation of dividing and by extension of U-Rank in the amalgamation limits of those constructions, and since we are not limited to finitary cases we can use our results for studying pseudo-exponentiation in the next two chapters. In Chapters 5 and 6 we thus narrow our focus even more and apply the tools developed above to the single class of exponential fields, illustrating the power of our general results through application to a topic which has been under close investigation ever since Boris Zilber proposed pseudo-exponentiation as a model-theoretic approach to complex exponentiation in the 2000s [42]. In passing we will also construct a non-almost-exponentially-algebraically closed quasiminimal exponential field, answering a question asked by Bays and Kirby in their recent [7].

In this work, we are going to find sufficient conditions on the initial metric on some 4-dimensional manifolds foliated by homogeneous S3 for a Type I singularity to occur when it is flowed under the Ricci flow. This work generalises the study on rotationally symmetric manifolds done by Angenent and Isenberg as well as the work of Isenberg, Knopf, and Sesum, in which they introduced some ansatz for the problem setup. In the latter study, a global frame for the tangent bundle called the Milnor frame was used to set up the problem. We begin with some discussions on the symmetries of the manifold and its ansatz metric derived from Lie groups as well as the global tangent bundle frame developed by Milnor. The curvature quantities and the Ricci flow equation of this ansatz metric will then be explicitly computed. Numerical simulations of the Ricci flow on these manifolds are done, which provides insight and conjectures for the main problems. Some analytic results will be proven for the manifolds S1xS3 and S4 using maximum principles from parabolic PDE theory and sufficiency conditions for a neckpinch singularity will be provided. Finally, a problem from general relativity with similar metric symmetries but endowed on manifolds homeomorphic to CPM2\B4 and B4 will be discussed, which will provide a possible direction for future work.

This thesis is split into three largely independent chapters. The first concerns the representation theory of Zp[G]-lattices. Specifically, we investigate regulator constants, due to Dokchitser-Dokchitser, which are isomorphism invariants of lattices whose extension of scalars to Qp is self-dual. We first show that when G has cyclic Sylow p-subgroups then regulator constants are strong invariants of permutation modules in a way that can be made precise. Our main result is then that, subject to an additional technical hypothesis on G, this can be combined with existing work of Yakovlev to provide an explicit list of accessible invariants which completely determine, up to isomorphism, any Zp[G]-lattice whose extension to Qp is self-dual. The second chapter is an application of this result in the context of number fields. Given a Galois extension of number fields K=F with Galois group G, the extension of scalars to Zp of the unit group of K modulo its torsion subgroup denes a Zp[G]-lattice. If we assume that G has cyclic Sylow p-subgroups and satisfies the aforementioned hypothesis, then the above result gives a list of invariants which determine the Galois module structure. The main result of this chapter is then that if p divides G at most once, we can explicate these invariants in terms of classical number theoretic objects. For example, in some cases this can be done in terms of capitulation of ideal classes and ramification information. The final (unrelated) topic concerns relative motives over Shimura varieties. Given a Shimura datum (G; Ӿ) and neat open compact subgroup K ≤ G(Af ), denote the corresponding Shimura variety ShK(G;Ӿ) by S. The canonical construction described by Pink shows how to associate variations of Hodge structure on San to representations of G. It is expected that this should be motivic in nature, i.e. that there is a motive over S for every representation of G whose Hodge realisation is the variation of Hodge structure given by the canonical construction. Using mixed Shimura varieties, we show that this can be done functorially for representations with Hodge type {(-1; 0); (0,-1)} and that this is compatible with change of S. When (G;Ӿ) has a chosen PEL-datum, existing work of Ancona allows us to associate a motive over S to any representation of G. We then give results to show that in some cases this compatible with change of S and independent of the choice of PEL-datum.

The construction introduced by Gross, Hacking and Keel in [28] allows one to construct a mirror family to (S, D) where S is a smooth rational projective surface and D a certain type of Weil divisor supporting an ample or anti-ample class. To do so one constructs a formal smoothing of a singular variety they call the n-vertex. By arguments of Gross, Hacking and Keel one knows that this construction can be lifted to an algebraic family if the intersection matrix for D is not negative semi-definite. In the case where the intersection matrix is negative definite the smoothing exists in a formal neighbourhood of a union of analytic strata. A proof of both of these is found in [GHK]. In our first project we use these ideas to find explicit formulae for the mirror families to low degree del Pezzo surfaces. In the second project we treat the remaining case of a negative semi-definite intersection matrix, corresponding to S being a rational elliptic surface and D a rational fibre. Using intuition from the first project we prove in the second project that in this case the formal family of their construction lifts to an analytic family.

