The purpose of this thesis is to study d-manifolds and d-orbifolds and their bordism groups. D-manifolds and d-orbifolds were recently introduced by Joyce as a new class of geometric objects to study moduli problems in algebraic and symplectic geometry. In the spirit of Joyce we will introduce the notion of (stable) nearly and homotopy complex structures on these 2-categories and study their unitary bordism groups. Fukaya and Ono proved that the moduli space of ,em>n-pointed, genus g, J-holomorphic curves M_g,n(M,J,β) carries a so called stably almost complex structure, and as Kuranishi spaces are closely related to d-orbifolds, the introduction of complex structures will be essential in studying symplectic Gromov-Witten invariants using d-orbifolds. We furthermore introduce the notion of representable d-orbifolds and prove that these representable d-orbifolds can be embedded into an orbifold. We will then explain how a result of Kresch can be used to show that many important moduli spaces in algebraic geometry, are representable and thus embeddable d-orbifolds. Moreover we will sketch how one could prove an analogous result in the symplectic case. We then prove as one of our main results, that for a compact manifold the unitary d-bordism group is isomorphic to its ‘classical’ unitary bordism group. This result extends a result by Joyce who proved a similar statement for oriented manifolds and d-manifolds. Furthermore we will introduce the notion of blowups in the 2-category of d-manifolds and prove that these d-blowups satisfy a universal property. Finally, we sketch how our results may be used to make a step towards a proof of the Gopakumar–Vafa integrality conjecture and a “resolution of singularities” theorem for d-orbifolds.

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This thesis investigates the birational geometry of a class of higher dimensional Fano varieties of index 1 with quadratic hypersurface singularities. The main investigating question is, what structures of a rationally connected fibre space can these varieties have? Two cases are considered: double covers over a hypersurface of degree two, known as double quadrics and double covers over a hypersurface of degree three, known as double cubics. This thesis extends the study of double quadrics and cubics, first studied in the non-singular case by Iskovskikh and Pukhlikov, by showing that these varieties have the property of birational superrigidity, under certain conditions on the singularities of the branch divisor. This implies, amongst other things, that these varieties admit no non-trivial structures of a rationally connected fibre space and are thus non-rational. Additionally, the group of birational automorphisms coincides with the group of regular automorphisms. This is shown using the ``Method of maximal singularities" of Iskovskikh and Manin, expanded upon by Pukhlikov and others, in conjunction with the connectedness principal of Shokurov and Kollar. These results are then used to give a lower bound on the codimension of the set of all double quadrics (and double cubics) which are either not factorial or not birationally superrigid, in the style of the joint work of Pukhlikov and Eckl on Fano hypersurfaces. Such a result has applications to the study of varieties which admit a fibration into double quadrics or cubics.

This thesis concerns deformations of maps into submanifolds of projective spaces and in par- ticular the deformable surfaces of Lie sphere geometry. Using a gauge theoretic approach we study the transformations of Lie applicable surfaces and characterise certain classes of surfaces in terms of polynomial conserved quantities. In particular we unify isothermic, Guichard and L-isothermic surfaces as certain Lie applicable surfaces and show how their well known trans- formations arise in this setting. Another class of surfaces that is highlighted in this thesis is that of linear Weingarten surfaces in space forms and their transformations.

Line geometry is a foundation of screw algebra in line coordinates that were created by Plücker as ray coordinates taking a line as a ray between two points and axis coordinates taking a line as the intersection of two planes. This Thesis reveals the geometrical meaning and intrinsic relationship between these ray coordinates and axis coordinates, leading to an in-depth understanding of conformability and duality of these two sets of screw coordinates, and their related vector space and dual vector space. Based on the study of screw algebra, the resultant twist of a serial manipulator is presented geometrically by an assembly of unit joint screws with the corresponding velocity amplitudes. This leads to the geometrical interpretation for the resultant twist with its instantaneous screw axis (ISA) that is formulated by a combination of weighted position vectors of joint screws. The screw-based Jacobian is then derived after recognizing the resultant twist of a serial manipulator. The case leads to a revelation for the first time the relationship of a Jacobian matrix acquired by using screw algebra and a derivative Jacobian matrix using differential functions, and to an in-depth investigation of transformation between these two Jacobians. To extend the application of screw algebra and this derivative Jacobian, kinematics analysis of a novel reconfigurable base-integrated parallel mechanism is proposed and its screw-based Jacobian is derived, leading to its equivalent model, the Metamorphic hand with a reconfigurable palm. The method is then applied to the investigation of the Metamorphic Hand, while manipulating an object, based on the product sub-manifolds and the exponential method. Evaluation of the functionality of the Metamorphic hand is further analysed, with the Anthropomorphism Index (AI) and palmar shape modulation as the criteria, evaluating performance enhancement of the Metamorphic hand in comparison to other robotic hands with a fixed palm. The Thesis presents novel discoveries in the intrinsic geometry of screw coordinates and the coherent connection between Jacobian formed by screw algebra and the Jacobian using the derivative method. This intrinsic geometry insight is then used to investigate for the first time the parallel mechanism with a reconfigurable base, paving a way for an in-depth investigation of the Metamorphic hand on its reconfigurability and grasp affordability and for the first time using Anthropomorphic Index to evaluate the Metamorphic hand.

