Let K be a number field and S a finite set of prime ideals of K. By a classical result of Shafarevich ([Sil86]) we know that there are finitely many isomorphism classes ƸK;S of elliptic curves de ned over K with good reduction outside S. Many people have developed methods of computing ƸK;S explicitly. At the end, all the methods ask for solutions of specific Diophantine equations in order to determine ƸK;S. In this thesis we develop a new algorithmic method of computing EK;S by solving S{unit equations. The method is implemented in the mathematical software Sage ([Dev16]) and examples are included in this thesis.

A Seifert surface for a knot in ℝ³ is a compact orientable surface whose boundary is the knot. Seifert surfaces are not unique. In 1934 Herbert Seifert provided a construction of such a surface known as the Seifert Algorithm, using the combinatorics of a projection of the knot onto a plane. This thesis presents another construction of a Seifert surface, using differential geometry and a projection of the knot onto a sphere. Given a knot K : S¹⊂ R³, we construct canonical maps F : ΛdiffS² → ℝ=4πZ and G : ℝ³ - K(S¹) → ΛdiffS² where ΛdiffS² is the space of smooth loops in S². The composite FG : ℝ³ - K(S¹) → ℝ=4πZ is a smooth map defined for each u∈2 ℝ³ - K(S¹) by integration of a 2- form over an extension D² → S² of G(u) : S1 → S². The composite FG is a surjection which is a canonical representative of the generator 1∈H¹(ℝ³- K(S¹)) = Z. FG can be defined geometrically using the solid angle. Given u ∈ ℝ³ - K(S¹), choose a Seifert surface Σu for K with u ∉ Σu. It is shown that FG(u) is equal to the signed area of the shadow of Σu on the unit sphere centred at u. With this, FG(u) can be written as a line integral over the knot. By Sard's Theorem, FG has a regular value t ∈ ℝ=4πZ. The behaviour of FG near the knot is investigated in order to show that FG is a locally trivial fibration near the knot, using detailed differential analysis. Our main result is that (FG)-¹(t)⊂ ℝ³ can be closed to a Seifert surface by adding the knot.

It is proven that a solution to the Einstein-Maxwell equations whose gravitational and electromagnetic radiation fields vanish at infinity is in fact stationary in a neighbourhood of spatial infinity. That is, if in adapted coordinates the Weyl and Faraday tensors decay suitably fast and there is an asymptotically-to-all-orders Killing vector field, then this is indeed a Killing vector field in the region outside the bifurcate horizon of a sphere of sufficiently large radius. In particular, electrovacuum time-periodic spacetimes, which are truly dynamical, do not exist. This can be interpreted as a mild form of the statement: “Gravitational waves carry energy away from an isolated system". This is an extension of earlier work by Alexakis and Schlue, and Bičák, Scholtz and Tod, to include matter/energy models, in this case electromagnetism. It is also shown that the same result holds when the Einstein's equations are coupled to a massless Klein-Gordon field.

The main focus of this thesis is to study the equation A(x4+y4+z4+w4)+Bxyzw+C(x2y2+z2w2)+D(x2z2+y2w2)+E(x2w2+y2z2) = 0. To do so, we view this equation as a family of quartic K3 surfaces in P3[x,y,z,w], parametrised by points [A,B,C,D,E] E P4. We pursue two directions. First we look at 320 conics that such a K3 surface contains. In particular, we explore the field of definition of these 320 conics and the Monodromy group linked to these conics. In the other direction we explore the quartic K3 surfaces which contain lines. We list all subfamilies of K3 surfaces for which a very general member contains 8, 16, 24, 32 or 48 lines. We combine the two directions, by using the lines and conics found, to explore the Picard group of the various families found. In particular, not only do we work out the Picard rank of a very general member of a family, but we also decompose the Picard lattice into known lattices. This thesis has a secondary focus on hyperelliptic curves of genus two with complex multiplication (CM). At the end of the thesis, we design an algorithm to find CM curves of genus two which are defined over quadratic extensions of the rationals. To do so we also develop an algorithm which makes the coefficients of a curve smaller.

A wide class of one-dimensional continuous-time discrete-space interacting particle systems are shown to be Pfaffian point processes at fixed times with kernels characterised by the solutions to associated two-dimensional ODEs. The models comprise instantaneously coalescing or annihilating random walks with fully spatially inhomogeneous jump rates and deterministic initial conditions, including additional pairwise immigration or branching in the pure interaction regimes. We formulate convergence of Pfaffian point processes via their kernels, enabling investigation of diffusive scaling limits, which boils down uniform convergence of lattice approximations to two-dimensional PDEs. Convergence to continuum point processes is developed for a subset of the discrete models. Finally, in the case of annihilating random walks with pairwise immigration we extend the picture to multiple times, establishing the extended Pfaffian property for the temporal process.

We introduce and study submanifold bridge processes. Our method involves proving a general formula for the integral over a submanifold of the minimal heat kernel on a complete Riemannian manifold. Our formula expresses this object in terms of a stochastic process whose trajectories terminate on the submanifold at a fixed positive time. We study this process and use the formula to derive lower bounds, an asymptotic relation and derivative estimates. Using these results we introduce and characterize Brownian bridges to submanifolds. Before doing so we prove necessary estimates on the Laplacian of the distance function and define a notion of local time on a hypersurface. These preliminary developments also lead to a study of the distance between Brownian motion and a submanifold, in which we prove exponential bounds and concentration inequalities. This work is motivated by the desire to extend the analysis of path and loop space to measures on paths which terminate on a submanifold.

This work is devoted to investigating the behaviour of invariant algebraic curves for the two dimensional Lotka-Volterra systems and examining almost a geometrical approach for finding invariant algebraic surfaces in three dimensional Lotka-Volterra systems. We consider the twenty three cases of invariant algebraic curves found in Ollagnier (2001) of the two dimensional Lotka-Volterra system in the complex plane and then we explain the geometric nature of each curve, especially at the critical points of the mentioned system. We also investigate the local integrability of two dimensional Lotka-Volterra systems at its critical points using the monodromy method which we extend to use the behaviour of some of the invariant algebraic curves mentioned above. Finally, we investigate invariant algebraic surfaces in three dimensional Lotka- Volterra systems by a geometrical method related with the intersection multiplicity of algebraic surfaces with the axes including the lines at infinity. We will classify both linear and quadratic invariant algebraic surfaces under some assumptions and commence a study of the cubic surfaces.

This thesis is concerned with the spectral theory of the Laplacian on non-Euclidean spaces and its intimate links with harmonic analysis and the theory of special functions. More specifically, it studies the spectral theory of the Laplacian on the quotients M = Γ\G/K and X = G/K, where G is a connected semisimple Lie group, K is a maximal compact subgroup of G and Γ is a discrete subgroup of G.

In this thesis we present a range of different knot theories and then generalise them. Working with this, we focus on biquandles with linear and quadratic biquandle functions (in the quadratic case we restrict ourselves to functions with commutative coefficients). In particular, we show that if a biquandle is commutative, the biquandle function must have non-commutative coefficients, which ties in with the Alexander biquandle in the linear case. We then describe some computational work used to calculate rack and birack homology.

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