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Positive geometries provide a purely geometric point of departure for studying scattering amplitudes in quantum field theory. A positive geometry is a specific semi-algebraic set equipped with a unique rational top form-the canonical form. There are known examples where the canonical form of some positive geometry, defined in some kinematic space, encodes a scattering amplitude in some theory. Remarkably, the boundaries of the positive geometry are in bijection with the physical singularities of the scattering amplitude. The Amplituhedron, discovered by Arkani-Hamed and Trnka, is a prototypical positive geometry. It lives in momentum twistor space and describes tree-level (and the integrands of planar loop-level) scattering amplitudes in maximally supersymmetric Yang-Mills theory. In this dissertation, we study three positive geometries defined in on-shell momentum space: the Arkani-Hamed-Bai-He-Yan (ABHY) associahedron, the Momentum Amplituhedron, and the orthogonal Momentum Amplituhedron. Each describes tree-level scattering amplitudes for different theories in different spacetime dimensions. The three positive geometries share a series of interrelations in terms of their boundary posets and canonical forms. We review these relationships in detail, highlighting the author's contributions. We study their boundary posets, classifying all boundaries and hence all physical singularities at the tree level. We develop new combinatorial results to derive rank-generating functions which enumerate boundaries according to their dimension. These generating functions allow us to prove that the Euler characteristics of the three positive geometries are one. In addition, we discuss methods for manipulating canonical forms using ideas from computational algebraic geometry.

In this thesis, we develop a method to investigate the geometric quantisation of a hypertoric variety from an equivariant viewpoint, in analogy with the equivariant Verlinde formula for Higgs bundles. We do this by first using the residual circle action on a hypertoric variety to construct its symplectic cut that results in a compact cut space, which is needed for the localisation formulae to be well-defined and for the quantisation to be finite-dimensional. The hyperplane arrangement corresponding to the hypertoric variety is also affected by the symplectic cut, and to describe its effect we introduce the notion of a moment polyptych that is associated to the cut space. Also, we see that the prerequisite isotropy data that is needed for the localisation formulae can be read off from the combinatorial features of the moment polyptych. The equivariant Kawasaki-Riemann-Roch formula is then applied to the pre-quantum line bundle over each cut space, producing a formula for the equivariant character for the torus action on the quantisation of the cut space. Finally, using the quantisation of each cut space, we derive a formula expressing the dimension of each circle weight subspace of the quantisation of the hypertoric variety.

This thesis studies geometric and algebraic aspects of 3d N = 4 theories. We first focus on 3d N = 4 gauge theories compactified on an elliptic curve, and provide physical realisations of the equivariant elliptic cohomology of symplectic resolutions, and recent constructions therein. The Berry connection for supersymmetric ground states in the presence of mass parameters and flat connections for flavour symmetries is analysed, resulting in a natural construction of the equivariant elliptic cohomology variety of the Higgs branch. Supersymmetric boundary conditions are investigated in this set-up. From an analysis of boundary 't Hooft anomalies, their boundary amplitudes are demonstrated to represent equivariant elliptic cohomology classes. We then investigate two distinguished classes of N = (2,2) boundary conditions, each in 1-1 correspondence with the set of isolated massive vacua, known as exceptional Dirichlet and enriched Neumann. The former mimic isolated vacua at infinity in the presence of real mass and FI parameters. The two classes are further shown to be exchanged under mirror symmetry, via collision with a mirror symmetry interface. By computing boundary amplitudes, the enriched Neumann boundary conditions reproduce the elliptic stable envelopes of Aganagic-Okounkov, and the mirror symmetry interface the mother function in equivariant elliptic cohomology. Finally, correlation functions of Janus interfaces for varying mass parameters are considered, recovering the chamber R-matrices of elliptic integrable systems. We then study the factorisation of partition functions of N = 4 theories on closed 3-manifolds (such as the superconformal index, twisted index and S³ partition function) in terms of S¹ x HS² partition functions. We demonstrate the latter, equipped with exceptional Dirichlet boundary conditions, realise this factorisation exactly, and can be unambiguously defined and computed using supersymmetric localisation. We show certain limits of these hemisphere partition functions yield characters of lowest weight Verma modules over the quantised Higgs and Coulomb branch chiral rings. This leads to expressions for the closed 3-manifold partition functions in terms of such characters. On the way we uncover new connections between boundary 't Hooft anomalies, hemisphere partition functions and lowest weights of Verma modules.

In this thesis, we explore several parts of the geometry of the moduli spaces of genus one logarithmic stable maps. In Chapter 2, we exhibit a smooth compactification of the moduli space of elliptic curves in a product of projective spaces with tangency along a subset of its toric boundary divisors. This is a Vakil-Zinger type of desingularization for maps to a product of projective spaces using ideas of elliptic singularities and logarithmic geometry. However, we also give an explicit example showing that, when the target is P¹ × P¹, the Vakil-Zinger operations alone are not enough to desingularize the space; we do so by comparing the absolute geometry with the logarithmic one. Using the smoothness result, we construct the virtual fundamental classes of moduli spaces of maps to a special class of simple normal crossings pairs. We then prove a consistency result that relates the virtual classes when we remove certain "fictitious" markings, suggesting the theory has good recursive properties. In Chapter 3, we study the desingularized moduli spaces of genus 1 maps to projective spaces in closer detail. This allows us to describe the class of a stratum in the logarithmic moduli spaces by performing tautological operations on the corresponding stratum in the moduli spaces of genus 1 stable maps. We show there is an analogue of the splitting formula in the context of genus 1 logarithmic Gromov-Witten theory, which lays the groundwork for logarithmic degeneration formulae in this reduced genus one logarithmic Gromov-Witten theory.

