Given a reduced cooperad C, we consider the 2-colored operad Cyl(C) which governs diagrams U: V -> W, where V, W are Cobar(C)-algebras, and U is an infinity-morphism. We then investigate the deformation complexes of Cyl(C) and Cobar(C). Our main result is that the restriction maps between between the deformation complexes Der'(Cyl(C)) and Der'(Cobar(C)) are homotopic quasi-isomorphisms of filtered Lie algebras. We show how this result may be applied to modifying diagrams of homotopy algebras by derived automorphism. We then recall that Tamarkin's construction gives us a map from the set of Drinfeld associators to the homotopy classes of Lie infinity quasi-isomorphisms for Hochschild cochains of a polynomial algebra. Due to results of V. Drinfeld and T. Willwacher, both the source and the target of this map are equipped with natural actions of the Grothendieck-Teichmueller group GRT. We use our earlier results to prove that this map from the set of Drinfeld associators to the set of homotopy classes of Lie infinity quasi-isomorphisms for Hochschild cochains is GRT-equivariant. / Mathematics

<p dir="ltr">We first introduce several families of monoidal categories with plethysm products as their monoidal products and use this to describe operadic structures as plethysm monoids. In order to link this approach with the classical theory, we give a generalization of the Baez-Dolan plus construction. We then show that an operadic structure can be defined as a plethysm monoid if its associated Feynman category is a plus construction of a unique factorization category.</p>

Ce mémoire a deux objectifs principaux. Premièrement de développer et interpréter les groupes de cohomologie de Hochschild de basse dimension et deuxièmement de borner la dimension cohomologique des k-algèbres par dessous; montrant que presque aucune k-algèbre commutative est quasi-libre. / The aim of this master’s thesis is two-fold. Firstly to develop and interpret the low dimensional Hochschild cohomology of a k-algebra and secondly to establish a lower bound for the Hochschild cohomological dimension of a k-algebra; showing that nearly no commutative k-algebra is quasi-free.

We introduce the definition of 1-cotilting object in a Grothendieck category and investigate its relation to the analogue of the standard definition of 1-cotilting module. The 1-cotilting quasi-coherent sheaves on a Noetherian scheme are stud- ied in particular: using the classification of hereditary torsion pairs in the category of quasi-coherent sheaves on a Noetherian scheme X, to each hereditary torsion- free class F that is generating we assign a 1-cotilting quasi-coherent sheaf whose 1-cotilting class is F. This provides a family of pairwise non-equivalent 1-cotilting quasi-coherent sheaves which are parametrized by specialization closed subsets of X avoiding the set of associated points of a chosen generator of the category of quasi-coherent sheaves. In many cases (e.g. for separated schemes), this set of avoided points can be chosen as the set of associated points of the scheme. 1

This thesis is on the representation theory of finite groups. Specifically, it is about finding connections between fusion and universal deformation rings. Two elements of a subgroup N of a finite group Γ are said to be fused if they are conjugate in Γ, but not in N. The study of fusion arises in trying to relate the local structure of Γ (for example, its subgroups and their embeddings) to the global structure of Γ (for example, its normal subgroups, quotient groups, conjugacy classes). Fusion is also important to understand the representation theory of Γ (for example, through the formula for the induction of a character from N to Γ). Universal deformation rings of irreducible mod p representations of Γcan be viewed as providing a universal generalization of the Brauer character theory of these mod p representations of Γ. It is the aim of this thesis to connect fusion to this universal generalization by considering the case when Γ is an extension of a finite group G of order prime to p by an elementary abelian p-group N of rank 2. We obtain a complete answer in the case when G is a dihedral group, and we also consider the case when G is abelian. On the way, we compute for many absolutely irreducible FpΓ-modules V, the cohomology groups H2(Γ,HomFp(V,V) for i = 1, 2, and also the universal deformation rings R(Γ,V).

O objetivo dessa Dissertação é detalhar resultados conhecidos de Cohomologia em Topos de Grothendieck. Para isso, apresentamos a Álgebra Homológica em seu contexto mais geral, através de Categorias Abelianas, introduzindo as principais noções da área como funtores derivados e sequências espectrais. Desenvolvemos também o essencial da Teoria de Topos, explicando como um topos de Grothendieck surge como uma certa generalização dos feixes de conjuntos e fornecemos aspectos lógicos dos topos elementares. Focamos sobretudo nos Topos de Grothendieck pois a partir deles podemos construir categorias abelianas com suficientes injetivos, as quais são necessárias para expressar os grupos de cohomologia. / The final objective of this Dissertation is to detail known results of Cohomology in Grothendieck Topos. For this, we present Homological Algebra in its more general context, through Abelian Categories, introducing the main notions of the area as derived functors and spectral sequences. We also develop the basics of the Topos Theory, explaining how a Grothendieck Topos arises as a certain generalization of sheafs and we provide logical aspects of the elementary topos. We focus mainly in the Grothendieck Topos because from them we can construct abelians categories.

