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Displays can be thought of as relative versions of Fontaine's notion of strongly divisible lattice from integral p-adic Hodge theory. In favourable circumstances, the crystalline cohomology of a smooth projective R-scheme X is endowed with a display-structure coming from complexes associated with the relative de Rham-Witt complex of Langer-Zink, and can be thought of as a kind of mixed characteristic Hodge structure. In this thesis, we show that under certain geometric conditions, deforming X over PD-thickenings of R gives a crystal of relative displays. We then apply the crystal of relative displays to prove Grothendieck-Messing type results for the deformation theory of Calabi-Yau threefolds. We also show that primitive crystalline cohomology often carries a display-structure, and we prove a Grothendieck-Messing type result for the deformation theory of smooth cubic fourfolds in terms of the crystal of relative displays on primitive crystalline cohomology. Finally, we investigate the deformation theory of ordinary smooth cubic fourfolds in terms of the displays on the cohomology of their Fano schemes of lines.

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We develop an application of Almgren-Pitts min-max theory to the study of minimal hypersurfaces in dimension 3 ≤ m + 1 ≤ 7 as well as computing the k-width of the round 2-sphere for k = 1,...,8. We show that the space of minimal hypersurfaces is non-compact for an analytic metric of positive curvature and construct a min-max unstable closed geodesic in S^2 with multiplicity 2.

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Given a closed Riemannian n-manifold M, its shortest closed geodesic is called its systole and the length of this geodesic is denoted syst_1(M). For any ε > 0 and any n at least 2 one may construct a closed hyperbolic n-manifold M with syst_1(M) at most equal to ε. Constructions are detailed herein. The volume of M is bounded from below, by A_n/syst_1(M)^(n−2) where A_n is a positive constant depending only on n. There also exist sequences of n-manifolds M_i with syst_1(M_i) → 0 as i → ∞, such that vol(M_i) may be bounded above by a polynomial in 1/syst_1(M_i). When ε is sufficiently small, the manifold M is non-arithmetic, so that its fundamental group is an example of a non-arithmetic lattice in PO(n,1). The lattices arising from this construction are also exhibited as examples of non-coherent groups in PO(n,1). Also presented herein is an overview of existing results in this vein, alongside the prerequisite theory for the constructions given.

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It is well known that the low energy dynamics of many types of soliton can be approximated by geodesic motion on Mn, the moduli space of static n-solitons, which is usually a Kähler manifold. This thesis presents a detailed study of magnetic geodesic motion on a Kähler manifold in the case where the magnetic field 2-form is the Ricci form. This flow, which we call Ricci Magnetic Geodesic (RMG) flow, is first studied in general. A symmetry reduction result is proved which allows one to localize the flow onto the fixed point set of any group of holomorphic isometries of a Kähler manifold M. A subtlety of this reduction, which was overlooked by Krusch and Speight, is pointed out. Since RMG flow occurs at constant speed, it follows immediately that the flow is complete if M is geodesically complete. We show, by means of an explicit counterexample that, contrary to a conjecture of Krusch and Speight, the converse is false: it is possible for a geodesically incomplete manifold to be RMG complete. RMG completeness of metrically incomplete manifolds is therefore a nontrivial issue, and one which will be addressed repeatedly in later chapters. We then specialize to the case where Mn is the moduli space of abelian Higgs n-vortices, which is the context in which RMG flow was first proposed, by Collie and Tong, as a low energy model of the dynamics of a certain type of Chern-Simons n-vortices on ℝ2. The unit vortex is constructed numerically, and its asymptotics is studied. It is shown that, contrary to an assertion of Collie and Tong, RMG flow does not coincide with an earlier proposed magnetic geodesic model of vortex motion due to Kim and Lee. It is further shown that Kim and Lee’s model is ill-defined on the vortex coincidence set. An asymptotic formula for the scattering angle of well-separated vortices executing RMG flow is computed. We then change the spatial geometry, placing the vortices on the hyperbolic plane of critical curvature. An explicit formula for the two-vortex metric is derived, extending the results of Strachan, who computed the metric on a submanifold of centred 2-vortices. The RMG flow localized on this submanifold is compared with its intrinsic RMG flow, revealing strong qualitative differences. We then study the moduli space Hn,k(∑) of degree n ℂPk lumps on a compact Riemann surface ∑. It is shown that Rat1 = H1,1(S2) is RMG complete (despite being metrically incomplete). The Einstein-Hilbert action of H1,k(S2) is computed, supporting (for k > 1) a conjecture of Baptista. A natural class of topologically cylindrical submanifolds of Hn,1(∑), called dilation cylinders, is studied: their volumes are computed, and it is shown that they are all isometrically embeddable as surfaces of revolution in R3. Conditions under which they are totally geodesic, for ∑ = S2 and T2, are found, and RMG flow on some examples is studied. Finally, a new metric on Hn,1(∑), derived from the Baby-Skyrme model, is introduced. On Rat1, this metric is determined explicitly and some geometric aspects such as the volume, geodesic flow and the spectral problem with respect to this metric are studied.

This thesis is motived by the wish to understand the structure of the moduli space of monopoles on R^5. Our approach to define monopoles is twistorial and we start by developing the twistor theory of R^5, which is an analogue of the twistor theory for R^3 developed by Hitchin. Using this, we describe a Hitchin-Ward transform for R^5, giving monopoles for the group SU(2). In order for us to construct monopoles we make use of spectral curves. Then, using those spectral curves we find a new system of equations, analogue to the Nahm's equations. Lastly, we prove that the geometry of the moduli space of solutions to this Nahm's equations carries a 2-symplectic structure.