Inspired by concepts in string theory, the notion of stability conditions on triangulated categories was introduced by Bridgeland in 2002. Its impact across mathematics includes the solutions of classical problems in algebraic geometry, which were hard to tackle directly. This concept leads to a wall-crossing machinery: there is a manifold of stability conditions, with a wall-and-chamber decomposition, such that the moduli space of stable objects only changes as we cross a wall. This has many geometrical applications. In the first part, we show that wall-crossing transformations can be more involved than was previously known, by proving the existence of a wall-crossing with unexpected behaviour. In particular, it fails an expected correspondence between wall-crossing and birational transformations. This significantly complicates the overall picture in this fundamentally important correspondence to applications of stability conditions to algebraic geometry. In the second part, we apply the machinery to answer some basic questions about the classical Hilbert scheme of canonical genus four curves in P 3 via an effective control over its wall-crossing. The strategy uses the space of PT-stable pairs as an intermediate step.

Within the area of birational geometry one of the most frequent and complicated operations that occur are flips. They have been proven to exist in all dimensions, but their explicit classification and terminality still remain as some of the most difficult open questions in geometry. Explicit description of flips are fundamental within the Minimal Model Program, introduced by Reid and Mori around 40 years ago, as well as in Sarkisov Program. This thesis works towards the classification of 3-fold flips in codimension 2, that is flips that arise from a complete intersection of two hypersurfaces in C⁶.

Affine q-Schur algebras admit a geometric construction as a convolution algebra on the set of functions with constructible support on double affine flag varieties, due to Ginzburg and Vasserot [17], Varagnolo and Vasserot [36] and Lusztig [28]. Using the geometry of affine flag varieties we define a generic product of orbits in the double flag varieties which leads to the construction of a new associative Z-algebra Ĝ(n, r), called the generic affine algebra. We then study the link between Ĝ(n, r) and the affine 0-Schur algebra Ŝ₀(n, r) and we conjecture that these two algebras are isomorphic when r < n. This conjecture will give a monomial basis in Ŝ₀(n, r) and a presentation by a quiver with relations. This work generalises the work of Jensen and Su [25] in the type A case, which gives a geometric realisation of 0-Schur algebras.

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.

In this thesis we consider problems related to Joyce’s vertex algebra construction and the topology of stabilized moduli spaces. We first compute the homology of the moduli stack of objects in the derived category of a smooth complex projective variety X in class D, showing that the rational cohomology ring is freely generated by tautological classes. This is used to identify Joyce’s construction with a generalized super-lattice vertex algebra on the rational K-theory of X^an. Then, we prove orientability of moduli spaces of (complexes of) coherent sheaves on projective Calabi–Yau 4-folds–this has applications to defining a C-linear enumerative invariant theory for Calabi–Yau 4-folds. This result is based on joint work with Yalong Cao and Dominic Joyce. Lastly, we consider a connective even complex-oriented homology theory E with associated formal group law F and show that, given a Künneth isomorphism, replacing ordinary homology with E in Joyce’s construction yields a vertex F -algebra–this result is based on joint work with Markus Upmeier.

This thesis aims to generalise the theory of complements to log canonical Fano varieties and relate theory of complements to the index conjecture of log Calabi-Yau varieties. We mainly work over an algebraically closed field of characteristic zero, more specifically over C. We will first introduce some basic background theory for birational geometry, including notion of singularities, pairs, complements. We will then cover some backgrounds of (log) Fano and Calabi-Yau varieties. We will also state the main new results in the introduction. The majority of work is then split into the following 4 sections: complements on surfaces, complements on log canonical 3-fold, index conjecture for log Calabi-Yau varieties, relative 3-fold complements. For the section about complements on surfaces, we will firstly cover the known result about the theory for complements and then prove new results about semi-dlt surfaces. We will extend the notion of complements to semidlt surfaces and then prove a result for "gluing" complements for semi-dlt surfaces. Then we will move on to prove results about boundedness of complements for global log canonical Fano 3-fold. In other direction, we will prove the index conjecture for log Calabi-Yau varieties in dimension 3 in full generality and then prove some new inductive results towards the conjecture. In the last chapter, I will include proof of the boundedness of complements for log Fano in the relative case in dimension 3. The last chapter is a part of joint work, with Stefano Filipazzi and Joaquin Moraga, where we proved a general theorem of boundedness of complements in dimension 3.

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.

