In this bachelor’s thesis an introduction to the fundamentals of algebraic geometry is given. Some concepts in algebraic geometry are introduced such as Spec of a ring and Proj of a graded ring and several results related to these are either proven or stated. Special focus is directed towards defining the so called ”Hilbert scheme” which is the main topic in a lot of modern algebraic geometry research.

In this paper, the study of k-fold branched coverings for which the branch set is a stratified set is considered. First of all, the existence of universal k-fold branched coverings over CW-complexes with stratified branch set is proved using Brown's Representability Theorem. Next, an explicit construction of universal k-fold branched coverings over manifolds is given. Finally, some homotopy and homology groups are computed for some specific examples of Universal k-fold branched coverings.

Este estudo tem como objetivo investigar os efeitos de sentidos produzidos sobre a representabilidade política produzidos no discurso de campanha de Barack H. Obama e no discurso de recepção da mídia (institucional e auto comandada) durante as eleições presidenciais nos Estados Unidos em 2008. Nesta pesquisa, seguimos os pressupostos teóricos da análise do discurso de vertente pecheutiana e articulamos os conceitos de enunciação e de formação imaginária com o conceito de identidade advindos dos estudos culturais (Rose 2001; Wardward 2000). A análise dos materiais que compõem nosso corpus mostraram que, em relação ao processo de representabilidade, o discurso da mídia institucional se vale da crise econômica estadunidense para delinear discursivamente os aspectos necessários que o novo representante do país deve ter para gerenciar a crise que o país enfrentava. Neste trajeto discursivo, pudemos encontrar semelhanças do discurso da mídia institucional com o de Obama no que tange às representações sobre como governar sustentadas em uma FD que concebe como prática administrativa ideal aquela que emana da concepção de sujeito de direito. Em relação ao discurso da mídia autocomandada, a análise do boato permitiu que viessem à tona dizeres pautados em uma FD que estabelece interditos no que se refere ao tripé candidato-religião-presidência. O estudo concluiu também que, no dircurso de campanha de Barack H. Obama, a representabilidade é construída por dizeres que estabelecem ligações com elementos do discurso fundador dos E.U.A., o que acaba fazendo com que o acontecimento discursivo seja produzido por meio da materialidade do corpo do candidato. / The aim of this study was to investigate what meanings effects regarding to representability were produced in Barack Obamas election campaign speeches and in the texts produced by the American media (mass communication and also mass- selfcommunication) in 2008. This study has been informed by concepts introduced by Michel Pêcheux in Discourse Analysis such as enunciation and imaginary formations, which were also articulated in my discussion of the concept of identity from Cultural Studies (Rose, 2001; Woodward 2000). The analysis of the materials that make up our corpus showed that, in relation to the representability process, the mass media discourse focused in its discourse on the aspects required of a new representative of the country in order to manage the economic crisis that the country was facing at that time. In this case, the discourse of the institutional media shows similarities with Obama\'s speech on the representations of how to govern. Both samples hold their speeches on a discursive formation which conceives that the ideal administrative practice derives from the conception of the legal subject. Regarding the discourse of mass self-communication, the analysis of rumors about the candidates religious beliefs allowed the presence of a discursive formation that sets up as meaning effects restricted ideas of representability to certain elements. With regard to Obamas 2008 campaign speech, the analysis showed that representability is built through the values that base on the founding discourse of the American nation. By emphasizing this aspect, the candidate\'s 2008 campaign speeches negate the possibility of new networks of meaning that could produce an original discursive event.

Este estudo tem como objetivo investigar os efeitos de sentidos produzidos sobre a representabilidade política produzidos no discurso de campanha de Barack H. Obama e no discurso de recepção da mídia (institucional e auto comandada) durante as eleições presidenciais nos Estados Unidos em 2008. Nesta pesquisa, seguimos os pressupostos teóricos da análise do discurso de vertente pecheutiana e articulamos os conceitos de enunciação e de formação imaginária com o conceito de identidade advindos dos estudos culturais (Rose 2001; Wardward 2000). A análise dos materiais que compõem nosso corpus mostraram que, em relação ao processo de representabilidade, o discurso da mídia institucional se vale da crise econômica estadunidense para delinear discursivamente os aspectos necessários que o novo representante do país deve ter para gerenciar a crise que o país enfrentava. Neste trajeto discursivo, pudemos encontrar semelhanças do discurso da mídia institucional com o de Obama no que tange às representações sobre como governar sustentadas em uma FD que concebe como prática administrativa ideal aquela que emana da concepção de sujeito de direito. Em relação ao discurso da mídia autocomandada, a análise do boato permitiu que viessem à tona dizeres pautados em uma FD que estabelece interditos no que se refere ao tripé candidato-religião-presidência. O estudo concluiu também que, no dircurso de campanha de Barack H. Obama, a representabilidade é construída por dizeres que estabelecem ligações com elementos do discurso fundador dos E.U.A., o que acaba fazendo com que o acontecimento discursivo seja produzido por meio da materialidade do corpo do candidato. / The aim of this study was to investigate what meanings effects regarding to representability were produced in Barack Obamas election campaign speeches and in the texts produced by the American media (mass communication and also mass- selfcommunication) in 2008. This study has been informed by concepts introduced by Michel Pêcheux in Discourse Analysis such as enunciation and imaginary formations, which were also articulated in my discussion of the concept of identity from Cultural Studies (Rose, 2001; Woodward 2000). The analysis of the materials that make up our corpus showed that, in relation to the representability process, the mass media discourse focused in its discourse on the aspects required of a new representative of the country in order to manage the economic crisis that the country was facing at that time. In this case, the discourse of the institutional media shows similarities with Obama\'s speech on the representations of how to govern. Both samples hold their speeches on a discursive formation which conceives that the ideal administrative practice derives from the conception of the legal subject. Regarding the discourse of mass self-communication, the analysis of rumors about the candidates religious beliefs allowed the presence of a discursive formation that sets up as meaning effects restricted ideas of representability to certain elements. With regard to Obamas 2008 campaign speech, the analysis showed that representability is built through the values that base on the founding discourse of the American nation. By emphasizing this aspect, the candidate\'s 2008 campaign speeches negate the possibility of new networks of meaning that could produce an original discursive event.

