<p>This thesis has three major topics. The first is on generalized multiplicities. The second is on height bounds for ideals of minors of matrices with a given rank. The last topic is on the ideal of minors of generic generalized diagonal matrices.</p> <p>In the first part of this thesis, we discuss various generalizations of Hilbert-Samuel multiplicity. These include the Buchsbaum-Rim multiplicity, mixed multiplicities, $j$-multiplicity, and $\varepsilon$-multiplicity. For $(R,m)$ a Noetherian local ring of dimension $d$ and $I$ a $m$-primary ideal in $R$, Lech showed the following bound for the Hilbert-Samuel multiplicity of $I$, $e(I) \leq d!\lambda(R/I)e(m)$. Huneke, Smirnov, and Validashti improved the bound to $e(mI) \leq d!\lambda(R/I)e(m)$. We generalize the improved bound to the Buchsbaum-Rim multiplicity and to mixed multiplicities. </p> <p>For the second part of the thesis we discuss bounds on heights of ideals of minors of matrices. A classical bound for these heights was shown by Eagon and Northcott. Bruns' bound is an improvement on the Eagon-Northcott bound taking into consideration the rank of the matrix. We prove an analogous bound to Bruns' bound for alternating matrices. We then discuss an open problem by Eisenbud, Huneke, and Ulrich that asks for height bounds for symmetric matrices given their rank. We show a few reduction steps and prove some small cases of this problem. </p> <p>Finally, for the last topic we explore properties of the ideal of minors of generic generalized diagonal matrices. Generalized diagonal matrices are matrices with two ladders of zeros in the bottom left and top right corners. We compute their initial ideals and give a description of the facets of their Stanley-Reisner complex. Using this description, we characterize when these ideals are Cohen-Macaulay. In the special case where the ladders of zeros are triangles, we compute the height and multiplicity</p>

We study Hessenberg varieties in type A via their local defining equations, called patch ideals. We focus on two main classes of Hessenberg varieties: those associated to a regular nilpotent operator and to those associated to a semisimple operator. In the setting of regular semisimple Hessenberg varieties, which are known to be smooth and irreducible, we determine that their patch ideals are triangular complete intersections, as defined by Da Silva and Harada. For semisimple Hessenberg varieties, we give a partial positive answer to a conjecture of Insko and Precup that a given family of set-theoretic local defining ideals are radical. A regular nilpotent Hessenberg Schubert cell is the intersection of a Schubert cell with a regular nilpotent Hessenberg variety. Following the work of the author with Da Silva, Harada, and Rajchgot, we construct an embedding of the regular nilpotent Hessenberg Schubert cells into the coordinate chart of the regular nilpotent Hessenberg variety corresponding to the longest-word permutation in Bruhat order. This allows us to use work of Da Silva and Harada to conclude that regular nilpotent Hessenberg Schubert cells are also local triangular complete intersections. / Thesis / Master of Science (MSc) / Algebraic varieties provide a generalization of curves in the plane, such as parabolas and ellipses. One such family of these varieties are called Hessenberg varieties, and they are known to have connections to other areas of pure and applied mathematics, including to numerical linear algebra, combinatorics, and geometric representation theory. In this thesis, we view Hessenberg varieties as a collection of subvarieties, called coordinate charts, and study the computational geometry of each coordinate chart. Although this is a local approach, we recover global geometric data on Hessenberg varieties. We also provide a partial positive answer to an open question in the area.

Neste trabalho é desenvolvido um estudo sobre a relação entre as variedades de Thom-Boardman e os ideais jacobianos iterados associados a estas variedades. Inicialmente são estudadas as singularidades de Thom-Boardman associadas a germes de aplicações analíticas com a finalidade de introduzir as varidades de Thom-Boardman no espaço dos jatos. Posteriormente são estudados os ideais jacobianos extendidos, seguindo a construção de Morin. Finalmente é definida a multiplicidade c_i(f) associada a um símbolo de Boardman i=(i_1,...,i_k) e ao extrato (Sigma)^1(f). / In this work we study the relation between the Thom-Boardman manifolds and the iterated jacobian ideals associate to these manifolds. First, we study the Thom-Boardman singularities associate to analitic map germs with the objective to introduce Thom-Boardman manifolds in the jet space. After, we study the extended jacobians ideals, following Morin's construction. We give the definition of the mulitiplicity c_i(f) associate to a Boadman symbol i=(i_1,...,i_k) and the stratum (Sigma)^i(f).

