Systems of polynomial equations often have symmetries. In solving such a system using Buchberger's algorithm, the symmetries are neglected. Incorporating symmetries into the solution process enables us to solve larger problems than with Buchberger's algorithm alone. This paper presents a method that shows how this can be achieved and also gives an algorithm that brings together continuously parameterized symmetries with Buchberger's algorithm.

We present some techniques in the construction of spectral triples for C*-algebras, in particular those which determine a compatible metric on the state space, which provides a noncommutative analogue of geodesic distance between points on a manifold. The main body of the thesis comprises three sections. In the first, we provide a further analysis on the existence of spectral triples on crossed products by discrete groups and their interplay with classical metric dynamics. Dynamical systems arising from non-unital C*-algebras and certain semidirect products of groups are considered. The second section is a construction of spectral triples for certain unital extensions by stable ideals, using the language of unbounded Kasparov theory as presented by Mesland, Kaad and others, These ideas can be implemented for both the equatorial Podle\'s spheres and quantum SU2 group. Finally, we investigate the potential of the construction of twisted spectral triples, as outlined by Connes and Moscovici. We achieve a construction of twisted spectral triples on all simple Cuntz-Krieger algebras, whose unique KMS state is obtained from the asymptotics of the Dirac.

This thesis is concerned with two different problems in harmonic analysis: the multilinear Kakeya theorem, and Wolff-type inequalities for paraboloids. Chapter 1 gives an overview of both of these problems. In Chapter 2 we investigate an important special case of the multilinear Kakeya theorem, the so-called “bush example”. While the endpoint case of the multilinear Kakeya theorem was recently proved by Guth, the proof is highly abstract; our aim is to provide a more elementary proof in this special case. This is achieved for a significant part of the three-dimensional case in the main result of the chapter. Chapter 3 is a study of the endpoint case of a mixed-norm Wolff-type inequality for the paraboloid. The main result adapts an example of Bourgain to show that the endpoint inequality cannot hold with an absolute constant; there must be a dependence on the thickening of the paraboloid. The remainder of the chapter is a series of case studies, through which we establish positive endpoint results for certain classes of function, as well as indicating specific examples which need to be better understood in order to obtain the full endpoint result.

Let K be the field of complex numbers. In this thesis we construct new examples of noncommutative surfaces of GK-dimension 4 using the language of formal and infinitesimal deformations as introduced by Gerstenhaber. Our approach is to find families of deformations of a certain well known GK-dimension 4 birationally commutative surface defined by Zhang and Smith in unpublished work cited in [YZ06], which we call A. Let B* and K* be respectively the bar and Koszul complexes of a PBW algebra C = KhV / (R) . We construct a graph whose vertices are elements of the free algebra KhV i and edges are relations in R. We define a map m2 : B2 ! K2 that extends to a chain map m* : B* → K*. This map allows the Gerstenhaber bracket structure to be transferred from the bar complex to the Koszul complex. In particular, m2 provides a mechanism for algorithmically determining the set of infinitesimal deformations with vanishing primary obstruction. Using the computer algebra package 'Sage' [Dev15] and a Python package developed by the author [Cam], we calculate the degree 2 component of the second Hochschild cohomology of A. Furthermore, using the map m2 we describe the variety U ⊆ HH2/2 (A) of infinitesimal deformations with vanishing primary obstruction. We further show that U decomposes as a union of 3 irreducible subvarieties Vg, Vq and Vu. More generally, let C be a Koszul algebra with relations R, and let E be a localisation of C at some (left and right) Ore set. Since R is homogeneous in degree two, there is an embedding R ,↪ C⊗C and in the following we identify R with its (nonzero) image under this map. We construct an injective linear map ~⋀ : HH²(C) → HH²(E) and prove that if f ∈ HH²(E) satisfies f(R) ⊆ C then f ∈ Im(~⋀). In this way we describe a relationship between infinitesimal deformations of C with those of E. Rogalski and Sierra [RS12] have previously examined a family of deformations of A arising from automorphism of the surface P1 X P1. By applying our understanding of the map ~⋀ we show that these deformations correspond to the variety of infinitesimal deformations Vg. Furthermore, we show that deformations defined similarly by automorphisms of other minimal rational surfaces also correspond to infinitesimal deformations lying in Vg. We introduce a new family of deformations of A, which we call Aq. We show that elements of this family have families of deformations arising from certain quantum analogues of geometric automorphisms of minimal rational surfaces, as defined by Alev and Dumas. Furthermore, we show that after taking the semi-classical limit q → 1 we obtain a family of deformations of A whose infinitesimal deformation lies in Vq. Finally, we apply a heuristic search method in the space of Hochschild 2-cocycles of A. This search yields another new family of deformations of A. We show that elements of this family are non-noetherian PBW noncommutative surfaces with GK-dimension 4. We further show that elements of this family can have as function skew field the division ring of the quantum plane Kq(u; v), the division ring of the first Weyl algebra D1(K) or the commutative field K(u; v).

