Noncommutative Riemannian geometry is an area that has seen intense activity over the past 25 years. Despite this, noncommutative complex geometry is only now beginning to receive serious attention. The theory of quantum groups provides a large family of very interesting potential examples, namely the quantum flag manifolds. Thus far, only the irreducible quantum flag manifolds have been investigated as noncommutative complex spaces. In a series of papers, Heckenberger and Kolb showed that for each of these spaces, there exists a q-deformed Dolbeault double complex. In this thesis a comprehensive framework for noncommutative complex geometry on quantum homogeneous spaces is introduced. The main ingredients used are covariant differential calculi and Takeuchi's categorical equivalence for faithfully at quantum homogeneous spaces. A number of basic results are established, producing a simple set of necessary and sufficient conditions for noncommutative complex structures to exist. It is shown that when applied to the quantum projective spaces, this theory reproduces the q-Dolbeault double complexes of Heckenberger and Kolb. Furthermore, the framework is used to q-deform results from Borel{Bott{ Weil theory, and to produce the beginnings of a theory of noncommutative Kahler geometry.

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.

This thesis explores the geometric principles underlying many of the known Trichotomy Theorems. The main aims are to unify the field construction in non-linear o-minimal structures and generalizations of Zariski Geometries as well as to pave the road for completely new results in this direction. In the first part of this thesis we introduce a new axiomatic framework in which all the relevant structures can be studied uniformly and show that these axioms are preserved under elementary extensions. A particular focus is placed on the study of a smoothness condition which generalizes the presmoothness condition for Zariski Geometries. We also modify Zilber's notion of universal specializations to obtain a suitable notion of infinitesimals. In addition, families of curves and the combinatorial geometry of one-dimensional structures are studied to prove a weak trichotomy theorem based on very weak one-basedness. It is then shown that under suitable additional conditions groups and group actions can be constructed in canonical ways. This construction is based on a notion of ``geometric calculus'' and can be seen in close analogy with ordinary differentiation. If all conditions are met, a definable distributive action of one one-dimensional type-definable group on another are obtained. The main result of this thesis is that both o-minimal structures and generalizations of Zariski Geometries fit into this geometric framework and that the latter always satisfy the conditions required in the group constructions. We also exhibit known methods that allow us to extract fields from this. In addition to unifying the treatment of o-minimal structures and Zariski Geometries, this also gives a direct proof of the Trichotomy Theorem for "type-definable" Zariski Geometries as used, for example, in Hrushovski's proof of the relative Mordell-Lang conjecture.

This work concerns the construction of Calabi-Yau threefolds in codimension 4. Based on a study of Hilbert series, we give a list of families of Calabi-Yau threefolds which may exist in codimension 3 and codimension 4. Using birational methods, we construct Calabi-Yau threefolds that realize several of the listed families. The main result is that the cases we consider in codimension 4 lie in two different deformation components.

Although introductory courses in special relativity give an introduction to the causal structure of Minkowski space, it is common for causal structure in general space- times to be regarded as an advanced topic, and omitted from introductory courses in general relativity, although the related topic of gravitational lensing is often included. Here a numerical approach to visualizing the light cones in exterior Schwarzschild space taking advantage of the symmetries of Schwarzschild space and the conformal invariance of null geodesics is formulated, and used to make some of these ideas more accessible. By means of the Matlab software developed, a user is able to produce figures showing how light cones develop in Schwarzschild space, starting from an arbitrary point and developing for any length of time. The user can then interact with the figure, changing their point of view, or zooming in or out, to investigate them. This approach is then generalised, using the symbolic manipulation facility of Matlab, to allow the user to specify a metric as well as an initial point and time of development. Finally, the software is demonstrated with a selection of metrics.

The geometry of surfaces of rotation in three dimensional Euclidean space has been studied widely. The rotational surfaces in three dimensional Euclidean space are generated by rotating an arbitrary curve about an arbitrary axis. Moreover, the geodesics on surfaces of rotation in three dimension Euclidean space have been considered and discovered. Clairaut's [1713-1765] theorem describes the geodesics on surfaces of rotation and provides a result which is very helpful in understanding all geodesics on these surfaces. On the other hand, the Minkowski spaces have shorter history. In 1908 Minkowski [1864-1909] gave his talk on four dimensional real vector space, with asymmetric form of signature (+,+,+,-). In this space there are different types of vectors/axes (space-like- time-like and null) as well as different types of curves (space-like- time-like and null). This thesis considers the different types of axes of rotations, then creates three different types of surfaces of rotation in three dimensional Minkowski space, and generates Clairaut's theorem to each type of these surfaces, it also explains the analogy between three dimensional Euclidean and Minkowskian spaces. Moreover, this thesis produces different types of surfaces of rotation in four dimensional Minkowski spaces. It also generalises Clairaut's theorem for these surfaces of rotations in four dimensional Minkowski space. Then we see how Clairaut's theorem characterization carries over to three dimensional and four dimensional Minkowski spaces.

