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In this thesis an extension of the classical intersection cohomology of Goresky and MacPherson, which we call multiperverse cohomology, is defined for a certain class of depth 1 controlled stratified spaces, which we call multicontrolled stratified spaces. These spaces are spaces with singularities -- this being their controlled structure -- with additional multicontrol data. Multiperverse cohomology is constructed using a cochain complex of tau-multiperverse forms, defined for each case tau of a parameter called a multiperversity. For the spaces that we consider these multiperversities, forming a lattice M, extend the general perversities of intersection cohomology. Multicontrolled stratified spaces generalise the structure of (the compactifications of) Q-rank 1 locally symmetric spaces. In this setting multiperverse cohomology generalises some of the aspects of the weighted cohomology of Harder, Goresky and MacPherson. We define two special cases of multicontrolled stratified spaces: the product-type case, and the flat-type case. In these cases we can calculate the multiperverse cohomology directly for cones and cylinders, this yielding the local calculation at a singular stratum of a multicontrolled space. Further, we obtain extensions of the usual Mayer-Vietoris sequences, as well as a partial Kunneth Theorem. Using the concept a dual multiperversity we are able to obtain a version of Poincare duality for multiperverse cohomology for both the flat-type and the product-type case. For this Poincare duality there exist self-dual multiperversities in certain cases, such as for non-Witt spaces, where there are no self-dual perversities. For certain cusps, called double-product cusps, which are naturally compactified to multicontrolled spaces, the multiperverse cohomology of the compactification of the double-product cusp for a certain multiperversity is equal to the L2-cohomology, analytically defined, for certain doubly-warped metrics.

This work constructs and compares different kinds of categorified cohomology of a locally contractible topological space X. Fix a commutative ring k of characteristic 0 and also denote by k the differential graded category with a single object and endomorphisms k. In the Morita model structure k is weakly equivalent to the category of perfect chain complexes over k. We define and compute derived global sections of the constant presheaf k considered as a presheaf of dg-categories with the Morita model structure. If k is a field this is done by showing there exists a suitable local model structure on presheaves of dg-categories and explicitly sheafifying constant presheaves. We call this categorified Cech cohomology Morita cohomology and show that it can be computed as a homotopy limit over a good (hyper)cover of the space X. We then prove a strictification result for dg-categories and deduce that under mild assumptions on X Morita cohomology is equivalent to the category of homotopy locally constant sheaves of k-complexes on X. We also show categorified Cech cohomology is equivalent to a category of ∞-local systems, which can be interpreted as categorified singular cohomology. We define this category in terms of the cotensor action of simplicial sets on the category of dg-categories. We then show ∞-local systems are equivalent to the category of dg-representations of chains on the loop space of X and find an explicit method of computation if X is a CW complex. We conclude with a number of examples.

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This thesis is an investigation of the difference between metabolic and hyperbolic objects in a variety of settings and how they interact with cobordism and 'double cobordism', both in the setting of algebraic L-theory and in the context of knot theory. Let A be a commutative Noetherian ring with involution and S be a multiplicative subset. The Witt group of linking forms W(A,S) is defined by setting metabolic linking forms to be 0. This group is well-known for many localisations (A,S) and it is a classical fact that it forms part of a localisation exact sequence, essential to many Witt group calculations. However, much of the deeper 'signature' information of a linking form is invisible in the Witt group. The beginning of the thesis comprises the first general definition and careful investigation of the double Witt group of linking forms DW(A,S), given by the finer equivalence relation of setting hyperbolic linking forms to be 0. The treatment will include invariants, structure theorems and localisation exact sequences for various types of rings and localisations. We also make clear the relationship between the double Witt groups of linking forms over a Laurent polynomial ring and the double Witt group of those forms over the ground ring that are equipped with an automorphism. In particular we prove the isomorphism between the double Witt group of Blanchfield forms and the double Witt group of Seifert forms. In the main innovation of the thesis, we next define chain complex generalisations of the double Witt groups which we call the double L-groups DLn(A,S). In double L-theory, the underlying objects are the symmetric chain complexes of algebraic L-theory but the equivalence relation is now the finer relation of double algebraic cobordism. In the main technical result of the thesis we solve an outstanding problem in this area by deriving a double L-theory localisation exact sequence. This sequence relates the DL-groups of a localisation to both the free L-groups of A and a new group analogous to a 'double' algebraic homology surgery obstruction group of chain complexes over the localisation. We investigate the periodicity of the double L-groups via skew-suspension and surgery 'above and below the middle dimension'. We then reconcile the double L-groups with the double Witt groups, so that we also prove a double Witt group localisation exact sequence. Finally, in a topological application of double Witt and double L-groups, we apply our results to the study of doubly-slice knots. A doubly-slice knot is a knot that is the intersection of an unknotted sphere and a plane. We show that the double knot-cobordism group has a well-defined map to the DL-group of Blanchfield complexes and easily reprove some classical results in this area using our new methods.