This thesis explores aspects of generalized geometry, a geometric framework introduced by Hitchin and Gualtieri in the early 2000s. In the first part, we introduce a new class of submanifolds in stable generalized complex manifolds, so-called Lagrangian branes with boundary. We establish a correspondence between stable generalized complex geometry and log symplectic geometry, which allows us to prove results on local neighbourhoods and small deformations of this new type of submanifold. We further investigate Lefschetz thimbles in stable generalized complex Lefschetz fibrations and show that Lagrangian branes with boundary arise in this context. Stable generalized complex geometry provides the simplest examples of generalized complex manifolds which are neither complex nor symplectic, but it is sufficiently similar to symplectic geometry for a multitude of symplectic results to generalize. Our results on Lefschetz thimbles in stable generalized complex geometry indicate that Lagrangian branes with boundary are part of a potential generalisation of the Wrapped Fukaya category to stable generalized complex manifolds. The work presented in this thesis should be seen as a first step towards the extension of Floer theory techniques to stable generalized complex geometry, which we hope to develop in future work. The second part of this thesis studies Dorfman brackets, a generalisation of the Courant- Dorfman bracket, within the framework of double vector bundles. We prove that every Dorfman bracket can be viewed as a restriction of the Courant-Dorfman bracket on the standard VB-Courant algebroid, which is in this sense universal. Dorfman brackets have previously not been considered in this context, but the results presented here are reminiscent of similar results on Lie and Dull algebroids: All three structures seem to fit into a more general duality between subspaces of sections of the standard VB-Courant algebroid and brackets on vector bundles of the form T M ⊕ E ∗ , E → M a vector bundle. We establish a correspondence between certain properties of the brackets on one, and the subspaces on the other side.

We study the relationship between geometric properties of toric varieties and combinatorial properties of the corresponding lattice polytopes. In particular, we give a bound for a very ample lattice polytope to be k-normal. Equivalently, we give a new combinatorial bound for the Castelnuovo-Mumford regularity of normal projective toric varieties. We also give a new combinatorial proof for a special case of Reider's Theorem for smooth toric surfaces.

Let Y be a smooth complex projective Calabi{Yau threefold. Donaldson-Thomas invariants [Tho00] are integer invariants that virtually enumerate curves on Y. They are organised in a generating series DT(Y) that is interesting from a variety of perspectives. For example, well-known series in mathematics and physics appear in explicit computations. Furthermore, closer to the topic of this thesis, the generating series of birational Calabi-Yau threefolds determine one another [Cal16a]. The crepant resolution conjecture for Donaldson-Thomas invariants [BCY12] conjectures another such comparison result. It relates the Donaldson{Thomas generating series of a certain type of three-dimensional Calabi-Yau orbifold to that of a particular resolution of singularities of its coarse moduli space. The conjectured relation is an equality of generating series. In this thesis, I first provide a counterexample showing that this conjecture cannot hold as an equality of generating series. I then verify that both generating series are the Laurent expansion about different points of the same rational function. This suggests a reinterpretation of the crepant resolution conjecture as an equality of rational functions. Second, following a strategy of Bridgeland [Bri11] and Toda [Tod10a, Tod13, Tod16a], I prove a wall-crossing formula in a motivic Hall algebra relating the Hilbert scheme of curves on the orbifold to that on the resolution. I introduce the notion of pair object associated to a torsion pair, putting ideal sheaves and stable pairs on the same footing, and generalise the wall-crossing formula to this setting, essentially breaking the former in many pieces. Pairs, and their wall-crossing formula, are fundamentally objects of the bounded derived category of the Calabi-Yau orbifold. Finally, I present joint work with J. Calabrese and J. Rennemo [BCR] in which we use the wall-crossing formula and Joyce's integration map to prove the crepant resolution conjecture for Donaldson-Thomas invariants as an equality of rational functions. A crucial ingredient is a result of J. Rennemo that detects when two generating functions related by a wall-crossing are expansions of the same rational function.

