El objetivo principal de este trabajo es presentar la evolución del principio del máximo y algunas aplicaciones de él a problemas geométricos. En este sentido, estudiamos el comportamiento de la curvatura escalar S de hipersuperficies de curvatura media constante inmersas en espacios forma, bajo hipótesis de no-compacidad como: la completitud y la completitud estocástica, obteniendo una estimación óptima para el ínfimo de S. Además, estudiamos estas hipersuperficies con las condiciones de dos curvaturas principales y que verifiquen el principio del máximo de Omori-Yau, derivando una estimación óptima para el supremo de S. Por último, damos un principio débil del máximo del operador diferencial L, introducido por Cheng y Yau [19] para el estudio de hipersuperficies completas de curvatura escalar constante, y presentamos una aplicación donde se estima el ínfimo de la curvatura media de estas hipersuperficies. Los resultados de este trabajo están recogidos en los artículos [5], [6] y [7]. / The goal of this work is to show the evolution of the maximum principle and several applications of this to geometric problems. In this sense, we study the behavior of the scalar curvature S of hypersurfaces immersed with constant mean curvature into a Riemannian space form, under non-compactness’s hypotheses as: the completeness and the stochastic completeness, obtaining a sharp estimate for the infimum of S. Moreover, we study these hypersurfaces with the conditions of two principal curvatures and satisfying the Omori-Yau maximum principle, deriving a sharp estimate for the supremum of S. Finally, we establish a weak maximum principle of differential operator L, introduced by Cheng and Yau [19] for study of complete hypersurfaces with constant scalar curvature , and give an application where we estimate the infimum of the mean curvature of these hypersurfaces . The results of this work are collected in the papers [5], [6] and [7].

In this thesis we study the space of compact connected oriented genus g subsurfaces of a fixed manifold M, and in particular its homological properties. We construct a “scanning map” which compares this space to the space of sections of a certain fibre bundle over M associated to its tangent bundle, and show that this map induces an isomorphism on homology in a range of degrees. Our results are analogous to McDuff’s theorem on configuration spaces, extended from 0-submanifolds to 2-submanifolds. / En esta tesis estudiamos el espacio de subsuperficies compactas, conexas y orientadas de género g de una variedad ambiente M, y en particular sus propiedades homológicas. En particular, construimos una aplicación scanning que compara este espacio con el espacio de secciones de un cierto fibrado sobre M asociado a su fibrado tangente, y mostramos que esta aplicación induce un isomorfismo en homología en cierto rango. Nuestros resultados son análogos al teorema de McDuff sobre espacios de configuraciones, generalizados de 0-subvariedades a 2-subvariedades.

L’objectiu principal del treball és identificar la geometria que va servir per replantejar el traçat de l’amfiteatre romà de Tarragona, seguint una metodologia que es correspon amb els diferents capítols en què s’estructura la investigació: historiografia de la recerca, identificació de les estructures originals, aixecament de les restes in situ, comprovació gràfica i estadística dels traçats proposats i anàlisi constructiva. Entre les diverses opcions possibles pel replanteig de l’amfiteatre de Tarragona, creiem que la més viable és una el•lipse traçada amb un mètode discontinu. Els òvals de 8 o més centres presenten desviacions equivalents, però requereixen operacions de replanteig molt complexes per aconseguir un traçat similar al de l’el•lipse. Hem descartat completament un replanteig mitjançant un òval de 4 centres. / The main objective of this work is to identify the geometry that was used to design the layout of Tarragona’s Roman amphitheater. It follows a methodology that corresponds to the chapters in which the investigations is structured: historiography research, identification of the original structures, survey of in situ remains and, finally, graphic, statistic and constructive analysis of the proposed figures. Among the various options for the layout of Tarragona’s amphitheater, we believe that the most feasible is an ellipse drawn with a discontinuous method. Eight-centered ovals exhibit equivalent deviations, but require complex operations to achieve a layout similar to that of the ellipse. We have completely ruled out a four-centered oval layout.

We study various quotient maps between frames which are defined by stipulating that they satisfy certain conditions on the cozero parts of their domains and codomains. By way of example, we mention that C-quotient and C -quotient maps (as defined by Ball and Walters- Wayland [7]) are typical of the types of homomorphisms we consider in the initial parts of the thesis. To be little more precise, we study uplifting quotient maps, C1- and C2-quotient maps and show that these quotient maps possess some properties akin to those of a C-quotient maps. The study also focuses on R - and G - quotient maps and show, amongst other things, that these quotient maps coincide with the well known C - quotient maps in mildly normal frames. We also study quasi-F frames and give a ring-theoretic characterization that L is quasi-F precisely when the ring RL is quasi-B´ezout. We also show that quasi-F frames are preserved and reflected by dense coz-onto R -quotient maps. We characterize normality and some of its weaker forms in terms of some of these quotient maps. Normality is characterized in terms of uplifting quotient maps, -normally separated frames in terms of C1-quotient maps and mild normality in terms of R - and G -quotient maps. Finally we define cozero complemented frames and show that they are preserved and reflected by dense z#- quotient maps. We end by giving ring-theoretic characterizations of these frames. / Mathematical Science / D. Phil. (Mathematics)