Neste trabalho é desenvolvido um estudo sobre a relação entre as variedades de Thom-Boardman e os ideais jacobianos iterados associados a estas variedades. Inicialmente são estudadas as singularidades de Thom-Boardman associadas a germes de aplicações analíticas com a finalidade de introduzir as varidades de Thom-Boardman no espaço dos jatos. Posteriormente são estudados os ideais jacobianos extendidos, seguindo a construção de Morin. Finalmente é definida a multiplicidade c_i(f) associada a um símbolo de Boardman i=(i_1,...,i_k) e ao extrato (Sigma)^1(f). / In this work we study the relation between the Thom-Boardman manifolds and the iterated jacobian ideals associate to these manifolds. First, we study the Thom-Boardman singularities associate to analitic map germs with the objective to introduce Thom-Boardman manifolds in the jet space. After, we study the extended jacobians ideals, following Morin's construction. We give the definition of the mulitiplicity c_i(f) associate to a Boadman symbol i=(i_1,...,i_k) and the stratum (Sigma)^i(f).

A mídia exerce um importante papel de produção e reprodução de conteúdos simbólicos para toda a sociedade, mudando as relações de comunicação e possibilitando novos sentidos para as pessoas em sua constituição identitária. A mídia se apropria de repertórios simbólicos que estão circulando na sociedade, os reproduz ou modifica e os devolve para a sociedade que por sua vez os (re) interpretará. Torna-se importante estudar as ideologias subjacentes à mídia direcionada aos jovens, pois este se encontra em fase importante da formação identitária. Nosso objetivo foi analisar como os ideais de mulher são veiculados pelas revistas, dirigidas ao público juvenil, Capricho e Todateen, considerando padrões estéticos; visão de corpo ideal e comportamentos desejáveis nas relações afetivo-sexuais. Realizamos um estudo qualitativo, fundamentado sob uma epistemologia social utilizando uma perspectiva de gênero proposta por Joan W. Scott. Identificamos quatro grupos para a análise, considerando as seguintes temáticas: 1- Sexualidade: o beijo e o amasso, ou seja, a troca de carícias sem o intercurso sexual; as práticas sexuais; e o início da vida sexual. 2-Relacionamento afetivo: o namoro, o ficar, o rolo e os sentimentos advindos desses tipos de relacionamentos. 3- Escolha e conquista do parceiro: modelos de procedimentos, técnicas e manuais passo a passo sobre quais condutas adotar para conquistar o parceiro afetivo-sexual; a escolha do parceiro, atributos desejados para uma garota que almeja conquistar um parceiro. Ideais de estética: ideais/ padrões de beleza e estética conjuntamente com a visão que o jovem possui sobre sua própria aparência física e auto10 estima relacionada à satisfação com o próprio corpo. Podemos considerar que tanto a revista Capricho quanto a revista Todateen trazem em suas matérias mais de um ideal de mulher, ou seja, ambas as revistas abordam em seus conteúdos idéias diversas e algumas vezes contrastantes do que é ser mulher e do que é feminilidade. Outro ponto comum a ser destacado, é que enquanto as matérias selecionadas abordam diversos atributos de feminilidades e ideais de mulher, é difundido, praticamente, apenas um ideal de masculino e de masculinidade. Desta forma, observamos um ideal de mulher e homem cuja relação é dicotômica, biologizante e essencializante Já o outro ideal de mulher em que o autoconhecimento, a auto-estima são valorizados e desejados numa relação entre gêneros igualitária. / The media plays an important role in the production and reproduction of symbolic contents for all the society, changing the relations of communication and making new meanings possible for people in their identity constitution. The media appropriates symbolic repertoires that are flowing in the society, reproduces or modifies them, and returns them to the society that will interpret them again. Since young people are in important phase of identity formation, studying medias underlying ideologies aimed at young people is important. Our objective was to analyze how the woman ideals are publicized by the youth magazines Capricho and Todateen, considering aesthetic standards; image of the ideal body and desirable behaviors in the affectivesexual relationships. This was a qualitative study, based on a social epistemology using Joan W. Scotts gender approach. We identified four groups for analysis, considering the following thematics: 1 - Sexuality: kissing and making out, in other words, caressing somebody as an expression of sexual desire without engaging in sexual intercourse, named amasso; sexual pratices; and the beginning of the sexual life. 2- Affective Relationship: dating, ficar, rolo, and the emotional reaction to affective relationships. 3 - Selection and seduction of the partner: models of procedures, techniques and step by step manuals on which behaviors to adopt to seduce the affective-sexual partner; the selection of the partner, desirable attributes for a girl who expects to seduce a partner. 4 Aesthetic ideals: ideal standards of aesthetic beauty and both the youngs image of his or her own physical appearance and self-esteem related to the satisfaction with his or her own body. We consider that the contents of Capricho and Todateen magazines show more than one ideal of woman. In other words, the magazines approach to being a woman and femininity are diverse and it sometimes displays disparities. One common point to be highlighted in this study is that while the approach of the selected material displays various attributes of femininity and ideals of woman, only one ideal of man and masculinity is presented. Therefore, we observe an ideal of woman and man whose relationship is dichotomist, biological and essentialist. On the other hand, there is the other ideal of woman whose self-knowledge and self-esteem are valued and desired in a more egalitarian gender relationship.