Building on the work of Longuet-Higgins in 1972 and Calderbank and Macpherson in 2009, we study the combinatorics of symmetric configurations of hyperplanes and points in projective space, called generalized Cox configurations. To do so, we use the formalism of morphisms between incidence systems. We notice that the combinatorics of Cox configurations are closely related to incidence systems associated to certain Coxeter groups. Furthermore, the incidence geometry of projective space P (V ), where V is a vector space, can be viewed as an incidence system of maximal parabolic subalgebras in a semisimple Lie algebra g, in the special case g = pgl (V ) the projective general linear Lie algebra of V . Using Lie theory, the Coxeter incidence system for the Coxeter group, whose Coxeter diagram is the underlying diagram of the Dynkin diagram of the g, can be embedded into the parabolic incidence system for g. This embedding gives a symmetric geometric configuration which we call a standard parabolic configuration of g. In order to construct a generalized Cox configuration, we project a standard parabolic configuration of type Dn into the parabolic incidence system of projective space using a process called parabolic projection, which maps a parabolic subalgebra of the Lie algebra to a parabolic subalgebra of a lower dimensional Lie algebra. As a consequence of this construction, we obtain Cox configurations and their analogues in higher dimensional projective spaces. We conjecture that the generalized Cox configurations we construct using parabolic projection are nondegenerate and, furthermore, any non-degenerate Cox configuration is obtained in this way. This conjecture yields a formula for the dimension of the space of non-degenerate generalized Cox configurations of a fixed type, which enables us to develop a recursive construction for them. This construction is closely related to Longuet-Higgins’ recursive construction of (generalized) Clifford configurations but our examples are more general and involve the extra parameters.

Joseph S. Perkell and Marc H. Cohen. / Bibliography: p. 41-43. / Supported in part by the National Institutes of Health under grant 5 R01 NS04332 Supported in part by the Clarence Lebel Professorship in the Department of Electrical Engineering and Computer Science at M.I.T. held by Professor Kenneth N. Stevens.

Sea M un módulo cuasi-inyectivo y S el anillo de endomorfimos de S. El objetivo de la tesis consiste en describir las propiedades que tiene que satisfacer M para que S satisfaga una propiedad dada. / Let M a quasi-inyective module and S the ring of endomorfisms of S. The goal of this thesis consists in describing the properties that M should satisfy for S to satisfy a given property.