1 Topological and Geometrical Combinatorics Martin Tancer Abstract The task of the thesis is to present several new results on topological methods in combinatorics. The results can be split into two main streams. The first stream regards intersection patterns of convex sets. It is shown in the thesis that finite projective planes cannot be intersection patterns of convex sets of fixed dimension which answers a question of Alon, Kalai, Matoušek and Meshulam. Another result shows that d-collapsibility (a necessary condition on properties of in- tersection patterns of convex sets in dimension d) is NP-complete for recognition if d ≥ 4. In addition it is shown that d-collapsibility is not a necessary condition on properties of intersection patterns of good covers, which disproves a conjecture of G. Wegner from 1975. The second stream considers algorithmic hardness of recognition of simplicial com- plexes embeddable into Rd . The following results are proved: It is algorithmically un- decidable whether a k-dimensional simplicial complex piecewise-linearly embeds into Rd for d ≥ 5 and k ∈ {d−1, d}; and this problem is NP-hard if d ≥ 4 and d ≥ k ≥ 2d−2 3 .

Práce se zabývá otázkami reprezentace zvolené částky pomocí libovolného množství mincí předep- saného typu. V první kapitole odvozujeme vzorce pro počet nereprezentovatelných částek a hodnotu největší nereprezentovatelné částky pro dvoumincové systémy. Dále ukazujeme grafový algoritmus pro výpočet Frobeniova čísla a d·kaz NP-úplnosti rozhodovacího problému reprezentovatelnosti zvolené částky v systému s více mincemi. V druhé kapitole se zabýváme výpočtem počtu reprezentací částky zvláš' v systémech o dvou nebo více mincích. Ve třetí kapitole se věnujeme otázce, zda lze ve zvoleném systému mincí použít hladový algoritmus pro nalezení reprezentace částky pomocí nejmenšího možného množství mincí. Poslední kapitola obsahuje sbírku řešených logických úloh o mincích. 1

The problem of polynomial representability of functions is central to many branches of mathematics. If the underlying set is a finite field, every function can be represented as a polynomial. In this thesis we consider polynomial representability over a special class of finite rings, namely, Zpn, where p is a prime and n is a positive integer. This problem has been studied in literature and the two notable results were given by Carlitz(1965) and Kempner(1921).While the Kempner’s method enumerates the set of distinct polynomial functions, Carlitz provides a necessary and sufficient condition for a function to be polynomial using Taylor series. Further, these results are existential in nature. The aim of this thesis is to provide an algorithmic characterization, given a prime p and a positive integer n, to determine whether a given function over Zpn is polynomially representable or not. Note that one can give an exhaustive search algorithm using the previous results. Our characterization involves describing the set of polynomial functions over Zpn with a ‘suitable’ generating set. We make use of this result to give an non-exhaustive algorithm to determine whether a given function over Zpn is polynomial representable.nβ