Let C be the boundary surface of a strictly convex d-dimensional body. Andrews obtained an upper bound in terms of M for the number of points on MC, the M-fold enlargement of C. We consider the integer points within a distance 5 of the hypersurface MC. Introducing S requires some uniform approximability condition on the surface C, involving determinants of derivatives. To obtain an asymptotic formula (main term the volume of the search region) requires the Fourier transform with conditions up to the Gd-th derivative. We obtain an upper bound subject to a Curvature Condition that re quires only first and second derivatives, that MC has a tangent hyperplane everywhere, and each two-dimensional normal section has radius of curvature in the range cqM +1/2 < p <C M 1/2, where cq and c are non-zero constants. Our main result is Theorem 2. THEOREM 2. Let C be a strictly convex hypersurface in d-dimensional space (d > 3), satisfying the Curvature Condition at size M. Then the total number, N, of integer points lying within a distance 6 of MC is bounded by the sum of two terms, one from Andrews's bound, the other from the hypervolume of the search region, with explicit constant factors involving 6, cq and c . In the body of the thesis, to simplify the notation, we use C for the enlarged surface called MC in this summary. In Part II we enumerate a class of special magic squares. We observe a new identity between values of the zeta functions at even integers.

In this dissertation we discuss two new constructions of Fano varieties, each directly inspired by ideas in Mirror Symmetry. The first recasts the Fanosearch program (Coates--Corti--Kasprzyk et al.) for surfaces in terms of a construction related to the SYZ conjecture. In particular we construct Q-Gorenstein smoothings of toric varieties via an application of the Gross-Siebert algorithm to certain affine manifolds. We recover the theory of combinatorial mutation, which plays a central role in the Fanosearch program, from these affine manifolds. Combining this construction and the work of Gross--Hacking--Keel on log Calabi--Yau surfaces we produce a cluster structure on the mirror to a log del Pezzo surface proposed by Coates--Corti--et al. We exploit the cluster structure, and the connection to toric degenerations, to prove two classification results for Fano polygons. This cluster variety is equipped with a superpotential defined on each chart by a so-called maximally mutable Laurent polynomial. We study an enumerative interpretation of this superpotential in terms of tropical disc counting in the example of the projective plane (with a general choice of boundary divisor). In the second part we develop a new construction of Fano toric complete intersections in higher dimensions. We first consider the problem of finding torus charts on the Hori--Vafa/Givental model, adapting the approach taken by Przyjalkowski. We exploit this to identify 527 new families of four-dimensional Fano manifolds. We then develop an inverse algorithm, Laurent Inversion, which decorates a Fano polytope P with additional information used to construct a candidate ambient space for a complete intersection model of the toric variety defined by P. Moving in the linear system defining this complete intersection allows us to construct new models of known Fano manifolds, and also to construct new examples of Fano manifolds from conjectured mirror Laurent polynomials. We use this algorithm to produce families simultaneously realising certain collections of 'commuting' mutations, extending the connection between polytope mutation and deformations of toric varieties.

We study overconvergent modular forms on certain unitary Shimura curves, dened for integral weights by Kassaei and general weights by Brasca. There are Hecke operators acting on these spaces of overconvergent modular forms, and there is a distinguished UP-operator which is a compact operator on these spaces. We construct a deformationtheoretic eigencurve in this setting, which comes with a projection to the weight space. Then we prove that its nilreduction is isomorphic to the Hecke eigencurve. In particular, for each weight in the weight space, the bre above it is idented with systems of Hecke eigenvalues arising from overconvergent eigenforms of that weight, whose UPeigenvalu e is not 0. Lastly, we prove that this eigencurve is proper (that is, the map to the weight space satises the valuative criterion of properness). This is done using padic Hodge theory, via interpreting the UPeigenvalues on the automorphic side as the eigenvalues of Frobenius on the padic Hodge theory side, for families of Galois representations attached to nite slope, overconvergent eigenforms.

In this thesis we develop a framework for constructing quotients of varieties by actions of linear algebraic groups which is similar in spirit to that of Mumford's geometric invariant theory. This is done by extending the work of Doran and Kirwan in the unipotent setting to deal with more general non-reductive groups. Given a linear algebraic group acting on an irreducible variety with a linearisation, an open subset of stable points is identified that admits a geometric quotient in the category of varieties. This lies within the enveloped quotient, which is a dense constructible subset of a scheme that is locally of finite type, called the enveloping quotient. Ways to compactify the enveloped quotient---and the quotient of the stable locus therein---are considered. In particular, the theory of reductive envelopes from Doran and Kirwan's work is extended to the more general non-reductive setting to give ways of constructing compactifications of the enveloped quotient by using the techniques of Mumford's geometric invariant theory for reductive groups. We then look at two ways in which this non-reductive geometric invariant theory can be used in practice. Firstly, we consider a procedure for constructing quotients inductively, using the extra data of a choice of appropriate subnormal series of a group. Related to this is a method for constructing an approximation of the stable set. Secondly, we study the actions of certain extensions of unipotent groups by multiplicative groups on projective varieties with very ample linearisation. Here we identify an open subset of points that admits a geometric quotient by the action of the extended group and which is explicitly computable via Hilbert-Mumford-like criteria.