In [Bri07], Bridgeland introduced the notion of stability conditions on the bounded derived category D(X) of coherent sheaves on an algebraic variety X. This topic is originally inspired by concepts in string theory and mathematical physics and has many interesting applications in algebraic geometry. In the first part of the thesis, we provide a direct proof of an important result in [Bri08, BMS16] which states there is a two dimensional family of weak Bridgeland stability conditions on the bounded derived category D(X) of coherent sheaves on a variety X. As a first application of this result, we prove an effective restriction theorem which provides sufficient conditions on a stable locally free sheaf on a projective variety such that its restriction to a hypersurface remains stable. Secondly, we extend and complete Mukai's program to reconstruct a K3 surface from a curve on that surface. We show that the K3 surface containing the curve can be obtained uniquely as a Fourier-Mukai partner of a suitable Brill-Noether locus of vector bundles on the curve.

Given a quiver flag variety Y equipped with a tilting bundle E, a construction ofCraw, Ito and Karmazyn [CIK18] produces a closed immersion f_E : Y -> M(E), where M(E) is the fine moduli space of cyclic modules over the algebra End(E).In this thesis we present two classes of examples where f_E is an isomorphism. Firstly, when Y is toric and E is the tilting bundle from [Cra11]; this generalises the well-known fact that P^n can be recovered from the endomorphism algebra of \oplus_{0\leq i \leq n} O_{P^n}(i). Secondly, when Y = Gr(n, 2), the Grassmannian of 2-dimensional quotients of k^n and E is the tilting bundle from [Kap84]. In each case, we give a presentation of the tilting algebra A = End(E) by constructing a quiver Q' such that there is a surjective k-algebra homomorphism \Phi: kQ' -> A, and then give an explicit description of the kernel.

One of the most important tools to study the geometry of irreducible holomorphic symplectic manifolds is the monodromy group. The first part of this dissertation concerns the construction and studyof monodromy operators on irreducible holomorphic symplectic manifolds which are deformation equivalent to the 10-dimensional example constructed by O'Grady. The second part uses the knowledge of the monodromy group to compute the number of connected components of moduli spaces of bothmarked and polarised irreducible holomorphic symplectic manifolds which are deformationequivalent to generalised Kummer varieties.

This thesis consists roughly of two parts. In the first part, we study various properties of Z2-equivariant Heegaard Floer cohomology. Then, in the second part, we apply the results of the first part to study transverse knot invariants in the standard contact S3. When (L,p) is a based link in S3, the branched double cover Sigma(L) is a rational homology sphere which carries a natural Z2-action. Hendricks, Lipshitz, and Sarkar studied this action on the Heegaard Floer setting to define HFZ2(Sigma(L), p), the Z2-equivariant Heegaard Floer cohomology of Sigma(L), using bridge diagrams of L drawn on a sphere. They also proved that its isomorphism class as an F2[theta]-module is an isotopy invariant of (L,p), which also implies that we can drop p from its notation when L is a knot. In this thesis, we prove that, when L=K is a knot, then HFZ2(Sigma(K)) satisfies naturality and admits cobordism maps for based knot cobordisms. Then we move on to show that HFZ2(Sigma(K)) can be calculated in terms of bridge diagrams of L, which are now drawn on any weakly admissible Heegaard diagrams representing S3. Based on this observation, we point out that, for knots K, we can consider HFZ2(Sigma(K)) as a weak Heegaard invariant, and prove that it is actually a strong Heegaard invariant. We also prove that the definition of HFZ2(Sigma(K)) in terms of bridge diagrams and its alternative definition as a strong Heegaard diagram are equivalent, by constructing an invertible natural transformation between the two TQFT functors induced by those two definitions. Using that fact, we construct Z2-equivariant knot Floer cohomology HFKZ2(Sigma(K)), and show that it is a refinement of both HFK(-S3,K) and HFZ2(Sigma(K)). We then construct the Z2-equivariant contact class cZ2(xiK) of a transverse knot K inside the standard contact 3-sphere (S3,xistd) std, as a well-defined element of HFZ2(Sigma(K)), and prove that it is a transverse knot invariant which refines the contact class c(xiK) of the contact branched double cover (Sigma(K),xiK) of (S3,xistd) along K. We also construct the Z2-equivariant LOSS invariant TZ2(K) of a transverse knot K in (S3,xistd), as an element of HFKZ2(Sigma(K),K) and prove that it is also a transverse knot invariant which refines both cZ2(xiK) and the LOSS invariant T(K). Lastly, we give an interesting topological application of cZ2(xiK), by showing that it can be used to prove the nonvanishing of c(xiK) when K achieves equality in a Bennequin-type inequality.