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Las t-estructuras en categorías trianguladas fueron introducidas a principios de los ochenta del último siglo por Beilinson, Bernstein y Deligne [BBD], en su estudio de los haces perversos sobre una variedad analítica o algebraica. El descubrimiento fundamental de este concepto era la existencia de una categoría abeliana “escondida”, llamada por ellos el corazón de la t-estructura, que permitía el desarrollo de una teoría de homología intrínseca dentro de la propia categoría triangulada en cuestión. Surge de manera natural las siguientes cuestiones: 1. ¿Cuándo el corazón de una t-estructura es una categoría de Grothendieck?. 2. ¿Cuándo es él una categoría de módulos?. Lo inabordable de la pregunta, ha hecho que sólo se estudien casos particulares de la misma, estableciendo condiciones en la t-estructura así como también en la categoría triangulada en cuestión. De hecho, todos los trabajos que conocemos en está dirección, están concentrados en la llamada t-estructura de Happel-Reiten-Smalo (ver [CGM], [CG], [CMT], [HKM] y [MT]). En esta tesis, se abordaron las cuestiones 1 y 2, para la t-estructura de Happel-Reiten-Smalo, solventado algunos casos que no fueron cubiertos en los trabajos [CGM], [CG], [CMT], [HKM] y [MT]. Por otra parte, una de las novedades de esta tesis, fue el estudiar las cuestiones 1 y 2, para t-estructuras más generales que el caso de Happel-Reiten-Smalo. En el capítulo 5 se estudia el corazón de las t-estructuras compactamente generadas en la categoría derivada de un anillo conmutativo Noetheriano. A continuación, daremos una lista de los resultados más relevantes de esta tesis. Resultados Cuestión 1, para la t-estructura de Happel-Reiten-Smalo: En este caso, fijamos una categoría de Grothendieck G y un par de torsión t = (T,F) en G y denotaremos por Ht el corazón de la t-estructura asociada en D(G). En primera instancia se mostró que Ht es una categoría abeliana AB5 si, y sólo si, los funtores de homologías Hk: Ht → G, conmutan con límites directos, para todo entero k. También probamos que si Ht es una categoría de Grothendieck entonces F es cerrada para límites directos en G. Como una consecuencia de nuestros resultados, para los pares de torsión inclinantes y coinclinantes, se logro dar resultados más allá de la condición de Grothendieck, generalizando algunos resultados de [CMT] y [BK]. Cuestión 2, para la t-estructura de Happel-Reiten-Smalo: En este caso, el par de torsión t = (T,F) se fija en la categoría de módulos R-Mod sobre un anillo asociativo con unidad R. En el capítulo 4, se da respuesta definitiva a esta cuestión, en términos de un progenerador de Ht. Aprovechando dicho resultado, en el caso de pares de torsión introducidos por Hoshino, Kato y Miyachi, llamados pares de torsión HKM en lo que sigue, establecimos la relación precisa entre un complejo HKM que define el par de torsión y progenerador de Ht. Como consecuencia, se muestra un ejemplo de un complejo HKM que no está en el corazón y otro ejemplo de un par de torsión que no es un par de torsión HKM, cuyo corazón es una categoría de módulos. Por otra parte, para los pares de torsión hereditarios las condiciones que deben exigirse a un complejo para ser un progenerador de Ht se simplifican, surgiendo de manera natural las ternas TTF(=torsion torsionfree). En el caso en que suponemos que t = (T,F) es la parte derecha de una terna TTF en R-Mod, bajo unas hipótesis suficientemente generales, las condiciones a exigir al complejo, quedan reducidas. Otra pregunta natural que surge es la de encontrar un progenerador de Ht que sea lo más sencillo posible. En la tesis se estudiamos cuándo dicho progenerador puede ser elegido de manera que sea una suma directa de tallos. En el caso de un solo tallo, se logra dar un ejemplo de un par de torsión no inclinante cuyo corazón es una categoría de módulos que admite un progenerador de la forma V[0] para algún módulo V en T. Cuestiones 1 y 2, para las t-estructuras compactamente generadas en la categoría derivada de un anillo conmutativo Noetheriano: Alonso, Jeremías y Saorín [AJS], clasifican las t-estructuras compactamente generada en D(R), donde R es un anillo conmutativo Noetheriano, en términos de filtraciones por soportes del espectro de R. Denotaremos por Ф tal filtración, y por HФ el corazón de la t-estructura asociada en D(R). Primero probamos que HФ siempre tiene un generador, así la cuestión 1 se reduce a determinar cuándo dicho corazón es una categoría abeliana AB5. Luego probamos que si Ф es una filtración acotada por la izquierda, entonces HФ es AB5 y por lo tanto, es una categoría de Grothendieck. A diferencia de la cuestión 1, la cuestión 2 es totalmente cubierta en la tesis. En esta repsuesta, la categoría cociente de R-Mod por una clase de torsión hereditaria juega un papel importante. Referencias [AJS] L. Alonso, A. Jeremías, M. Saorín, Compactly generated t-structures on the derived category of a Noetherian ring, Journal of Algebra, 324 (2010), 313-346. [BBD] A. Beilinson, J. Bernstein, P. Deligne, “Faisceaux Pervers”. Analysis and topology on singulas spaces, I, Luminy 1981, Astèrisque. 