Geometrical physics is a relatively young branch of applied mathematics that was initiated by the 60's and the 70's when A. Lichnerowicz, W.M. Tulczyjew and J.M. Souriau, among many others, began to study various topics in physics using methods of differential geometry. This "geometrization" provides a way to analyze the features of the physical systems from a global viewpoint, thus obtaining qualitative properties that help us in the integration of the equations that describe them. Since then, there has been a strong development in the intrinsic treatment of a variety of topics in theoretical physics, applied mathematics and control theory using methods of differential geometry. Most of the work done in geometrical physics since its first days has been devoted to study first-order theories, that is, those theories whose physical information depends on (at most) first-order derivatives of the generalized coordinates of position (velocities). However, there are theories in physics in which the physical information depends explicitly on accelerations or higher-order derivatives of the generalized coordinates of position, and thus more sophisticated geometrical tools are needed to model them acurately. In this Ph.D. Thesis we pretend to give a geometrical description of some of these higher-order theories. In particular, we focus on dynamical systems and field theories whose dynamical information can be given in terms of a Lagrangian function, or a Hamiltonian that admits Lagrangian counterpart. More precisely, we will use the Lagrangian-Hamiltonian unified approach in order to develop a geometric framework for autonomous and non-autonomous higher-order dynamical system, and for second-order field theories. This geometric framework will be used to study several relevant physical examples and applications, such as the Hamilton-Jacobi theory for higher-order mechanical systems, relativistic spin particles and deformation problems in mechanics, and the Korteweg-de Vries equation and other systems in field theory. / La física geomètrica és una branca relativament jove de la matemàtica aplicada que es va iniciar als anys 60 i 70 qua A. Lichnerowicz, W.M. Tulczyjew and J.M. Souriau, entre molts altres, van començar a estudiar diversos problemes en física usant mètodes de geometria diferencial. Aquesta "geometrització" proporciona una manera d'analitzar les característiques dels sistemes físics des d'una perspectiva global, obtenint així propietats qualitatives que faciliten la integració de les equacions que els descriuen. D'ençà s'ha produït un fort desenvolupamewnt en el tractament intrínsic d'una gran varietat de problemes en física teòrica, matemàtica aplicada i teoria de control usant mètodes de geometria diferencial. Gran part del treball realitzat en la física geomètrica des dels seus primers dies s'ha dedicat a l'estudi de teories de primer ordre, és a dir, teories tals que la informació física depèn en, com a molt, derivades de primer ordre de les coordenades de posició generalitzades (velocitats). Tanmateix, hi ha teories en física en les que la informació física depèn de manera explícita en acceleracions o derivades d'ordre superior de les coordenades de posició generalitzades, requerint, per tant, d'eines geomètriques més sofisticades per a modelar-les de manera acurada. En aquesta Tesi Doctoral ens proposem donar una descripció geomètrica d'algunes d'aquestes teories. En particular, estudiarem sistemes dinàmics i teories de camps tals que la seva informació dinàmica ve donada en termes d'una funció lagrangiana, o d'un hamiltonià que prové d'un sitema lagrangià. Per a ser més precisos emprarem la formulació unificada Lagrangiana-Hamiltoniana per tal de desenvolupar marcs geomètrics per a sistemes dinàmics d'ordre superior autònoms i no autònoms, i per a teories de camps de segon ordre. Amb aquest marc geomètric estudiarem alguns exemples físics rellevants i algunes aplicacions, com la teoria de Hamilton-Jacobi per a sistemes mecànics d'ordre superior, partícules relativístiques amb spin i problemes de deformació en mecànica, i l'equació de Korteweg-de Vries i altres sistemes en teories de camps.

This thesis investigates the topological reconstruction problem, which is inspired by the reconstruction conjecture in graph theory. We ask how much information about a topological space can be recovered from the homeomorphism types of its point-complement subspaces. If the whole space can be recovered up to homeomorphism, it is called reconstructible. In the first part of this thesis, we investigate under which conditions compact spaces are reconstructible. It is shown that a non-reconstructible compact metrizable space must contain a dense collection of 1-point components. In particular, all metrizable continua are reconstructible. On the other hand, any first-countable compactification of countably many copies of the Cantor set is non-reconstructible, and so are all compact metrizable h-homogeneous spaces with a dense collection of 1-point components. We then investigate which non-compact locally compact spaces are reconstructible. Our main technical result is a framework for the reconstruction of spaces with a maximal finite compactification. We show that Euclidean spaces ℝⁿ and all ordinals are reconstructible. In the second part, we show that it is independent of ZFC whether the Stone-Čech remainder of the integers, ω*, is reconstructible. Further, the property of being a normal space is consistently non-reconstructible. Under the Continuum Hypothesis, the compact Hausdorff space ω* has a non-normal reconstruction, namely the space ω*\{p} for a P-point p of ω*. More generally, the existence of an uncountable cardinal κ satisfying κ = κ^<κ implies that there is a normal space with a non-normal reconstruction. The final chapter discusses the Stone-Čech compactification and the Stone-Čech remainder of spaces ω*\{x}. Assuming the Continuum Hypothesis, we show that for every point x of ω*, the Stone-Čech remainder of ω*{x} is an ω₂-Parovičenko space of cardinality 2^{2^c} which admits a family of 2^c disjoint open sets. This implies that under 2^c = ω₂, the Stone-Čech remainders of ω*\{x} are all homeomorphic, regardless of which point x gets removed.