The classical discriminant D_n of degree n polynomials detects whether a given univariate polynomial f has a repeated root. It is itself a polynomial in the coefficients of f which, according to several classical results by Bézout, Sylvester, Cayley, and others, may be expressed as the determinant of a matrix whose entries are much simpler polynomials in the coefficients of f. This thesis is concerned with the construction and classification of such determinantal formulae for D_n. In particular, all of the formulae for D_n appearing in the classical literature are equivalent in the sense that the cokernels of their associated matrices are isomorphic as modules over the associated polynomial ring. This begs the question of whether there exist formulae which are not equivalent to the classical formulae and not trivial in the sense of having the same cokernel as the 1 x 1 matrix (D_n). The results of this thesis lie in two directions. First, we construct an explicit non-classical formula: the presentation matrix of the open swallowtail first studied by Arnol'd and Givental. We study the properties of this formula, contrasting them with the properties of the classical formulae. Second, for the discriminant of the polynomial x^4+a_2x^2+a_3x+a_4 we embark on a classification of determinantal formulae which are homogeneous in the sense that the cokernels of their associated matrices are graded modules with respect to the grading deg a_i=i. To this end, we use deformation theory: a moduli space of such modules arises from the base spaces of versal deformations of certain modules over the E_6 singularity {x^4-y^3}. The method developed here can in principle be used to classify determinantal formulae for all discriminants, and, indeed, for all singularities.

The classical discriminant D_n of degree n polynomials detects whether a given univariate polynomial f has a repeated root. It is itself a polynomial in the coefficients of f which, according to several classical results by Bézout, Sylvester, Cayley, and others, may be expressed as the determinant of a matrix whose entries are much simpler polynomials in the coefficients of f. This thesis is concerned with the construction and classification of such determinantal formulae for D_n. In particular, all of the formulae for D_n appearing in the classical literature are equivalent in the sense that the cokernels of their associated matrices are isomorphic as modules over the associated polynomial ring. This begs the question of whether there exist formulae which are not equivalent to the classical formulae and not trivial in the sense of having the same cokernel as the 1 x 1 matrix (D_n). The results of this thesis lie in two directions. First, we construct an explicit non-classical formula: the presentation matrix of the open swallowtail first studied by Arnol'd and Givental. We study the properties of this formula, contrasting them with the properties of the classical formulae. Second, for the discriminant of the polynomial x^4+a_2x^2+a_3x+a_4 we embark on a classification of determinantal formulae which are homogeneous in the sense that the cokernels of their associated matrices are graded modules with respect to the grading deg a_i=i. To this end, we use deformation theory: a moduli space of such modules arises from the base spaces of versal deformations of certain modules over the E_6 singularity {x^4-y^3}. The method developed here can in principle be used to classify determinantal formulae for all discriminants, and, indeed, for all singularities.

Splines are piecewise polynomial functions of a given order of smoothness r on a triangulated region (or polyhedrally subdivided region) of Rd. The set of splines of degree at most k forms a vector space Crk() Moreover, a nice way to study Cr k()is to embed n Rd+1, and form the cone b of with the origin. It turns out that the set of splines on b is a graded module Cr b() over the polynomial ring R[x1; : : : ; xd+1], and the dimension of Cr k() is the dimension o This dissertation follows the works of Billera and Rose, as well as Schenck and Stillman, who each approached the study of splines from the viewpoint of homological and commutative algebra. They both defined chain complexes of modules such that Cr(b) appeared as the top homology module. First, we analyze the effects of gluing planar simplicial complexes. Suppose 1, 2, and = 1 [ 2 are all planar simplicial complexes which triangulate pseudomanifolds. When 1 \ 2 is also a planar simplicial complex, we use the Mayer-Vietoris sequence to obtain a natural relationship between the spline modules Cr(b), Cr (c1), Cr(c2), and Cr( \ 1 \ 2). Next, given a simplicial complex , we study splines which also vanish on the boundary of. The set of all such splines is denoted by Cr(b). In this case, we will discover a formula relating the Hilbert polynomials of Cr(cb) and Cr (b). Finally, we consider splines which are defined on a polygonally subdivided region of the plane. By adding only edges to to form a simplicial subdivision , we will be able to find bounds for the dimensions of the vector spaces Cr k() for k 0. In particular, these bounds will be given in terms of the dimensions of the vector spaces Cr k() and geometrical data of both and . This dissertation concludes with some thoughts on future research questions and an appendix describing the Macaulay2 package SplineCode, which allows the study of the Hilbert polynomials of the spline modules.

Made available in DSpace on 2015-05-15T11:46:14Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 773765 bytes, checksum: b63ba4fb3ff15c5a4aef5a708fce596e (MD5) Previous issue date: 2012-12-20 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The goal this work is to understand the local cohomology functor, and some of its properties. We show that this functor has a relation with the functor Ext. Furthermore, we show the followings theorems: Grothendieck's Vanishing Theorem, Hartshorne's Vanishing Theorem, Grothendieck's Non-Vanishing Theorem and Hartshorne-Linchenbaum's Vanishing Theorem. / O objetivo desta dissertação é entender o funtor de Cohomologia Local, assim como algumas de suas propriedades. Mostramos que este funtor tem uma relação com o funtor Ext. Além disso, expomos os seguintes teoremas: Teorema do Anulamento de Grothendieck, Teorema do Anulamento de Hartshorne, Teorema do Não Anulamento de Grothendieck e o Teorema do Anulamento de Hartshorne-Linchtenbaum.