Homotopy type theory (HoTT) is a branch of mathematics that combines and benefits from a variety of fields, most importantly homotopy theory, higher dimensional category theory, and, of course, type theory. We present several original results in homotopy type theory which are related to the truncation level of types, a concept due to Voevodsky. To begin, we give a few simple criteria for determining whether a type is 0-truncated (a set), inspired by a well-known theorem by Hedberg, and these criteria are then generalised to arbitrary n. This naturally leads to a discussion of functions that are weakly constant, i.e. map any two inputs to equal outputs. A weakly constant function does in general not factor through the propositional truncation of its domain, something that one could expect if the function really did not depend on its input. However, the factorisation is always possible for weakly constant endofunctions, which makes it possible to define a propositional notion of anonymous existence. We additionally find a few other non-trivial special cases in which the factorisation works. Further, we present a couple of constructions which are only possible with the judgmental computation rule for the truncation. Among these is an invertibility puzzle that seemingly inverts the canonical map from Nat to the truncation of Nat, which is perhaps surprising as the latter type is equivalent to the unit type. A further result is the construction of strict n-types in Martin-Lof type theory with a hierarchy of univalent universes (and without higher inductive types), and a proof that the universe U(n) is not n-truncated. This solves a hitherto open problem of the 2012/13 special year program on Univalent Foundations at the Institute for Advanced Study (Princeton). The main result of this thesis is a generalised universal property of the propositional truncation, using a construction of coherently constant functions. We show that the type of such coherently constant functions between types A and B, which can be seen as the type of natural transformations between two diagrams over the simplex category without degeneracies (i.e. finite non-empty sets and strictly increasing functions), is equivalent to the type of functions with the truncation of A as domain and B as codomain. In the general case, the definition of natural transformations between such diagrams requires an infinite tower of conditions, which exists if the type theory has Reedy limits of diagrams over the ordinal omega. If B is an n-type for some given finite n, (non-trivial) Reedy limits are unnecessary, allowing us to construct functions from the truncation of A to B in homotopy type theory without further assumptions. To obtain these results, we develop some theory on equality diagrams, especially equality semi-simplicial types. In particular, we show that the semi-simplicial equality type over any type satisfies the Kan condition, which can be seen as the simplicial version of the fundamental result by Lumsdaine, and by van den Berg and Garner, that types are weak omega-groupoids. Finally, we present some results related to formalisations of infinite structures that seem to be impossible to express internally. To give an example, we show how the simplex category can be implemented so that the categorical laws hold strictly. In the presence of very dependent types, we speculate that this makes the Reedy approach for the famous open problem of defining semi-simplicial types work.

John Nash probó la existencia de inmersión e inmersión homeomorfa inyectiva isométricas de cualquier variedad Riemanniana en un espacio euclídeo de dimensión suficiente. Desde entonces el estudio de las inmersiones isométricas se ha orientado principalmente en cuatro direcciones:a) Encontrar propiedades propias de la variedad y de la inmersión que permitan establecer acotaciones o reducir la dimensión del espacio euclídeo ambiente.b) Estudio de la rigidez de las inmersiones isométricas.c) Estudio de las restricciones que puede dar el grupo o el álgebra de holonomía de la variedad ,a las inmersiones de ésta.d) Estudio de las inmersiones homeomorfas isométricas equivariantes, o sea aquellas para las que el grupo de isometrías de la variedad se incluye en el del espacio euclideo ambiente.En líneas generales, el objetivo de esta tesis consiste en aplicar el hecho ya conocido de que se puede efectuar el estudio de las inmersiones isométricas de variedades Riemannianas en espacios euclídeos, a partir del sistema de tensores obtenido por medio del fibrado normal a la variedad inmersa, a los temas siguientes:- Inmersiones en codimensión dos, con curvatura normal cero y álgebra local de holonomía no total.- Influencia del álgebra de Lie de las isometrias infinitesimales de la variedad sobre las inmersiones en codimensión dos con curvatura normal cero.- Reducción de la codimensión.- Estudio de la rigidez de la inmersión por medio de las isometrías infinitesimales de la variedad y su relación con el fibrado normal de la variedad.La estructura de la tesis consta de los siguientes capítulos:Capítulo 0.- Se da una demostración de la generalización del teorema de Bonet para inmersiones en codimensión cualquiera.Capítulo I.- Teniendo cuenta lo estudiado por Bishop, Alexander, Moore y Alexander-Maltz sobre las inmersiones isométricas para variedades producto, en codimensión igual al número de componentes de la variedad, viendo que la inmersión se puede descomponer en producto de inmersiones en codimensión uno, estos resultados nos han sugerido el estudio de las inmersiones isométricas en codimensión dos,con curvatura normal cero.Capítulo II.- Erbacher ha probado la posibilidad de reducir la codimensión de inmersiones isométricas, utilizando el paralelismo del primer espacio normal. Utilizando técnicas deducidas del capítulo cero ,en hipótesis como las de Erbacher y algunas más probamos con gran facilidad la posibilidad de reducir la codimensión. Por último damos condiciones suficientes para reducir la codimensión, utilizando los sucesivos espacios normales de la inmersión.Capítulo III.- En este capitulo se estudia la influencia del álgebra de Lie de isometrías infinitesimales de la variedad inmersa en la rigidez de dichas inmersiones. Dicha influencia puede observarse en Goldstein-Ryan al estudiar las deformaciones infinitesimales de las esferas. En concreto, nosotros estudiamos el concepto clásico de rigidez (dos inmersiones isométricas de una misma variedad son mutuamente rígidas cuando difieren en una isometría del espacio euclídeo ambiente).Capítulo IV.- En este capítulo se demuestra para variedades de dimensión dos y tres la posibilidad de obtener inmersiones isométricas, a partir de las isometrías infinitesimales de la variedad, de forma que dichas isometrías infinitesimales sean restricción de isometrías infinitesimales del espacio euclídeo ambiente.