Esta Tesis Doctoral está enmarcada dentro del área del Álgebra, concretamente, de los Anillos de Grupo. El objetivo principal de la misma es el estudio del grupo de unidades del anillo de grupo con coeficientes enteros de un grupo finito. El primer trabajo que se conoció sobre este grupo de unidades lo presento Higman en 1940 en su tesis doctoral. Éste describe el grupo de unidades para el anillo de grupo con coeficientes enteros de un grupo abeliano y como consecuencia de este resultado se tiene que las unidades centrales del anillo de grupo con coeficientes enteros no son mas que las triviales. A partir de este momento, muchos autores se interesaron en dicho grupo de unidades. Sobre este grupo de unidades se sabe que es finitamente generado, pero no se conoce un conjunto finito de generadores, y que en muchos casos contiene un subgrupo libre de rango dos. También se ha intentado describir las unidades de orden finito y los subgrupos finitos de este grupo de unidades. Justamente este es el propósito de las tres Conjeturas de Zassenhaus, planteadas por Hans Zassenhaus en los años sesenta. Unos años después se presento un contraejemplo para dos de ellas. Otro punto interesante sobre el grupo de unidades del anillo de grupo con coeficientes enteros de un grupo finito, es conocer sus subgrupos de índice finito. Problemas como el de los subgrupos de congruencia traducidos a este contexto son de gran ayuda para este propósito. En esta Tesis hemos abordado dos problemas clásicos como son la Conjetura de Zassenhaus y el Problema de los Subgrupos de Congruencia para anillos de grupo con coeficientes enteros. Durante la realización de la Tesis doctoral, localizamos y estudiamos en profundidad la bibliografía existente relacionada con nuestro objeto de estudio. Establecimos contacto continuo con expertos en la materia y realice una estancia de tres meses en la Universidad Libre de Bruselas. El fruto de este trabajo se vio reflejado en los artículos “Zassenhaus Conjecture for cyclic-by-abelian groups” el cual esta aceptado en “Journal of the London Mathematical Society” y “On the Congruence Subgroup Problem for integral group rings” el cual esta sometido. La monografía, que consta de una introducción, tres capítulos y las conclusiones, está dividida principalmente en dos partes. Uno de los tópicos centrales es la Conjetura de Zassenhaus. Esta pretende describir las unidades de orden finito del anillo de grupo con coeficientes enteros de un grupo finito. Nuestra aportación principal en este aspecto consiste en probar la Conjetura de Zassenhaus para grupos cíclicos-por-abelianos. El segundo problema que abordamos es el de clasificar los grupos finitos para los cuales el grupo de unidades del anillo de grupo con coeficientes enteros tiene núcleo de congruencia finito. Desafortunadamente en este problema encontramos un gran obstáculo, por lo que dimos una clasificación muy cercana a la planteada originalmente y que resulta de gran utilidad porque nos da mucha información sobre los subgrupos de índice finito del anillo de grupo con coeficientes enteros. / This thesis is placed in the general framework of Algebra, concretely, in Group Rings. The main aim of it is to study the group of units of the integral group ring of a finite group. Higman presented the first work about this group of units in 1940 in his thesis. It describe the group of units of the integral group ring of an abelian group, moreover shows that the central units are exactly the trivial ones. From here, it has attracted the interest of many authors. About the group of units of the integral group ring of a finite group we know that it is finite generated, however a finite set of generators is not known in general, and also it contains in many cases a free subgroup of rank two. On the other hand, many authors have attempted to describe the units of finite order and the finite subgroups of such group of units. This is just the goal of the three Zassenhaus conjectures, posed by Hans Zassenhaus in the 60s. Some years later, a counterexample for two of them appeared. Another interesting point on the group of units of the integral group ring of a finite group is to know its subgroups of finite index. One way to do so is to translate the Congruence Subgroup Problem to the context of integral group rings. In this thesis we have addressed two classical problems, namely Zassenhaus Conjecture and the Congruence Subgroup Problem for integral group rings of a finite group. During the realization of this thesis, we found and study in depth the existing literature concerning our subject. We established contact with experts and I did my stay in “Vrije Universiteit Brussel”. This work has given rise to my two papers “Zassenhaus Conjecture for cyclic-by-abelian groups ” which is accepted in “Journal of the London Mathematical Society” and “On the Congruence Subgroup Problem for integral group rings ” which is submitted. The monograph, consisting of an introduction, three chapters and the conclusions, is divided into two parts. A central topic is the Zassenhaus Conjecture. This tries to describe the units of finite order of the integral group ring of a finite group. Our main contribution consists in proving the Zassenhaus Conjecture for cyclic-by-abelian groups. Later on we deal with the problem of classifying the finite groups for which the group of units of the integral group ring has finite congruence kernel. Unfortunately, in this problem we encountered an obstacle. So we give a classification, which is very close to the original one, and it gives us relevant information on the subgroups of finite index of the group of units of the integral group ring of a finite group.

