The purpose of this thesis is to study link cobordisms. The main tool we use to do so is given by Juhász’s cobordism maps, although we sometimes make use of a recent construction by Zemke. Given a decorated link cobordism, that is, a link cobordism with additional structure, there is a map induced at the level of link Floer homology (HFL, also denoted HFK for knots). When X is a decorated concordance from K_0 to K_1, we prove that the cobordism map F preserves the natural bigrading of HFK, and that there is a morphism of spectral sequences from HFK(K_i) to HF(S^3) which agrees with F on the first page, and with the identity of HF(S^3) on the infinity page. We use this result to obstruct the existence of invertible concordances between given knots, and to define a non-vanishing element of HFK(K) associated to a slice disc for the knot K. We then give a full description of the maps induced by elementary decorated link cobordisms, which generate all decorated link cobordisms. In particular, we relate the map associated to a saddle cobordism to a skein exact triangle which generalises Ozsváth and Szabó’s oriented skein exact triangle in HFL to decorated links. Lastly, we use this description to prove that the TQFT defined by HFL on a category of unlinks and cobordisms between them agrees with Khovanov’s reduced TQFT.

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Topology optimization is a tool for finding a domain in which material is placed that optimizes a certain objective function subject to constraints. This thesis considers topology optimization for structural mechanics problems, where the underlying PDE is derived from linear elasticity. There are two main approaches for solving topology optimization: Solid Isotropic Material with Penalisation (SIMP) and Evolutionary Structural Optimization (ESO). SIMP is a continuous relaxation of the problem solved using a mathematical programming technique and so inherits the convergence properties of the optimization method. By contrast, ESO is based on engineering heuristics and has no proof of optimality. This thesis considers the formulation of the SIMP method as a mathematical optimization problem. Including the linear elasticity state equations is considered and found to be substantially less reliable and less efficient than excluding them from the formulation and solving the state equations separately. The convergence of the SIMP method under a regularising filter is investigated and shown to impede convergence. A robust criterion to stop filtering is proposed and demonstrated to work well in high-resolution problems (O(10^6)). The ESO method is investigated to fully explain its non-monotonic convergence behaviour. Through a series of analytic examples, the steps taken by the ESO algorithm are shown to differ arbitrarily from a linear approximation. It is this difference between the linear approximation and the actual value taken which causes ESO to occasionally take non-descent steps. A mesh refinement technique has been introduced with the sole intention of reducing the ESO step size and thereby ensuring descent of the algorithm. This is shown to work on numerous examples. Extending the classical topology optimization problem to included a global buckling constraint is considered. This poses multiple computational challenges, including the introduction of numerically driven spurious localised buckling modes and ill-defined gradients in the case of non-simple eigenvalues. To counter such issues that arise in a continuous relaxation approach, a method for solving the problem that enforces the binary constraints is proposed. The method is designed specifically to reduce the number of derivative calculations made, which is by far the most computationally expensive step in optimization involving buckling. This method is tested on multiple problems and shown to work on problems of size O(10^5).

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En esta tesis se desarrolla un sistema de tratamiento de informaciones geométricas con incertidumbre, basado en la propagación y fusión de regiones elipsoidales. Se adopta un modelo de incertidumbre acotada en el espacio de parámetros por conjuntos elipsoidales. Estos conjuntos pueden ser degenerados o no, permitiendo representar tanto informaciones parciales como completas de los elementos observados.Se ha obtenido una formula para calcular la fusión de dos elipsoides. Es decir, que dadas varias observaciones de un mismo elemento, cada una con su región elipsoidal de incertidumbre, la operación de fusión calcula la menor cota elipsoidal.La propagación de elipsoides se hace a través de las ecuaciones correspondientes a las relaciones geométricas entre los elementos del entorno. Se ha obtenido una formula que generaliza la propagación de elipsoides a los casos degenerados usando matrices pseudoinversas y permitiendo así también la propagación de informaciones parciales.Se presenta un algoritmo para la actualización global de las informaciones que se basa en la fusión y la propagación. Se implementa sobre un grafo cuyos nodos representan los elementos del entorno y cuyos arcos representan las relaciones entre ellos.Finalmente se expone, como ejemplo de aplicación del sistema desarrollado, un algoritmo para la imposición de condiciones de consistencia en dibujos poligonales con posición inserta de los vértices, para ser proyecciones bidimensionales de poliedros. Las condiciones de consistencia se definen de forma natural en el grafo y la imposición se hace con el algoritmo de propagación global.