Cette thèse étudie deux problèmes de nature géométrique et topologique associés au crochet de Poisson sur les variétés symplectiques. Le premier problème porte sur la notion de submersion symplectique que nous introduisons dans le présent texte et qui généralise la notion de symplectomorphisme. Il s'avère qu'une submersion symplectique est un morphisme de Poisson : il s'agit d'une application entre variétés symplectiques qui préserve le crochet de Poisson. Notre intérêt pour ces fonctions réside dans le fait que le théorème de non-tassement de Gromov porte sur l'aire minimale possible pour les images des submersions symplectiques (allant d'une boule symplectique vers le plan symplectique) obtenues comme compositions d'un plongement symplectique dans l'espace symplectique euclidien de dimension 2n et de la projection standard vers le plan de coordonnées conjuguées (p_1, q_1). Nous investiguons le problème inverse dit « de représentabilité » : nous obtenons des conditions nécessaires et suffisantes pour qu'une submersion symplectique comme ci-dessus se factorise comme précédemment à travers un plongement ou une immersion symplectique dans l'espace euclidien. Nous montrons par ailleurs qu'il existe une submersion symplectique qui ne se factorise pas de la sorte à travers une immersion et qu'il existe aussi une submersion symplectique qui se factorise de la sorte à travers une immersion, mais pas à travers un plongement. Le deuxième problème porte sur la conjecture du crochet de Poisson de Polterovich. Étant donné une variété symplectique (M, omega) et un recouvrement U de M, nous pouvons définir l'invariant pb(F) associé à une partition de l'unité F subordonnée à U, qui est une sorte de norme sur les crochets de Poisson entre les paires de fonctions de la partition. En dénotant e(U) l'énergie de disjonction de Hofer maximale d'un ouvert du recouvrement U, la conjecture demande s'il existe une constante positive C indépendante de U et de F telle que le produit de pb(F) et de e(U) soit supérieur à C. Cette conjecture a été établie récemment par Buhovski-Logunov-Tanny dans le cas des surfaces ; en nous inspirant de travaux antérieurs de Buhovski-Tanny, nous avons aussi démontré la conjecture pour les surfaces de genre plus grand que 1. Nous exposons notre approche dans le second chapitre de cette thèse. À l'aide des submersions symplectiques, nous généralisons nos méthodes afin d'attaquer la conjecture en dimensions supérieures ; nous obtenons ainsi une nouvelle preuve d'un théorème de Polterovich et de Buhovski-Tanny concernant l'invariant pb pour des recouvrements formés de petits ouverts. Afin de rendre cette thèse aussi accessible et auto-suffisante que possible, nous débutons par une introduction à la topologie symplectique. Des annexes recueillent les faits plus particuliers que nous utilisons tout au long de ce travail. / This thesis studies two problems of geometric and topological nature associated to the Poisson bracket on symplectic manifolds. The first problem concerns the notion of "symplectic submersion" that we introduce here and which generalizes the concept of symplectomorphism. A symplectic submersion turns out to be a Poisson morphism, namely a map between symplectic manifolds which preserves the Poisson bracket. Our interest in those maps stems from the fact that Gromov's nonsqueezing theorem is a statement about the minimal area possible for the images of the symplectic submersions (going from a symplectic ball to a symplectic plane) which are compositions of a symplectic embedding into the Euclidean symplectic space and of the standard projection onto the plane of conjugated variables (p_1, q_1). We investigate the inverse "representability" problem: we give necessary and sufficient conditions for a symplectic submersionas above to factorize in the previous way either through a symplectic embedding or through a symplectic immersion into Euclidean space. We show moreover that there exists a symplectic submersion which does not factorize in this way through an immersion, and also that there exists a symplectic submersion which does factorize in this way through an immersion, but not through an embedding. The second problem concerns Polterovich's Poisson bracket conjecture. Given a symplectic manifold (M, omega) and an open cover U of M, we can define the invariantpb(F) of a partition of unity F subordinated to U, which is a sort of norm on the pairwise Poisson brackets of the functions in F. Denoting e(U) the maximal Hofer displacement energy of a set in U, the conjecture asks whether there exists a positive constant C independent of U and F such that the product of pb(F) and e(U) is greater than C. This conjecture was proved recently by Buhovsky-Logunov-Tanny in the case of surfaces; based on earlier work of Buhovsky-Tanny , we also proved the conjecture for surfaces of genus one and above. We present our approach in the second chapter of this thesis. Using symplectic submersions, we generalize our methods in order to tackle the conjecture in higher dimensions; in particular, we obtain a new proof of a theorem of Polterovich and Buhovsky-Tanny about the pb invariant of covers made up of small open sets. In order to make this thesis as accessible and self-contained as possible, we first give an introduction to symplectic topology. The appendices also collect the more specialized facts we use throughout this work.

Network reconstruction is the process of recovering a unique structured representation of some dynamic system using input-output data and some additional knowledge about the structure of the system. Many network reconstruction algorithms have been proposed in recent years, most dealing with the reconstruction of strictly proper networks (i.e., networks that require delays in all dynamics between measured variables). However, no reconstruction technique presently exists capable of recovering both the structure and dynamics of networks where links are proper (delays in dynamics are not required) and not necessarily strictly proper.The ultimate objective of this dissertation is to develop algorithms capable of reconstructing proper networks, and this objective will be addressed in three parts. The first part lays the foundation for the theory of mathematical representations of proper networks, including an exposition on when such networks are well-posed (i.e., physically realizable). The second part studies the notions of abstractions of a network, which are other networks that preserve certain properties of the original network but contain less structural information. As such, abstractions require less a priori information to reconstruct from data than the original network, which allows previously-unsolvable problems to become solvable. The third part addresses our original objective and presents reconstruction algorithms to recover proper networks in both the time domain and in the frequency domain.