The advancements in additive manufacturing techniques enable novel designs using lattice structures in mechanical parts, lightweight materials, biomaterials and so forth. Lattice-skin structures are a class of structures that couple thin-shells with lattices, which potentially combine the advantages of the thin-shell and the lattice structure. A new and systematic isogeometric analysis approach that integrates the geometric design, structural analysis and optimisation of lattice-skin structures is proposed in the dissertation. In the geometric design of lattice-skin structures, a novel shape interrogation scheme for splines, specifically subdivision surfaces, is proposed, which is able to compute the line/surface intersection efficiently and robustly without resorting to successive refinements or iterations as in Newton-Raphson method. The line/surface intersection algorithm involves two steps: intersection detection and intersection computation. In the intersection detection process, a bounding volume tree of k-dops (discrete oriented polytopes) for the subdivision surface is first created in order to accelerate the intersection detection between the line and the surface. The spline patches which are detected to be possibly intersected by the line are converted to Bézier representations. For the intersection computation, a matrix-based algorithm is applied, which converts the nonlinear intersection computation into solving a sequence of linear algebra problems using the singular value decomposition (SVD). Finally, the lattice-skin geometry is generated by projecting selected lattice nodes to the nearest intersection points intersected by the lattice edges. The Stanford bunny example demonstrates the efficiency and accuracy of the developed algorithm. The structural analysis of lattice-skin structures follows the isogeometric approach, in which the thin-shell is discretised with spline basis functions and the lattice structure is modelled with pin-jointed truss elements. In order to consider the lattice-skin coupling, a Lagrange multiplier approach is implemented to enforce the displacement compatibility between the coupled lattice nodes and the thin-shell. More importantly, the parametric coordinates of the coupled lattice nodes on the thin-shell surface are obtained directly from the lattice-skin geometry generation, which integrates the design and analysis process of lattice-skin structures. A sandwich plate example is analysed to verify the implementation and the accuracy of the lattice-skin coupling computation. In addition, a SIMP-like lattice topology optimisation method is proposed. The topology optimisation results of lattice structures are analysed and compared with several examples adapted from the benchmark examples commonly used in continuum topology optimisation. The SIMP-like lattice topology optimisation proposed is further applied to optimise the lattice in lattice-skin structures. The lattice-skin topology optimisation is fully integrated with the lattice-skin geometry design since the sensitivity analysis in the proposed method is based on lattice unit cells which are inherited from the geometry design stage. Finally, shape optimisation of lattice-skin structures using the free-form deformation (FFD) technique is studied. The corresponding shape sensitivity of lattice-skin structures is derived. The geometry update of the lattice-skin structure is determined by the deformation of the FFD control volume, and in this process the coupling between lattice nodes and the thin-shell is guaranteed by keeping the parametric coordinates of coupled lattice nodes which are obtained in the lattice-skin geometry design stage. A pentagon roof example is used to explore the combination of lattice topology optimisation and shape optimisation of lattice-skin structures.

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.

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.

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.

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.

The birational variant of Grothendieck's section conjecture proposes a characterisation of the rational points of a curve over a finitely generated field over Q in terms of the sections of the absolute Galois group of its function field. While the p-adic version of the birational section conjecture has been proven by Jochen Koenigsmann, and improved upon by Florian Pop, the conjecture in its original form remains very much open. One hopes to deduce the birational section conjecture over number fields from the p-adic version by invoking a local-global principle, but if this is achieved the problem remains to deduce from this that the conjecture holds over all finitely generated fields over Q. This is the problem that we address in this thesis, using an approach which is inspired by a similar result by Mohamed Saïdi concerning the section conjecture for étale fundamental groups. We prove a conditional result which says that, under the condition of finiteness of certain Shafarevich-Tate groups, the birational section conjecture holds over finitely generated fields over Q if it holds over number fields.

Recently, there has been interest in the tropicalization of the Grassmannian, known as the tropical Grassmannian. It has been shown that the tropical Grassmannian, and a related tropical space called the Dressian, can be used to construct subdivisions of matroid polytopes into matroid polytopes. There has also been interest ``nonnegative" and ``flag" versions of the tropical Grassmannian and the Dressian, which can be used to construct subdivisions of ``nonnegative" and ``flag" versions of matroid polytopes. This thesis combines these perspectives, exploring ``nonnegative flag" versions of these objects. We show that, for flags whose ranks consist of consecutive integers, many nice results that hold in the previously studied settings have ``nonnegative flag" analogues. The nonnegative flag variety of rank r=(r_1,..., r_k), defined by Lusztig, can be described as the set of rank r flags of linear subspaces in R^n which can be represented by a matrix whose minors are all positive. We show that, for flag varieties of consecutive rank, this equals the subset of the flag variety with nonnegative Plücker coordinates. This generalizes the well-established fact, proven independently by many authors including Rietsch, Talaska and Williams, Lam, and Lusztig, that the nonnegative Grassmannian equals the subset of the Grassmannian with nonnegative Plücker coordinates, and yields a straightforward condition to determine whether a flag of consecutive rank lies in the nonnegative flag variety. A flag F in any nonnegative flag variety determines a combinatorial object called a flag positroid, defined in terms of the set of nonzero Plücker coordinates of F. We characterize flag positroids of consecutive ranks as oriented flag matroids whose constituents are positively oriented matroids. We also explore flag varieties using tropical geometry, which is the study of algebraic geometry over the (min,+) semifield. The tropical flag variety and the flag Dressian are tropical spaces parameterizing realizable and abstract flags of tropical linear spaces, respectively. In general, the flag Dressian strictly contains the tropical flag variety. However, we show that the nonnegative tropical flag variety and the nonnegative flag Dressian, which parameterize positively realizable and abstract flags of positive tropical linear spaces, respectively, are equal for flags of consecutive ranks. This generalizes the equality of the nonnegative Dressian and the nonnegative tropical Grassmannian, proven by Speyer and Williams. The flag Dressian is closely related to coherent subdivisions of flag matroid polytopes: it is a polyhedral complex whose cones parameterize coherent subdivisions of flag matroid polytopes into smaller flag matroid polytopes. We specialize this relationship to the nonnegative flag Dressian. We prove that the restriction of this parameterization to the cones of the nonnegative flag Dressian instead parameterizes coherent subdivisions of flag positroid polytopes into smaller flag positroid polytopes. In the special case of the positive (non-flag) Dressian, this recovers the description of coherent subdivisions of hypersimplices into positroid polytopes by Lukowski, Parisi, and Williams. In the special case of the nonnegative complete flag Dressian, this describes coherent subdivisions of Bruhat interval polytopes into Bruhat interval polytopes.