100. Soc. Math. France, Paris. (1982), 5-171. [BK] A.B. Buan, H. Krause, Cotilting modules over tame hereditary algebras. Pacific J. Math 211(1)(2003), 41-59. [CG] R. Colpi, E. Gregorio, The Heart of cotilting theory pair is a Grothendieck category, Preprint. [CGM] R. Colpi, E. Gregorio, F. Mantese, On the Heart of a faithful torsion theory, Journal of Algebra, 307 (2007), 841-863 [CMT] R. Colpi, F. Mantese, A. Tonolo, When the heart of a faithful torsion pair is a module category, Journal of Pure and Applied Algebra, 215 (2011) 2923-2936. [HRS] D. Happel, I. Reiten, S.O. Smalo, Tilting in abelian categories and quasitilted algebras, Mem. Amer. Math. Soc. 120 (1996). [HKM] M. Hoshino, Y. Kato, J-I. Miyachi, On t-structures and torsion theories induced by compact objects, Journal of Pure and Applied Algebra, 167 (2002), 15-35. [MT] F. Mantese, A. Tonolo, On the heart associated with a torsion pair, Topology and its Applications, 159 (2012), 2483-2489. / T-structures on triangulated categories were introduced in the early eighties of last century by Beilinson, Bernstein and Deligne [BBD] in their study of perverse sheaves on an analytic or an algebraic variety. The main discovery of this concept was the existence of a 'hidden' abelian category, called by them the heart of the t-structure, which allowed the development of a homology theory that is intrinsic to the triangulated category. In a natural way the following questions arise: 1. When is the heart of a t-structure a Grothendieck category? 2. When is it a category of modules? The intractability of the questions has led to study only particular cases of them, by establishing conditions on the t-structure as well as on the triangulated category in question. In fact, all the works that we know of in this respect are focused on the so-called t-structure of Happel-Reiten-Smalo (see [CGM], [CG], [CMT], [HKM] and [MT]). In this thesis, we tackled questions 1 and 2 above for the t-structure of Happel-Reiten-Smalo and solved some cases that were not covered in the work [CGM], [CG], [CMT], [HKM] and [MT]. On the other hand, one of the novelties of the thesis was to study questions 1 and 2, for t-structures more general than the Happel-Reiten-Smalo case. In chapter 5 we study the heart of compactly generated t-structures in the derived category of a commutative Noetherian ring. In the sequel we give a list of the most relevant results in the thesis. Results Question 1, for the t-structure of Happel-Reiten-Smalo: In this case, we fix a Grothendieck category G and a torsion pair t=(T,F) in G and we will denote by Ht the heart of the associated t-structure in D(G). First, we proved that Ht is an AB5 abelian category if, and only if, the homology functors Hk: Ht → G commute with direct limits, for each integer k. We also proved that if Ht is a Grothendieck category then F is closed under taking direct limits in G. As a consequence of our results, for tilting and cotilting torsion pairs we managed to give results further than the Grothendieck case, generalizing results from [CMT] and [BK]. Question 2, for the t-structure of Happel-Reiten-Smalo: In this case, the torsion pair t=(T,F) is fixed in the module category R-Mod over an associative ring with unit R. In chapter 4 a definitive answer to the question is given, in terms of a progenerator of Ht. Taking advantage of this result, in the case of the torsion pairs introduced by Hoshino, Kato and Miyachi, called HKM torsion pairs in the sequel, we established the precise relationship between an HKM complex which defines the torsion pair and the progenerator of Ht. As a consequence, we show an example of an HKM complex which is not in the heart and another example of a non-HKM torsion pair whose heart is a module category. On the other hand, for hereditary torsion pairs, the conditions to impose to a complex in order for it to be a progenerator of Ht get simplified, appearing in a natural way the TTF (=torsion torsionfree) triples. When we assume that t=(T,F) is the right constituent pair of a TTF triple in R-Mod, under sufficiently general hypotheses, the conditions to impose to the complex get reduced. Another natural by-side question which arises is that of finding a progenerator of Ht which is the simplest possible. In the thesis we study when such a progenerator can be chosen to be a direct sum of stalk complexes. In the case of a unique stalk complexes, we manage to give an example of a non-tilting torsion pair whose heart is a module category which admits a progenerator of the form V[0], for some module V in T. Questions 1 y 2, for compactly genrated t-structures in the derived category of a commutative Noetherian ring: Alonso, Jeremías and Saorín [AJS] classify the compactly generated t-structres in D(R), where R is a commutative Noetherian ring, in terms of filtrations by supports of the spectrum of R. We will denote by Φ such a filtration and by HΦ the heart of the associated t-structure in D(R). We first proved that HΦ always has a generator, so that question 1 reduces to determine when this heart is an AB5 abelian category. We then proved that if Φ is a left bounded filtration, then HΦ is AB5 and, hence, a Grothendieck category. Unlike question 1, question 2 has been completely answered in the thesis. In this answer the quotient category of R-Mod by a hereditary torsion class plays a very important role. References [AJS] L. Alonso, A. Jeremías, M. Saorín, Compactly generated t-structures on the derived category of a Noetherian ring, Journal of Algebra, 324 (2010), 313-346. [BBD] A. Beilinson, J. Bernstein, P. Deligne, “Faisceaux Pervers”. Analysis and topology on singulas spaces, I, Luminy 1981, Astèrisque. 