The aim of this thesis is to study dynamical properties of pseudo -Anosov braids on the n-times punctured disk Dn making use of a particular coordinate system called the Dynnikov coordinate system. The Dynnikov coordinate system gives a homeomorphism from the space of measured foliations MFn on Dn (up to a certain equivalence relation) to Sn = R2n-4\ {O}, and restricts to a bijection from the set of integral laminations (disjoint unions of finitely many essential simple closed curves) on Dn to Cn = Z2n-4 \ {O}. In the first part of the thesis, we introduce a new method for computing the topological entropy of each member of an infinite family of pseudo -Anosov braids making use of Dynnikov's coordinates. The method is developed using the results in Thurston's seminal paper on the geometry and dynamics of surface automorphisms and builds on, more recent work of Moussafir. To be more spe- cific, the method gives a so-called Dynnikov matrix which describes the action of a given pseudo-Anosov braid B near its invariant unstable measured foliation [F, u] on the projective space PSn, and the eigenvalue \ > 1 of this matrix gives the topological entropy of B. In the second part of the thesis, we compare the spectra of Dynnikov matrices with the spectra of the train track transition matrices of a given pseudo-Anosov braid, and show that these matrices are isospectral up to roots of unity and zeros under some particular conditions.

This thesis presents some new results on Dehn surgery. The overarching theme of the thesis is to find restrictions on obtaining a 3-manifold by a Dehn surgery on a knot in another 3-manifold (although we also find new examples in chapter 5) and most of these restrictions are obtained by exploring the consequences of the mapping cone formula in Heegaard Floer homology. In particular, we show that only finitely many alternating knots can yield a given 3-manifold by Dehn surgery and confirm the knot complement conjecture for many classes of knots.

In this thesis we generalise three theorems from the literature on Heegaard Floer homology and Dehn surgery: one by Ozsv ́ath and Szab ́o on deficiency symmetries in half-integral L -space surgeries, and two by Greene which use Donaldson’s diagonali- sation theorem as an obstruction to integral and half-integral L -space surgeries. Our generalisation is two-fold: first, we eliminate the L -space conditions, opening these techniques up for use with much more general 3-manifolds, and second, we unify the integral and half-integral surgery results into a broader theorem applicable to non- zero rational surgeries in S 3 which bound sharp, simply connected, negative-definite smooth 4-manifolds. Such 3-manifolds are quite common and include, for example, a huge number of Seifert fibred spaces. Over the course of the first three chapters, we begin by introducing background material on knots in 3-manifolds, the intersection form of a simply connected 4- manifold, Spin- and Spin c -structures on 3- and 4-manifolds, and Heegaard Floer ho- mology (including knot Floer homology). While none of the results in these chapters are original, all of them are necessary to make sense of what follows. In Chapter 4, we introduce and prove our main theorems, using arguments that are predominantly algebraic or combinatorial in nature. We then apply these new theorems to the study of unknotting number in Chapter 5, making considerable headway into the extremely difficult problem of classifying the 3-strand pretzel knots with unknotting number one. Finally, in Chapter 6, we present further applications of the main theorems, ranging from a plan of attack on the famous Seifert fibred space realisation problem to more biologically motivated problems concerning rational tangle replacement. An appendix on the implications of our theorems for DNA topology is provided at the end.