Let X be a smooth complex projective variety. In 2002, [Bri07] defined a notion of stability for the objects in Db(X), the bounded derived category of coherent sheaves on X, which generalized the notion of slope stability for vector bundles on curves. There are many nice connections between stability conditions on X and the geometry of the variety. In 2012, [BMT14] gave a conjectural stability condition for threefolds. In the case that X is a complete intersection threefold, the existence of this stability condition would imply a Castelnuovo-type inequality for curves on X. I give a new Castelnuovo-type inequality for curves on complete intersection surfaces of high degree. I then show how this bound would imply the bound conjectured in [BMT14] if a weaker bound could be shown for curves of lower degree. I also construct new stability conditions for surfaces containing a curve C whose self-intersection is negative. I show that these stability conditions lie on a wall of the geometric chamber of Stab(X), the stability manifold of X. I then construct the moduli space Mσ (OX) of σ-semistable objects of class [OX] in K0(X) after wall-crossing.

Donaldson and Thomas defined Donaldson-Thomas (DT) invariants for moduli spaces of sheaves on proper Calabi-Yau threefolds in [DT,T]. The definition has been extended to more general moduli spaces of objects in 3-Calabi-Yau categories, and in the last few years there has been an increasing interest in studying refinements of DT invariants. This thesis is devoted to the computation of the generating series of refined DT invariants in some geometric examples. In Chapter 2 we compute the generating series of refined DT invariants for the moduli space Mn,r of higher rank framed sheaves on P3, generalizing the result of [BBS] for the Hilbert scheme of n points on C3. We describe Mn,r as a global critical locus by means of the Beilinson spectral sequence, and compute the generating series using motivic wallcrossing. We interpret the result as a count of r-tuples of weighted 3D partitions in analogy with the refined topological vertex [IKV]. In Chapter 3 we study the refined DT invariants for the moduli space Prn,d of higher rank (framed) stable pairs on the resolved conifold. Applying twice the Beilinson spectral sequence, we identify Prn,d with the moduli space of PT-stable representations of the r-framed conifold quiver with superpotential, generalizing a result of [NN11]. We compute a product expansion for the generating series of refined DT invariants of Prn,d, and we compare our results with the partition function counting U(1)-instantons on the minimal resolution of an Ar-1 surface singularity. In Chapter 4 we study the universal generating series of motivic DT invariants for deformations of graded 3-Calabi-Yau algebras. We consider Jacobi algebras which are derived equivalent to the resolved conifold and the resolution of a line of An-singularities, and investigate their marginal deformations. The universal series of motivic DT invariants are affected by the deformations, and their variation is interpreted in terms of virtual classes of some special representation varieties associated to the quiver.

This thesis constructs tilting bundles obtained from full strong exceptional collections of line bundles on all smooth toric Fano fourfolds. The tilting bundles lead to a large class of explicit Calabi-Yau-5 algebras, obtained as the corresponding rolled-up helix algebra. We provide two different methods to show that a collection of line bundles is full, whilst the strong exceptional condition is checked using the package QuiversToricVarieties for the computer algebra system Macaulay2, written by the author. A database of the full strong exceptional collections can also be found in this package.

This thesis concerns the study of chiral algebras over schemes of arbitrary dimension n. In Chapter I, we construct a chiral algebra over each smooth variety X of dimension n. We do this via the Hilbert scheme of points of X, which we use to build a factorisation space over X. Linearising this space produces a factorisation algebra over X, and hence, by Koszul duality, the desired chiral algebra. We begin the chapter with an overview of the theory of factorisation and chiral algebras, before introducing our main constructions. We compute the chiral homology of our factorisation algebra, and show that the D-modules underlying the corresponding chiral algebras form a universal D-module of dimension n. In Chapter II, we discuss the theory of universal D-modules and OO- modules more generally. We show that universal modules are equivalent to sheaves on certain stacks of étale germs of n-dimensional varieties. Furthermore, we identify these stacks with the classifying stacks of groups of automorphisms of the n-dimensional disc, and hence obtain an equivalence between the categories of universal modules and the representation categories of these groups. We also define categories of convergent universal modules and study them from the perspectives of the stacks of étale germs and the representation theory of the automorphism groups.

We investigate the action of the Atkin operator on spaces of overconvergent p-adic modular forms. Our contributions are both computational and geometric. We present several algorithms to compute the spectrum of the Atkin operator, as well as its p-adic variation as a function of the weight. As an application, we explicitly construct Heegner-type points on elliptic curves. We then make a geometric study of the Atkin operator, and prove a potential semi-stability theorem for correspondences. We explicitly determine the stable models of various Hecke operators on quaternionic Shimura curves, and make a purely geometric study of canonical subgroups.

We define and study the fundamental pro-finite 2-groupoid of varieties X defined over a field k. This is a higher algebraic invariant of a scheme X, analogous to the higher fundamental path 2-groupoids as defined for topological spaces. This invariant is related to previously defined invariants, for example the absolute Galois group of a field, and Grothendieck’s étale fundamental group. The special case of Brauer-Severi varieties is considered, in which case a “sections conjecture” type theorem is proved. It is shown that a Brauer-Severi variety X has a rational point if and only if its étale fundamental 2-groupoid has a special sort of section.

In this thesis, we consider q-deformations of multiplicative Hypertoric varieties, where q∈&Kopf;x for &Kopf; an algebraically closed field of characteristic 0. We construct an algebra Dq of q-difference operators as a Heisenberg double in a braided monoidal category. We then focus on the case where q is specialized to a root of unity. In this setting, we use Dq to construct an Azumaya algebra on an l-twist of the multiplicative Hypertoric variety, before showing that this algebra splits over the fibers of both the moment and resolution maps. Finally, we sketch a derived localization theorem for these Azumaya algebras.