100. Soc. Math. France, Paris. (1982), 5-171. [BK] A.B. Buan, H. Krause, Cotilting modules over tame hereditary algebras. Pacific J. Math 211(1)(2003), 41-59. [CG] R. Colpi, E. Gregorio, The Heart of cotilting theory pair is a Grothendieck category, Preprint. [CGM] R. Colpi, E. Gregorio, F. Mantese, On the Heart of a faithful torsion theory, Journal of Algebra, 307 (2007), 841-863 [CMT] R. Colpi, F. Mantese, A. Tonolo, When the heart of a faithful torsion pair is a module category, Journal of Pure and Applied Algebra, 215 (2011) 2923-2936. [HRS] D. Happel, I. Reiten, S.O. Smalo, Tilting in abelian categories and quasitilted algebras, Mem. Amer. Math. Soc. 120 (1996). [HKM] M. Hoshino, Y. Kato, J-I. Miyachi, On t-structures and torsion theories induced by compact objects, Journal of Pure and Applied Algebra, 167 (2002), 15-35. [MT] F. Mantese, A. Tonolo, On the heart associated with a torsion pair, Topology and its Applications, 159 (2012), 2483-2489.

This work considers the problem of Equations Over Groups and settles the KL- conjecture for equations of length five. Firstly, the problem of equations over groups is stated and discussed and the results, which were up to now obtained, are presented. Then, by way of contradiction, it is assumed that for the remaining cases of equations of length five a solution does not exist. The methodology adopted uses the combinatorial and topological arguments of relative diagrams. If D is a relative diagram representing the counter example, all types of interior regions of positive curvature are listed for each type of equation of length five. For each interior region of positive curvature, one region of negative curvature is found and the positive curvature is added to it to obtain the total curvature in the interior of diagram D. In the final chapter the curvature of the interior of D is added to the curvature of the boundary regions to obtain the total curvature of the diagram. It is proved that the total curvature of 4pi cannot be achieved, our desired contradiction, and therefore equations of length five have a solution.

In this thesis, we study a non-analytic growth bound $\zeta(f)$ associated with an exponentially bounded measurable function $f: \mathbb{R}_{+} \to \mathbf{X},$ which measures the extent to which $f$ can be approximated by holomorphic functions. This growth bound is related to the location of the domain of holomorphy of the Laplace transform of $f$ far from the real axis. We study the properties of $\zeta(f)$ as well as two associated abscissas, namely the non-analytic abscissa of convergence, $\zeta_{1}(f)$ and the non-analytic abscissa of absolute convergence $\kappa(f)$. These new bounds may be considered as non-analytic analogues of the exponential growth bound $\omega_{0}(f)$ and the abscissas of convergence and absolute convergence of the Laplace transform of $f,$ $\operatorname{abs}(f)$ and $\operatorname{abs}(\|f\|)$. Analogues of several well known relations involving the growth bound and abscissas of convergence associated with $f$ and abscissas of holomorphy of the Laplace transform of $f$ are established. We examine the behaviour of $\zeta$ under regularisation of $f$ by convolution and obtain, in particular, estimates for the non-analytic growth bound of the classical fractional integrals of $f$. The definitions of $\zeta, \zeta_{1}$ and $\kappa$ extend to the operator-valued case also. For a $C_{0}$-semigroup $\mathbf{T}$ of operators, $\zeta(\mathbf{T})$ is closely related to the critical growth bound of $\mathbf{T}$. We obtain a characterisation of the non-analytic growth bound of $\mathbf{T}$ in terms of Fourier multiplier properties of the resolvent of the generator. Yet another characterisation of $\zeta(\mathbf{T}) $ is obtained in terms of the existence of unique mild solutions of inhomogeneous Cauchy problems for which a non-resonance condition holds. We apply our theory of non-analytic growth bounds to prove some results in which $\zeta(\mathbf{T})$ does not appear explicitly; for example, we show that all the growth bounds $\omega_{\alpha}(\mathbf{T}), \alpha >0,$ of a $C_{0}$-semigroup $\mathbf{T}$ coincide with the spectral bound $s(\mathbf{A})$, provided the pseudo-spectrum is of a particular shape. Lastly, we shift our focus from non-analytic bounds to sun-reflexivity of a Banach space with respect to $C_{0}$-semigroups. In particular, we study the relations between the existence of certain approximations of the identity on the Banach space $\xspace$ and that of $C_{0}$-semigroups on $\mathbf{X}$ which make $\mathbf{X}$ sun-reflexive.