Esta memoria pretende contribuir al estudio de las singularidades de los gérmenes de curva alabeada en los tres aspectos siguientes:(A) Propiedades de los gérmenes de curva que quedan determinadas por una de sus truncaciones.(B) Existencia de esquemas que parametrizan gérmenes de curva, o truncaciones de gérmenes, con ciertos invariantes prefijados.(C) Número de ecuaciones requeridas por un germen de curva y su relación con las propiedades del germen.

This thesis contains new results about the representability of cohomological functors defined on a subcategory of compact objects (with respect to a fixed cardinal) of a well generated triangulated category. Classical theorems of Adams for the stable homotopy category and Neeman for compactly generated triangulated categories are extended to the first uncountable cardinal. The case of derived categories of rings and the stable motivic category are studied in detail. These results contribute to answering negatively a question raised by Rosický of whether all cohomological functors defined on a subcategory of compact objects with respect to a large enough cardinal are representable. Some of the findings in this thesis are based on new results about abelian categories, the most relevant being a generalization of the Auslander Lemma for non Grothendieck categories. / TESI "Representabilitat d'Adams en categories triangulades"TEXT:En aquesta tesi s'obtenen resultats nous sobre la representabilitat de functors cohomològics definits en subcategories d'objectes compactes (respecte a un cardinal fixat) d'una categoria triangulada ben generada. S'estenen al primer cardinal no numerable teoremes antics d'Adams per a la categoria d'homotopia estable i de Neeman per a categories compactament generades. S'estudien en detall els casos de la categoria derivada d'un anell i la categoria motívica estable. Aquests resultats contribueixen a respondre negativament una pregunta de Rosický sobre si tots els functors cohomològics definits en una subcategoria d'objectes compactes respecte a un cardinal suficientment gran són representables. Alguns dels avenços d'aquesta tesi es basen en nous resultats sobre categories abelianes, el més rellevant dels quals és una generalització del lema d'Auslander per a categories que no són de Grothendieck.

A lo largo de esta tesis estudiamos la geometría global de las superficies espaciales, y maximales en particular, en espacios producto lorentzianos. En primer lugar generalizamos el teorema de Calabi-Bernstein al caso de superficies maximales en un producto lorentziano. También estudiamos algunos problemas locales, que a posteriori tendrán importantes repercusiones globales. Los producto lorentzianos forman parte de la familia de los espacios de Robertson-Walker generalizados, al igual que los espacios tipo steady state. Las superficies equivalentes a las superficies maximales en un espacio tipo steady state son las superficies espaciales con H=1. En este contexto damos un resultado de unicidad para superficies espaciales completas con curvatura media constante acotadas del infinito en un espacio tipo steady state. Por último consideramos superficies espaciales con curvatura de Gauss constante en espacios producto, tanto lorentzianos como riemannianos. En este caso obtenemos algunos resultados de tipo Calabi-Bernstein cuando M es la esfera S2. / Along this PhD thesis we study the global geometry of spacelike surfaces, and in particular maximal surfaces, in Lorentzian product spaces. Firstly, we generalize the Calabi-Bernstien theorem when considering maximal surfaces in a Lorentzian product. We also study some local problems, which a posteriori will have important global consequences. The Lorentzian products are part of the family of the generalized Robertson-Walker spaces. Also the steady state type spaces form a subfamily of such spaces. The equivalent surfaces to the maximal ones in a steady state type space are the spacelike surfaces with H=1. In this context, we give a uniqueness result for complete spacelike surfaces with constant mean curvature bounded from the infinity of a steady state type space. Finally, we consider spacelike surfaces with constant Gaussian curvature in Riemannian and Lorentzian product spaces. In this case, we obtain some Calabi-Bernstein type results when M is the sphere S2