This thesis presents a series of results obtained in [13, 18, 19, 23{25, 87]. In [19], we prove a Darboux theorem for derived schemes with symplectic forms of degree k < 0, in the sense of [142]. We use this to show that the classical scheme X = t0(X) has the structure of an algebraic d-critical locus, in the sense of Joyce [87]. Then, if (X, s) is an oriented d-critical locus, we prove in [18] that there is a natural perverse sheaf P·X,s on X, and in [25], we construct a natural motive MFX,s, in a certain quotient ring MμX of the μ-equivariant motivic Grothendieck ring MμX, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants [102]. In [13], we obtain similar results for k-shifted symplectic derived Artin stacks. We apply this theory to categorifying Donaldson-Thomas invariants of Calabi-Yau 3-folds, and to categorifying Lagrangian intersections in a complex symplectic manifold using perverse sheaves, and to prove the existence of natural motives on moduli schemes of coherent sheaves on a Calabi-Yau 3-fold equipped with 'orientation data', as required in Kontsevich and Soibelman's motivic Donaldson-Thomas theory [102], and on intersections L??M of oriented Lagrangians L,M in an algebraic symplectic manifold (S,ω). In [23] we show that if (S,ω) is a complex symplectic manifold, and L,M are complex Lagrangians in S, then the intersection X= L??M, as a complex analytic subspace of S, extends naturally to a complex analytic d-critical locus (X, s) in the sense of Joyce [87]. If the canonical bundles KL,KM have square roots K1/2L, K1/2M then (X, s) is oriented, and we provide a direct construction of a perverse sheaf P·L,M on X, which coincides with the one constructed in [18]. In [24] we have a more in depth investigation in generalized Donaldson-Thomas invariants DTα(τ) defined by Joyce and Song [85]. We propose a new algebraic method to extend the theory to algebraically closed fields K of characteristic zero, rather than K = C, and we conjecture the extension of generalized Donaldson-Thomas theory to compactly supported coherent sheaves on noncompact quasi-projective Calabi-Yau 3-folds, and to complexes of coherent sheaves on Calabi-Yau 3-folds.

This thesis studies derived equivalences between total spaces of vector bundles and dg-quivers. A dg-quiver is a graded quiver whose path algebra is a dg-algebra. A quiver with superpotential is a dg-quiver whose differential is determined by a "function" Φ. It is known that the bounded derived category of representations of quivers with superpotential with finite dimensional cohomology is a Calabi- Yau triangulated category. Hence quivers with superpotential can be viewed as noncommutative Calabi- Yau manifolds. One might then ask if there are derived equivalences between Calabi-Yau manifolds and quivers with superpotential. In this thesis, we answer this question and, generalizing Bridgeland [15], give a recipe on how to construct such derived equivalences.

The goal of this dissertation is twofold. First, we give a proof of the geometric cone conjecture in relative dimension at most $2$ in the full generality of klt CY pairs. This generalizes previous work done by Kawamata in \cite{Kaw97}, Totaro in \cite{Tot10}, and Filipazzi, Hacon, and Svaldi in \cite{FHS21}. A corollary to this result is that all klt pairs $(X.\Delta)$ with $\kappa(X,\Delta)\geq \dim X - 2$ have only finitely many isomorphism classes of minimal models. In the second part, we prove that the property of Bott vanishing fails for 68 deformation families of smooth Fano threefolds in characteristic $0$. This result is sharp, as subsequent work by Totaro in \cite{Tot24} shows that Bott vanishing is satisfied by the remaining 37 deformation families.

We investigate the count of meromorphic differentials on the Riemann sphere pos-sessing a single zero, multiple poles with prescribed orders, and fixed residues at each pole. Gendron and Tahar previously examined this problem with respect to general residues using flat geometry, while Sugiyama approached it from the perspective of fixed-point multipliers of polynomial maps in the case of simple poles. In our study, we employ intersection theory on compactified moduli spaces of differentials, enabling us to handle arbitrary residues and pole orders, which provides a complete solution to this problem. We also determine interesting combinatorial properties of the solution formula. This thesis is organized as follows: In Chapter 1 we give an introduction to the problem and summarize the main results obtained. In Chapter 2 we review the compactification of moduli spaces of differentials and introduce various divisor classes. In Section 2.3 we explain how to identify the universal line bundle class with the divisor class of the locus of differentials satisfying a general given residue tuple and prove Theorem 1.0.1 (i). In Section 2.4 we impose exactly one independent partial sum vanishing condition to the residues and prove Theorem 1.0.1 (ii). In Section 2.5 we give a polynomial expression in terms of the zero order for the degree of mixed products between powers of the dual tautological class and the psi-class of the zero. Finally in Chapter 3 we prove Theorem 1.0.2 for arbitrary residues and investigate combinatorial properties of the solution formula. We have also verified our formula numerically for a number of cases by using the software package [CMZ2].

This thesis is concerned with the study of the geometry and derived categories associated to the moduli problems of local systems and Higgs bundles in positive characteristic. As a cornerstone of our investigation, we establish a local system analogue of the BNR correspondence for Higgs bundles. This result (Proposition 4.3.1) relates flat connections to certain modules of an Azumaya algebra on the family of spectral curves. We prove properness over the semistable locus of the Hitchin map for local systems introduced by Laszlo–Pauly (Theorem 4.4.1). Moreover, we show that with respect to this Hitchin map, the moduli stack of local systems is étale locally equivalent to the moduli stack of Higgs bundles (Theorem 4.6.3) (with or without stability conditions). Subsequently, we study two-dimensional examples of moduli spaces of parabolic Higgs bundles and local systems (Theorem 5.2.1), given by equivariant Hilbert schemes of cotangent bundles of elliptic curves. Furthermore, the Hilbert schemes of points of these surfaces are equivalent to moduli spaces of parabolic Higgs bundles, respectively local systems (Theorem 5.3.1). The proof for local systems in positive characteristic relies on the properness results for the Hitchin fibration established earlier. The Autoduality Conjecture of Donagi–Pantev follows from Bridgeland–King–Reid’s McKay equivalence in these examples. The last chapter of this thesis is concerned with the con- struction of derived equivalences, resembling a Geometric Langlands Correspondence in positive characteristic, generalizing work of Bezrukavnikov–Braverman. Away from finitely many primes, we show that over the locus of integral spectral curves, the derived category of coherent sheaves on the stack of local systems is equivalent to a derived category of coherent D-modules on the stack of vector bundles. We conclude by establishing the Hecke eigenproperty of Arinkin’s autoduality and thereby of the Geometric Langlands equivalence in positive characteristic.