In this thesis we apply methods from homological algebra to the study of finite $p$-groups. Let $G$ be a finite $p$-group and let $\mathbb{F}_p$ be the field of $p$ elements. We consider the cohomology groups $\operatorname{H}^1(G,\mathbb{F}_p)$ and $\operatorname{H}^2(G,\mathbb{F}_p)$ and the Massey product structure on these cohomology groups, which we use to deduce properties about $G$. We tie the classical theory of Massey products in with a general method from deformation theory for constructing hulls of functors and see how far the strictly defined Massey products can take us in this setting. We show how these Massey products relate to extensions of modules and to relations, giving us cohomological presentations of $p$-groups. These presentations will be minimal pro-$p$ presentations and will often be different from the presentations we are used to. This enables us to shed some new light on the classification of $p$-groups, in particular we give a `tree construction' illustrating how we can `produce' $p$-groups using cohomological methods. We investigate groups of exponent $p$ and some of the families of groups appearing in the tree. We also investigate the limits of these methods. As an explicit example illustrating the theory we have introduced, we calculate Massey products using the Yoneda cocomplex and give 0-deficiency presentations for split metacyclic $p$-groups using strictly defined Massey products. We also apply these methods to the modular isomorphism problem, i.e. the problem whether (the isomorphism class of) $G$ is determined by $\F_pG$. We give a new class $\mathcal{C}$ of finite $p$-groups which can be distinguished using $\mathbb{F}_pG$.

The author studies two coupled reaction-diffusion equations of 'λ-w' type, on an open, bounded, convex domain Ω C R(^d) (d ≤ 3), with a boundary of class C², and homogeneous Neumann boundary conditions. The equations are close to a supercritical Hopf bifurcation in the reaction kinetics, and are model equations for oscillatory reaction-diffusion equations. Global existence, uniqueness and continuous dependence on initial data of strong and weak solutions are proved using the classical Faedo-Galerkin method of Lions and compactness arguments. The work provides a complete case study for the application of this method to systems of nonlinear reaction-diffusion equations. The author also undertook the numerical analysis of the reaction-diffusion system. Results are presented for a fully-practical piecewise linear finite element method by mimicking results in the continuous case. Semi-discrete and fully-discrete error estimates are proved after establishing a priori bounds for various norms of the approximate solutions. Finally, the theoretical results are illustrated and verified via the numerical simulation of periodic plane waves in one space dimension, and preliminary results representing target patterns and spiral solutions presented in two space dimensions.

Chapter 1 briefly describes the motivation for the thesis and presents some background material. Chapter 2 develops the foundations of the theory of unprojection in the local and projective settings. Chapter 3 develops methods that calculate the unprojection ring for two important families of unprojection, Tom & Jerry. Finally, Chapter 4 proves some algebraic results concerning Catanese’s rank condition for symmetric matrices of small size.

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We begin in Chapter 1 by considering the original framework in which most work in stable homotopy theory has taken place, namely the stable homotopy category. We introduce the idea of structured ring and module spectra with the definition of ring spectra and their modules. We then proceed by considering the category of S-modules MS constructed in [19]. The symmetric monoidal category structure of MS allows us to discuss the notions of S-algebras and their modules, leading to modules over an S-algebra R. In Section 2.5 we use results of Strickland [43] to prove a result relating to the products on ko/? as a ko-module. A survey of results on nuclear and minimal atomic complexes from [5] and [23] is given in the context of MS in Chapter 3. We give an account of basic results for topological André-Quillen homology (HAQ) of commutative S-algebras in Chapter 4. In Section 4.2 we are able to set up a framework on HAQ for cell commutative S-algebras which allows us to extend results reported in Chapter 3 to the case of commutative S-algebras in Chapter 5. In particular, we consider the notion of a core of commutative S-algebras. We give examples of non-cores of MU, MSU, MO and MSO in Chapter 6. We construct commutative MU-algebra MU//x2 in Chapter 7 and consider various calculations associated to this construction.