Esta Tesis Doctoral está enmarcada dentro del área del Álgebra, concretamente, de la Teoría de Representación de Álgebras. El objetivo principal de la misma es el estudio de propiedades homológicas de una clase de álgebras finito dimensionales conocidas como Álgebras de malla finito dimensionales. Dichas álgebras, introducidas por primera vez por K. Erdmann y A. Skowronski en 2008, surgieron como generalización de las álgebras preproyectivas y han suscitado un gran interés en los últimos años en el contexto general de las álgebras finito dimensionales. Entre otras, cabe destacar su aplicación en problemas de álgebras de conglomerado, grupos cuánticos, clasificación de ecuaciones diferenciales, singularidades de Klenian y geometría diferencial. Durante el primer periodo de la realización de la Tesis doctoral, localizamos y estudiamos en profundidad la bibliografía existente relacionada con nuestro objeto de estudio. Establecimos contacto continuo con expertos en la materia e incluso tuve la oportunidad de trabajar con Karin Erdmann durante mis tres meses de estancia en la Universidad de Oxford. El fruto de este trabajo se vio reflejado en mis dos primeros artículos publicados “The Hochschild cohomology ring of preprojective algebras of type Ln” y “The Hochschild cohomology ring of preprojective algebras of type Ln over a field of characteristic 2”. La monografía, que consta de un total de 6 capítulos, está dividida en dos partes. Uno de los tópicos centrales es el anillo de cohomología de Hochschild de un álgebra. Este anillo tiene una notable influencia en diversas partes de las matemáticas como el Álgebra Conmutativa, la Teoría de Anillos, la Geometría Conmutativa y no Conmutativa, la Teoría de Representación, la Física Matemática, … Además, su estructura multiplicativa está estrechamente relacionada con el estudio de las variedades de módulos y su álgebra de Yoneda. La definición del anillo de cohomología de Hochschild es bien sencilla, sin embargo, muy poco se sabe acerca de él. Es más, en la gran mayoría de los casos resulta extremadamente difícil calcularlo. Nuestra aportación principal en este aspecto consiste en la descripción explícita, mediante generadores y relaciones, de la estructura multiplicativa del anillo de cohomología de Hochschild de las álgebras de malla finito dimensionales de tipo Ln y Bn. Nuestras conclusiones resultan sorprendentes pues muestran grandes diferencias en el comportamiento de dicho anillo asociado no sólo a distintas álgebras, sino a una misma álgebra. Por otra parte, abordamos las propiedades homológicas de simetría, periodo y dimensión de Calabi-Yau de las álgebras de malla finito dimensionales. Consideramos primeramente la simetría y, como resultado, identificamos aquellas álgebras que son débilmente simétricas y las que son a su vez simétricas. A pesar de que una de las características más conocidas de estas álgebras es que son periódicas, sólo en muy pocos casos se ha conseguido calcular su periodo. En esta Tesis calculamos explícitamente el periodo de cada una de ellas. Finalmente tratamos la noción Calabi-Yau, definida por M.Kontsevich a finales de los años 90 y que ha sido intensamente estudiada por muchos matemáticos en los últimos años. Nuestro resultado principal es la caracterización de las álgebras de malla finito dimensionales que son establemente Calabi-Yau y Calaib-Yau Frobenius. Además, en tal caso, calculamos ambas dimensiones probando que, a pesar de que en la mayoría de los casos coinciden, no siempre son iguales, hecho que a día de hoy era desconocido. / This thesis is placed in the general framework of Algebra, concretely, in Representation Theory of Algebras. The main aim of it is to study homological properties of a class of finite dimensional algebras known as finite dimensional mesh algebras. Such algebras, first introduced by K. Erdmann and A. Skowronski in 2008, arise as a generalization of preprojective algebras and have attracted great interest in the general context of finite dimensional algebras in recent years. Among others, it is worth mentioning their application to problems related with cluster algebras, quantum groups, classification of differential equations, Klenian singularities and differential geometry. During the first period of the realization of this thesis, we found and study in depth the existing literature concerning our subject. We established contact with experts and I even had the opportunity of working with K. Erdmann for three months during my stay at University of Oxford. This work have given rise to my two first published papers “The Hochschild cohomology ring of preprojective algebras of type Ln” y “The Hochschild cohomology ring of preprojective algebras of type Ln over a field of characteristic 2”. The monograph, consisting of six chapters, is divided into two parts. A central topic is the Hochschild cohomology ring of an algebra. This ring has great influence in many diverse areas of mathematics such as commutative algebra, ring theory, commutative and noncommutative geometry, representation theory, mathematical physics, … Also, its multiplicative structure is closely related to the study of module varieties and its Yoneda algebra. The definition of the ring is quite simple , however, only little information is known. Moreover, in most of the cases is extremely difficult the computation. Our main contribution consists of an explicit description, by means of generators and relators, of the multiplicative structure of the Hochschild cohomology ring of the finite dimensional mesh algebras of type Ln and Bn. Our conclusions are surprising since they show big differences in the behavior of this ring associated not only to two different algebras but also to the same one. On the other hand, we deal with the homological properties of symmetry, period and Calabi-Yau dimension of finite dimensional mesh algebras. We first consider the symmetry and, as a result, we identify those algebras being weakly symmetric and those which are in turn symmetric. Despite of the fact that it is well known that finite dimensional mesh algebras are periodic, the precise calculation of the period is only known in a few cases. In this thesis, we explicitly compute the period of any of this algebras. Finally, we deal with the Calabi-Yau notion, defined by M. Kontsevich in the late 90s and that has been intensively studied by many mathematicians in recent years. Our main result is the characterization of the stably Calabi-Yau and Calabi-Yau Frobenius finite dimensional mesh algebras. Moreover, in this case, we compute both dimensions showing that they need not to be equal, an unknown fact so far.

In this thesis we will be exclusively considering uncountable groups and semigroups. Roughly speaking the underlying problem is to find “large” subgroups (or subsemigroups) of the object in question, where we consider three different notions of “largeness”: we classify all the subsemigroups of the set of all mapping from a countable set back to itself which contains a specific uncountable subsemigroup; we investigate topological “largeness”, in particular subgroups which are finitely generated and dense; we investigate if it is possible to find an integer r such that any countable collection of elements belongs to some r-generated subsemigroup, and more precisely can these elements be obtained by multiplying the generators in a prescribed fashion.