This thesis contains two main results. The first is a comparison formula for the Donaldson-Thomas invariants of two (complex, smooth and projective) Calabi-Yau threefolds related by a flop; the second is a proof of the projective case of the Crepant Resolution Conjecture for Donaldson-Thomas invariants, as stated by Bryan, Cadman and Young. Both results rely on Bridgeland’s category of perverse coherent sheaves, which is the heart of a t-structure in the derived category of the given Calabi-Yau variety. The first formula is a consequence of various identities in an appropriate motivic Hall algebra followed by an implementation of the integration morphism (using the technology of Joyce and Song). Our proof of the crepant resolution conjecture is a quick and elegant application of the first formula in the context of the derived McKay correspondence of Bridgeland, King and Reid. The first chapter is introductory and is followed by two chapters of background material. The last two chapters are devoted to the proofs of the main results.

In this thesis we study various ways in which every two general points on a variety can be connected by curves of a fixed genus, thus mimicking the notion of a rationally connected variety but for arbitrary genus. We assume the existence of a covering family of curves which dominates the product of a variety with itself either by allowing the curves in the family to vary in moduli, or by assuming the family is trivial for some fixed curve of genus g. A suitably free curve will be one with a large unobstructed deformation space, the images of whose deformations can join any number of points on a variety. We prove that, at least in characteristic zero, the existence of such a free curve of higher genus is equivalent to the variety being rationally connected. If one restricts to the case of genus one, similar results can be obtained even allowing the curves in the family to vary in moduli. In later chapters we study algebraic properties of such varieties and discuss attempts to prove the same rational connectedness result in positive characteristic.

The thesis deals with definability of certain Zariski geometries, introduced by Zilber, in the theory of algebraically closed fields. I axiomatise a class of structures, called 'abstract linear spaces', which are a common reduct of these Zariski geometries. I then describe what an interpretation of an abstract linear space in an algebraically closed field looks like. I give a new proof that the structure "quantum harmonic oscillator", introduced by Zilber and Solanki, is not interpretable in an algebraically closed field. I prove that a similar structure from an unpublished note of Solanki is not definable in an algebraically closed field and explain the non-definability of both structures in terms of geometric interpretation of the group law on a Galois cohomology group H1(k(x), μn). I further consider quantum Zariski geometries introduced by Zilber and give necessary and sufficient conditions that a quantum Zariski geometry be definable in an algebraically closed field. Finally, I take an attempt at extending the results described above to complex-analytic setting. I define what it means for quantum Zariski geometry to have a complex analytic model, an give a necessary and sufficient conditions for a smooth quantum Zariski geometry to have one. I then prove a theorem giving a partial description of an interpretation of an abstract linear space in the structure of compact complex spaces and discuss the difficulties that present themselves when one tries to understand interpretations of abstract linear spaces and quantum Zariski geometries in the compact complex spaces structure.

The topic of this thesis is the moduli theory of (parabolic) sheaves on stable curves. Using geometric invariant theory (GIT), universal moduli spaces of semistable parabolic sheaves on stable marked curves are constructed: `universal' indicates that these are moduli spaces of pairs where the underlying marked curve may vary as well as the parabolic sheaf (as in the Pandharipande moduli space for pairs of stable curves and torsion-free sheaves without augmentations). As an intermediate step in this construction, we construct moduli spaces of semistable parabolic sheaves on flat families of arbitrary projective schemes (of any dimension or singularity type): this is the technical core of this thesis. These moduli spaces are projective, since they are constructed as GIT quotients of projective parameter spaces. The stability condition for parabolic sheaves depends on a choice of polarisation and is derived from the Hilbert-Mumford criterion. It is not quite the same as traditional stability with respect to parabolic Hilbert polynomials, but it is closely related to it, and the resulting moduli spaces are always compactifications of moduli of slope-stable parabolic sheaves. The construction works over algebraically closed fields of arbitrary characteristic.

This thesis is on moduli spaces of complexes of sheaves and diagrams of such moduli spaces. The objects in these diagrams are constructed as geometric invariant theory quotients and the points in these quotients correspond to certain equivalence classes of complexes. The morphisms in these diagrams are constructed by taking direct sums with acyclic complexes. We then study the colimit of such a diagram and in particular are interested in studying the images of quasi-isomorphic complexes in the colimit. As part of this thesis we construct categorical quotients of a group action on unstable strata appearing in a stratification associated to a complex projective scheme with a reductive group action linearised by an ample line bundle. We study this stratification for a closed subscheme of a quot scheme parametrising quotient sheaves over a complex projective scheme and relate the Harder-Narasimhan types of unstable sheaves with the unstable strata in the associated stratification. We also study the stratification of a parameter space for complexes with respect to a linearisation determined by certain stability parameters and show that a similar result holds in this case. The objects in these diagrams are indexed by different Harder-Narasimhan types for complexes and are quotients of parameter schemes for complexes of this fixed Harder-Narasimhan type. This quotient is given by a choice of linearisation of the action and so the diagrams depend on these choices. We conjecture that these choices can be made so that for any quasi-isomorphism between complexes representing points in this diagram both complexes are identified in the colimit of this diagram.