Let $S$ be a $2n$-dimensional complex manifold equipped with a line bundle with a real-analytic complex connection such that its curvature is of type $(1,1)$, and with a real analytic h-projective structure such that its h-projective curvature is of type $(1,1)$. For $n=1$ we assume that $S$ is equipped with a real-analytic M\"obius structure. Using the structure on $S$, we construct a twistor space of a quaternionic $4n$-manifold $M$. We show that $M$ can be identified locally with a neighbourhood of the zero section of the twisted (by a unitary line bundle) tangent bundle of $S$ and that $M$ admits a quaternionic $S^1$ action given by unit scalar multiplication in the fibres. We show that $S$ is a totally complex submanifold of $M$ and that a choice of a connection $D$ in the h-projective class on $S$ gives extensions of a complex structure from $S$ to $M$. For any such extension, using $D$, we construct a hyperplane distribution on $Z$ which corresponds to the unique quaternionic connection on $M$ preserving the extended complex structure. We show that, in a special case, the construction gives the Feix--Kaledin construction of hypercomplex manifolds, which includes the construction of hyperk\"ahler metrics on cotangent bundles. We also give an example in which the construction gives the quaternion-K\"ahler manifold $\mathbb{HP}^n$ which is not hyperk\"ahler. We show that the same construction and results can be obtained for $n=1$. By convention, in this case, $M$ is a self-dual conformal $4$-manifold and from Jones--Tod correspondence we know that the quotient $B$ of $M$ by an $S^1$ action is an asymptotically hyperbolic Einstein--Weyl manifold. Using a result of LeBrun \cite{Le}, we prove that $B$ is an asymptotically hyperbolic Einstein--Weyl manifold. We also give a natural construction of a minitwistor space $T$ of an asymptotically hyperbolic Einstein--Weyl manifold directly from $S$, such that $T$ is the Jones--Tod quotient of $Z$. As a consequence, we deduce that the Einstein--Weyl manifold constructed using $T$ is equipped with a distinguished Gauduchon gauge.

In this thesis we study the self-Floer theory of a monotone Lagrangian submanifold $L$ of a closed symplectic manifold $X$ in the presence of various kinds of symmetry. First we consider the group $\mathrm{Symp}(X, L)$ of symplectomorphisms of $X$ preserving $L$ setwise, and extend its action on the Oh spectral sequence to coefficients of arbitrary characteristic, working over an enriched Novikov ring. This imposes constraints on the differentials in the spectral sequence which force them to vanish in certain situations. We then specialise to the case where $L$ is $K$-homogeneous for a compact Lie group $K$, meaning roughly that $X$ is Kaehler, $K$ acts on $X$ by holomorphic automorphisms, and $L$ is a Lagrangian orbit. By studying holomorphic discs with boundary on $L$ we compute the image of low codimension $K$-invariant subvarieties of $X$ under the length zero closed-open string map. This places restrictions on the self-Floer cohomology of $L$ which generalise and refine the Auroux-Kontsevich-Seidel criterion. These often result in the need to work over fields of specific positive characteristics in order to obtain non-zero cohomology. The disc analysis is then developed further, with the introduction of the notion of poles and a reflection mechanism for completing holomorphic discs into spheres. This theory is applied to two main families of examples. The first is the collection of four Platonic Lagrangians in quasihomogeneous threefolds of $\mathrm{SL}(2, \mathbb{C})$, starting with the Chiang Lagrangian in $\mathbb{CP}^3$. These were previously studied by Evans and Lekili, who computed the self-Floer cohomology of the latter. We simplify their argument, which is based on an explicit construction of the Biran-Cornea pearl complex, and deal with the remaining three cases. The second is a family of $\mathrm{PSU}(n)$-homogeneous Lagrangians in products of projective spaces. Here the presence of both discrete and continuous symmetries leads to some unusual properties: in particular we obtain non-displaceable monotone Lagrangians which are narrow in a strong sense. We also discuss related examples including applications of Perutz's symplectic Gysin sequence and quilt functors. The thesis concludes with a discussion of directions for further research and a collection of technical appendices.

Let G be a semisimple linear algebraic group over an algebraically closed field of prime characteristic. In this thesis we outline the theory of such groups and their cohomology. We then concentrate on algebraic groups in rank 1 and 2, and prove some new results in their bounding cohomology.

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