Thesis (M.Sc. (Mathematics)) -- University of Limpopo, 2021 / We study balanced filters and balanced z-filters considered by Carlson in [20] and [21] in topological spaces. We consider closed filters which are open-generated and open filters which are closed-generated. We show that a closed filter is open-generated precisely if it is a minimal balanced closed filter and that an open filter is closed-generated precisely when it is a minimal balanced open filter. For a completely regular topological space X, we study balanced z-filters and show that there is a one-to-one correspondence between the nonempty closed sets of βX and the balanced z-filter on X. By dualising closed filters we obtain ideals which then enables us to put some of the results in the context of frames. Dube in [28] has shown that a frame is normal if and only if its closed-generated filters are precisely the stably closed-generated ones. By dualisation we show that a frame is extremally disconnected if and only if its open-generated ideals are precisely the stably open-generated ones. We show that there is one-to-one correspondence between points of βL and the balanced ideals of Coz L. Furthermore we study nearness frames and show that the locally finite nearness frames strictly contain the Pervin nearness frames and the two coincide if the locally finite nearness frames are totally bounded. For perfect extension h : M → L of L, we show that a point p of M is a remote point if and only if Ip = {a ∈ L | h∗(a) ≤ p}. / University of Limpopo and DST - NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE - MaSS)

<p>Consider the standard graded polynomial ring in $n$ variables over a field $k$ and fix the Hilbert function of a homogeneous ideal. In the nineties Bigatti, Hulett, and Pardue showed that the Hilbert scheme consisting of all the homogeneous ideals with such a Hilbert function contains an extremal point which simultaneously maximizes all the graded Betti numbers. Such a point is the unique lexsegment ideal associated to the fixed Hilbert function.</p> <p> For such a scheme, we consider the individual strata defined by all ideals with Castelnuovo-Mumford regularity bounded above by <em>m</em>. In 1997 Mall showed that when <em>k </em>is of characteristic 0 there exists an ideal in each nonempty strata with maximal possible Betti numbers among the ideals of the strata. In chapter 4 of this thesis we provide a new construction of Mall's ideal, extend the result to fields of any characteristic, and show that these ideals have other extremal properties. For example, Mall's ideals satisfy an equation similar to Green's hyperplane section theorem.</p> <p> The key technical component needed to extend the results of Mall is discussed in Chapter 3. This component is the construction of a new invariant called the distraction-generic initial ideal. Given a homogeneous ideal <em>I C S</em> we construct the associated distraction-generic initial ideal, D-gin_< (<em>I</em>), by iteratively computing initial ideals and general distractions. The result is a monomial ideal that is strongly stable in any characteristic and which has many properties analogous to the generic initial ideal of <em>I</em>.</p>