A suitable subcategory of affine Azumaya algebras is defined and a functor from this category to the category of Zariski structures is constructed. The rudiments of a theory of presheaves of topological structures is developed and applied to construct examples of structures at a generic parameter. The category of equivariant algebras is defined and a first-order theory is associated to each object. For those theories satisfying a certain technical condition, uncountable categoricity and quantifier elimination results are established. Models are shown to be Zariski structures and a functor from the category of equivariant algebras to Zariski structures is constructed. The two functors obtained in the thesis are shown to agree on a nontrivial class of algebras.

For a Fano variety V with at most Kawamata log terminal (klt) singularities and a finite group G acting bi-regularly on V , we say that V is G-exceptional (resp., G-weakly-exceptional) if the log pair (V,∆) is klt (resp., log canonical) for all G-invariant effective Q-divisors ∆ numerically equivalent to the anti-canonical divisor of V. Such G-exceptional klt Fano varieties V are conjectured to lie in finitely many families by Shokurov ([Sho00, Pro01]). The only cases for which the conjecture is known to hold true are when the dimension of V is one, two, or V is isomorphic to n-dimensional projective space for some n. For the latter, it can be shown that G must be primitive—which implies, in particular, that there exist only finitely many such G (up to conjugation) by a theorem of Jordan ([Pro00]). Smooth G-weakly-exceptional Fano varieties play an important role in non-rationality problems in birational geometry. From the work of Demailly (see [CS08, Appendix A]) it follows that Tian’s αG-invariant for such varieties is no smaller than one, and by a theorem of Tian such varieties admit G-invariant Kähler-Einstein metrics. Moreover, for a smooth G-exceptional Fano variety and given any G-invariant Kähler formin the first Chern class, the Kähler-Ricci iteration converges exponentially fast to the Kähler form associated to a Kähler- Einsteinmetric in the C∞(V)-topology. The termexceptional is inherited from singularity theory, to which this study enjoys strong links. We classify two-dimensional smooth G-exceptional Fano varieties (del Pezzo surfaces) and provide a partial list of all G-exceptional and G-weakly-exceptional pairs (S,G), where S is a smooth del Pezzo surface and G is a finite group of automorphisms of S. Our classification confirms many conjectures on two-dimensional smooth exceptional Fano varieties.

This thesis is concerned with Calabi-Yau threefolds and vector bundles upon them, which are the basic mathematical objects at the centre of smooth supersymmetric compactifications of heterotic string theory. We begin by explaining how these objects arise in physics, and give a brief review of the techniques of algebraic geometry which are used to construct and study them. We then turn to studying multiply-connected Calabi-Yau threefolds, which are of particular importance for realistic string compactifications. We construct a large number of new examples via free group actions on complete intersection Calabi-Yau manifolds (CICY's). For special values of the parameters, these group actions develop fixed points, and we show that, on the quotient spaces, this leads to a particular class of singularities, which are quotients of the conifold. We demonstrate that, in many cases at least, such a singularity can be resolved to yield another smooth Calabi-Yau threefold, with different Hodge numbers and fundamental group. This is a new example of the interconnectedness of the moduli spaces of distinct Calabi-Yau threefolds. In the second part of the thesis we turn to a study of two new `three-generation' manifolds, constructed as quotients of a particular CICY, which can also be represented as a hypersurface in dP6 x dP6, where dP6 is the del Pezzo surface of degree six. After describing the geometry of this manifold, and especially its non-Abelian quotient, in detail, we show how to construct on the quotient manifolds vector bundles which lead to four-dimensional heterotic models with the standard model gauge group and three generations of particles. The example described in detail has the spectrum of the minimal supersymmetric standard model plus a single vector-like pair of colour triplets.

Let Y be a quartic hypersurface in P⁴ with terminal singularities. The Grothendieck-Lefschetz theorem states that any Cartier divisor on Y is the restriction of a Cartier divisor on P⁴. However, no such result holds for the group of Weil divisors. More generally, let Y be a terminal Gorenstein Fano 3-fold with Picard rank 1. Denote by s(Y )=h_4 (Y )-h² (Y ) = h_4 (Y )-1 the defect of Y. A variety is Q-factorial when every Weil divisor is Q-Cartier. The defect of Y is non-zero precisely when the Fano 3-fold Y is not Q-factorial. Very little is known about the topology of non Q-factorial terminal Gorenstein Fano 3-folds. Q-factoriality is a subtle topological property: it depends both on the analytic type and on the position of the singularities of Y . In this thesis, I endeavour to answer some basic questions related to this global topolgical property. First, I determine a bound on the defect of terminal quartic 3-folds and on the defect of terminal Gorenstein Fano 3-folds that do not contain a plane. Then, I state a geometric motivation of Q-factoriality. More precisely, given a non Q-factorial quartic 3-fold Y , Y contains a special surface, that is a Weil non-Cartier divisor on Y . I show that the degree of this special surface is bounded, and give a precise list of the possible surfaces. This question has traditionally been studied in the context of Mixed Hodge Theory. I have tackled it from the point of view of Mori theory. I use birational geometric methods to obtain these results.