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.

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.

We present a uniform framework generalising and extending the classical theories of projective differential geometry, c-projective geometry, and almost quaternionic geometry. Such geometries, which we call \emph{projective parabolic geometries}, are abelian parabolic geometries whose flat model is an R-space $G\cdot\mathfrak{p}$ in the infinitesimal isotropy representation $\mathbb{W}$ of a larger self-dual symmetric R-space $H\cdot\mathfrak{q}$. We also give a classification of projective parabolic geometries with $H\cdot\mathfrak{q}$ irreducible which, in addition to the aforementioned classical geometries, includes a geometry modelled on the Cayley plane $\mathbb{OP}^2$ and conformal geometries of various signatures. The larger R-space $H\cdot\mathfrak{q}$ severely restricts the Lie-algebraic structure of a projective parabolic geometry. In particular, by exploiting a Jordan algebra structure on $\mathbb{W}$, we obtain a $\mathbb{Z}^2$-grading on the Lie algebra of $H$ in which we have tight control over Lie brackets between various summands. This allows us to generalise known results from the classical theories. For example, which riemannian metrics are compatible with the underlying geometry is controlled by the first BGG operator associated to $\mathbb{W}$. In the final chapter, we describe projective parabolic geometries admitting a $2$-dimensional family of compatible metrics. This is the usual setting for the classical projective structures; we find that many results which hold in these settings carry over with little to no changes in the general case.

La motivation initiale de cette thèse est l'étude d'une discrétisation volumique de surface (Chapitre 2) naturellement liée à la structure de varifold. Le point clé est qu'il est possible de munir d'une structure de varifold la plupart des objets utilisés pour représenter ou discrétiser des surfaces c'est-à-dire aussi bien des objets tels que les sous variétés ou les ensembles rectifiables que des objets tels que des nuages de points ou encore la discrétisation volumique proposée, ce qui permet d'étudier dans un cadre unifié une surface et sa discrétisation. Une difficulté essentielle est que, généralement, ces structures discrètes ne sont pas rectifiables, ce qui soulève la question suivante : comment assurer qu'un varifold, obtenu comme limite de discrétisations volumiques, soit une surface, au moins en un sens faible ? De façon plus précise : quelles conditions sur une suite de varifolds quelconques assurent que le varifold limite est rectifiable (Chapitre 3) ou encore qu'il est à variation première bornée (Chapitre 5) ? On obtient des conditions quantitatives assurant la rectifiabilité grâce à des énergies liées aux nombres beta de Jones. On s'intéresse ensuite à la régularité du varifold limite en termes de courbure (variation première). On a essayé de contrôler la variation première en utilisant des techniques de construction de mesures de type packing (Chapitre 4), une forme régularisée de la variation première d'un varifold. Cette régularisation permet de définir des énergies de Willmore approchées qui Gamma convergent dans l'espace des varifolds vers l'énergie de Willmore ainsi qu'une approximation de la courbure qui est testée numériquement dans le Chapitre 6 / The starting point of this work is the study of a volumetric surface discretization model naturally connected to the varifolds structure introduced in Chapter 2. The point is that not only the discretization we propose can be endowed with a structure of varifold but also a great part of objects used for surface representation and discretization (triangulation, cloud points, level sets etc.) so that we can use varifolds tools to study in some unified setting different ways of discretizing surfaces. An important point to overcome is that these structures are generally not rectifiable so that we address the following question: how to ensure that the limit of a sequence of such discrete surfaces is regular? More precisely, what conditions on a sequence of varifolds (not necessarily rectifiable nor with bounded variation) ensure that the limit varifold is rectifiable (Chapter 3) or has bounded first variation (Chapter 5)? We obtain quantitative conditions of rectifiability for variflods considering energies linked to Jones' beta numbers. We then address the question in terms of first variation (generalized curvature) of a limit varifold. We first try a packing measure construction of the first variation of a varifold V (Chapter 4), then we define a regularized form of the classical first variation, allowing us to exhibit an energetic condition ensuring that a limit of a sequence of varifolds has bounded first variation. We use this regularized form to build an approximate Willmore energy Gamma-converging in the class of varifolds to the Willmore energy. In Chapter 6, we test numerically a notion of approximate curvature derived from the regularized first variation