We study partially ordered sets of prime ideals as found in commutative Noetherian rings. These structures, commonly known as prime spectra, have long been a popular topic in the field of commutative algebra. As a consequence, there are many related questions that remain unanswered. Among them is the question of what partially ordered sets appear as Spec(A) of some Noetherian ring A, asked by Kaplansky during the 1950's. As a partial case of Kaplansky's question, we consider finite posets that are ring spectra of commutative Noetherian rings. Specifically, we show that finite spectra of such rings are always order-isomorphic to a bipartite graph. However, the most significant undertaking of this study is that of devising a constructive methodology for finding a ring with prime spectrum that is order-isomorphic to an arbitrary bipartite graph. As a result, we prove that any complete bipartite graph is order-isomorphic to the prime spectrum of some ring of essentially finite type over the field of rational numbers. Moreover, a series of potential generalizations and extensions are proposed to further enhance the constructive methodology. Ultimately, the results of this study constitute an original contribution and perspective on questions related to commutative ring spectra.

The concept of unique factorization was first recognized in the 1840s, but even then, it was still fairly believed to be automatic. The error of this assumption was exposed largely through attempts to prove Pierre de Fermat's, 1601-1665, last theorem. Once mathematicians discovered that this property did not always hold, it was only natural for them to try to search for the strongest available alternative. Thus began the attempt to generalize unique factorization. Using the ascending chain condition on principle ideals, we will show the conditions under which a ring is a unique factorization domain.

Ha Minh Lam et M. Morales ont introduit une classe d'idéaux binomiaux qui est une extension binomiale d'idéaux monomiaux libres de carrés.Étant donné I un idéal monomial quadratique de k[x] libre de carrés et J une somme d'idéaux de scroll de k[z] qui satisfont certaines conditions, nous définissons l'extension binomiale de I comme B=I+J. Le sujet de cette thèse est d'étudier le nombre p plus grand tel que les sizygies de B son linéaires jusqu'au pas p-1. Sous certaines conditions d'ordre imposées sur les facettes du complexe de Stanley-Reisner de I nous obtiendrons un ordre > pour les variables de l'anneau de polynomes k[z]. Ensuite nous prouvons pour un calcul des bases de Gröbner que l'idéal initial in(B), sous l'ordre lexicographique induit par l'ordre de variables >, est quadratique libre de carrés. Nous montrerons que B est régulier si et seulement si I est 2-régulier. Dans le cas géneral, lorsque I n'est pas 2-régulier nous trouverons une borne pour l'entier q maximal qui satisfait que les premier q-1 sizygies de B son linéaires. En outre, en supossant que J est un idéal torique et en imposant des conditions supplémentaires, nous trouveron une borne supérieure pour l'entier q maximal qui satisfait que les premier q-1 sizygies de B son linéaires. En imposant des conditions supplémentaires, nous prouverons que les deux bornes sont égaux. / Ha Minh Lam et M. Morales introduced a family of binomial ideals that are binomial extensions of square free monomial ideals. Let I be a square free monomial ideal of k[x] and J a sum of scroll ideals in k[z] with some extra conditions, we define the binomial extension of $I$ as $B=I+Jsubset sis$. The aim of this thesis is to study the biggest number p such that the syzygies of B are linear until the step p-1. Due to some order conditions given to the facets of the Stanley-Reisner complex of I we get an order > for the variables of the polynomial ring k[z]. By a calculation of the Gröbner basis of the ideal $B$ we obtain that the initial ideal in(B) is a square free monomial ideal. We will prove that B is 2-regular iff I is 2-regular. In the general case, wheter I is not 2-regular we will find a lower bound for the the maximal integer q which satisfies that the first q-1 sizygies of B are linear. On the other hand, wheter J is toric and supposing other conditions, we will find a upper bound for the integer q which satisfies that the first q-1 syzygies of B are linear. By given more conditions we will prove that the twobounds are equal.