A recent series of papers, starting with the paper of Cook, Harbourne, Migliore, and Nagel on the projective plane in 2018, studies a notion of unexpectedness for finite sets Z of points in N-dimensional projective space. Say the complete linear system L of forms of degree d vanishing on Z has dimension t yet for any general point P the linear system of forms vanishing on Z with multiplicity m at P is nonempty. If the dimension of L is more than the expected dimension of t−r, where r is N+m−1 choose N, we say Z has unexpected hypersurfaces of degree d and multiplicity m. We extend the definition of unexpectedness to include the possibility that the unexpectedness occurs only for P on a subvariety of positive codimension. We begin our study of this stratified unexpectedness by analyzing sets of points in the plane. We are able to give a characterization for unexpectedness for sets of points which lie on a degenerate conic. We then further analyze the techniques of Faenzi and Vallés and how they were used in the 2018 paper of Cook, Harbourne, Migliore, and Nagel. In the fourth chapter, we study lower bounds on the dimension of the space L. Lastly, we study the connection between Lefschetz properties and unexpectedness. Advisor: Brian Harbourne

In this thesis we apply methods from homological algebra to the study of finite $p$-groups. Let $G$ be a finite $p$-group and let $\mathbb{F}_p$ be the field of $p$ elements. We consider the cohomology groups $\operatorname{H}^1(G,\mathbb{F}_p)$ and $\operatorname{H}^2(G,\mathbb{F}_p)$ and the Massey product structure on these cohomology groups, which we use to deduce properties about $G$. We tie the classical theory of Massey products in with a general method from deformation theory for constructing hulls of functors and see how far the strictly defined Massey products can take us in this setting. We show how these Massey products relate to extensions of modules and to relations, giving us cohomological presentations of $p$-groups. These presentations will be minimal pro-$p$ presentations and will often be different from the presentations we are used to. This enables us to shed some new light on the classification of $p$-groups, in particular we give a `tree construction' illustrating how we can `produce' $p$-groups using cohomological methods. We investigate groups of exponent $p$ and some of the families of groups appearing in the tree. We also investigate the limits of these methods. As an explicit example illustrating the theory we have introduced, we calculate Massey products using the Yoneda cocomplex and give 0-deficiency presentations for split metacyclic $p$-groups using strictly defined Massey products. We also apply these methods to the modular isomorphism problem, i.e. the problem whether (the isomorphism class of) $G$ is determined by $\F_pG$. We give a new class $\mathcal{C}$ of finite $p$-groups which can be distinguished using $\mathbb{F}_pG$.

We study complex analogues on Calabi-Yau manifolds of gauge theories on low dimensional real manifolds. In particular we define a holomorphic analogue of the Casson invariant, counting coherent sheaves on a Calabi-Yau 3-fold.

We compute the nef cones in codimensions 2 & 3 and the effective cones in dimensions 2 & 3 for the Hilbert scheme of three points in $\mathbb{P}^3$. This involves studying the orbits of the PGL action on the Hilbert scheme, as well as extending Mallavibarrena and Sols' bases for the Chow groups of Hilbert schemes of points on $\mathbb{P}^2$ to the case of the Hilbert scheme of 3 points in $\mathbb{P}^3$. This work builds on results of Ryan and Stathis, and utilizes results of Rossello-Llompart on intersection products in the Hilbert scheme of three points in $\mathbb{P}^3$.

The geproci property is a recent development in the world of geometry. We call a set of points Z\subseq\P_k^3 an (a,b)-geproci set (for GEneral PROjection is a Complete Intersection) if its projection from a general point P to a plane is a complete intersection of curves of degrees a and b. Examples known as grids have been known since 2011. Previously, the study of the geproci property has taken place within the characteristic 0 setting; prior to the work in this thesis, a procedure has been known for creating an (a,b)-geproci half-grid for 4\leq a\leq b, but it was not known what other examples there can be. Furthermore, before the work in this thesis, only a few examples of geproci nontrivial non-grid non-half-grids were known and there was no known way to generate more. Here, we use geometry in the positive characteristic setting to give new methods of producing geproci half-grids and non-half-grids. We also pick up work that had been done in 2017 by Solomon Akesseh, who had proven that there are no unexpected cubics in characteristic 3 with distinct points and gave examples involving infinitely near points based on quasi-elliptic fibrations in characteristic 2. Each quasi-elliptic fibration has a Dynkin diagram. Here, in contrast, for each possible Dynkin diagram for a quasi-elliptic fibration in characteristic 3, we give an example of the fibration but show it does not give rise to an unexpected cubic. [Equations Omitted] Adviser: Brian Harbourne

Given an ordinary elliptic curve E over a field of characteristic p, there is an elliptic curve E over the Witt vectors W() for which we can lift the Frobenius morphism, called the canonical lifting of E. The Weierstrass coefficients and the elliptic Teichmüller lift of E are given by rational functions over _p that depend only on the coefficients and points of E. Finotti studied the properties of these rational functions over fields of characteristic p ≥ 5. We investigate the same properties for fields of characteristic 2 and 3, make progress on some conjectures of Finotti, and introduce some conjectures of our own. We also investigate the structure of rings of Witt vectors over arbitrary commutative rings and give a computationally useful isomorphism for Witt vectors over ℤ/p^αℤ [alpha].

Twisted Edwards Curves are a representation of Elliptic Curves given by the solutions of bx^2 + y^2 = 1 + ax^2y^2. Due to their simple and unified formulas for adding distinct points and doubling, Twisted Edwards Curves have found extensive applications in fields such as cryptography. In this thesis, we study the Canonical Liftings of Twisted Edwards Curves and the associated lift of points Elliptic Teichmu ̈ller Lift. The coordinate functions of the latter are proved to be polynomials, and their degrees and derivatives are computed. Moreover, an algorithm is described for explicit computations, and some properties of the general formulas are given. Additionally, we define a notion of Minimal and Absolute Minimal Degree Liftings modulo pn and an associated lift of points we call an Edwards Lift. We show how the Canonical Lift may be used to find examples of Absolute Minimal Degree Liftings, at least modulo p3.