My thesis is mostly concerned with algebraic and combinatorial aspects of the theory of hyperplane arrangements. More specifically, I study the Orlik-Terao algebra of a hyperplane arrangement and the broken circuit complex of a matroid. The Orlik-Terao algebra is a useful tool for studying hyperplane arrangements, especially for characterizing some non-combinatorial properties. The broken circuit complex, on the one hand, is closely related to the Orlik-Terao algebra, and on the other hand, plays a crucial role in the study of many combinatorial problem: the coefficients of the characteristic polynomial of a matroid are encoded in the f-vector of the broken circuit complex of the matroid. Among main results of the thesis are characterizations of the complete intersection and Gorenstein properties of the broken circuit complex and the Orlik-Terao algebra. I also study the h-vector of the broken circuit complex of a series-parallel network and relate certain entries of that vector to ear decompositions of the network. An application of the Orlik-Terao algebra in studying the relation space of a hyperplane arrangement is also included in the thesis.

Este trabalho tem como objetivo apresentar um estudo detalhado dos fundamentos da Geometria Complexa, ressaltando seus aspectos geométricos, topológicos e analíticos. Começando com materiais preliminares, como resultados básicos sobre funções holomorfas de uma ou mais variáveis e a definição e primeiros exemplos de variedades complexas, passamos a uma introdução à teoria de feixes e sua cohomologia, ferramenta indispensável para o restante do trabalho. Após um estudo sobre fibrados de linha e divisores damos atenção à Geometria de Kähler e alguns de seus resultados centrais, como por exemplo o Teorema da Decomposição de Hodge, o Teorema ``Difícil\'\' e o Teorema das $(1,1)$-classes de Lefschetz. Em seguida, nos dedicamos ao estudo dos fibrados vetoriais complexos e sua geometria, abordando os conceitos de conexões, curvatura e Classes de Chern. Terminamos o trabalho descrevendo alguns aspectos da topologia de variedades complexas, como o Teorema dos Hiperplanos de Lefschetz e algumas de suas consequências. / The main goal of this work is to present a detailed study of the foundations of Complex Geometry, highlighting its geometric, topological and analytical aspects. Beginning with a preliminary material, such as the basic results on holomorphic functions in one or more variables and the definition and first examples of a complex manifold, we move on to an introduction to sheaf theory and its cohomology, an essential tool to the rest of the work. After a discussion on divisors and line bundles we turn attention to Kähler Geometry and its central results, such as the Hodge Decomposition Theorem, the Hard Lefschetz Theorem and the Lefschetz Theorem on $(1,1)$-classes. After that, we study complex vector bundles and its geometry, focusing on the concepts of connections, curvature and Chern classes. Finally, we finish by describing some aspects of the topology of complex manifolds, such as the Lefschetz Hyperplane Theorem and some of its consequences.

Este trabalho tem como objetivo apresentar um estudo detalhado dos fundamentos da Geometria Complexa, ressaltando seus aspectos geométricos, topológicos e analíticos. Começando com materiais preliminares, como resultados básicos sobre funções holomorfas de uma ou mais variáveis e a definição e primeiros exemplos de variedades complexas, passamos a uma introdução à teoria de feixes e sua cohomologia, ferramenta indispensável para o restante do trabalho. Após um estudo sobre fibrados de linha e divisores damos atenção à Geometria de Kähler e alguns de seus resultados centrais, como por exemplo o Teorema da Decomposição de Hodge, o Teorema ``Difícil\'\' e o Teorema das $(1,1)$-classes de Lefschetz. Em seguida, nos dedicamos ao estudo dos fibrados vetoriais complexos e sua geometria, abordando os conceitos de conexões, curvatura e Classes de Chern. Terminamos o trabalho descrevendo alguns aspectos da topologia de variedades complexas, como o Teorema dos Hiperplanos de Lefschetz e algumas de suas consequências. / The main goal of this work is to present a detailed study of the foundations of Complex Geometry, highlighting its geometric, topological and analytical aspects. Beginning with a preliminary material, such as the basic results on holomorphic functions in one or more variables and the definition and first examples of a complex manifold, we move on to an introduction to sheaf theory and its cohomology, an essential tool to the rest of the work. After a discussion on divisors and line bundles we turn attention to Kähler Geometry and its central results, such as the Hodge Decomposition Theorem, the Hard Lefschetz Theorem and the Lefschetz Theorem on $(1,1)$-classes. After that, we study complex vector bundles and its geometry, focusing on the concepts of connections, curvature and Chern classes. Finally, we finish by describing some aspects of the topology of complex manifolds, such as the Lefschetz Hyperplane Theorem and some of its consequences.

Efficient memory usage is crucial for data-intensive applications as a smaller memory footprint ensures better cache performance and allows one to run a larger problem size given a axed amount of main memory. The solutions found by existing techniques for automatic storage optimization for arrays in a new loop-nests, which minimize the storage requirements for the arrays, are often far from good or optimal and could even miss nearly all storage optimization potential. In this work, we present a new automatic storage optimization framework and techniques that can be used to achieve intra-array as well as inter-array storage reuse within a new loop-nests with a pre-determined schedule. Over the last two decades, several heuristics have been developed for achieving complex transformations of a new loop-nests using the polyhedral model. However, there are no comparably strong heuristics for tackling the problem of automatic memory footprint optimization. We tackle the problem of storage optimization for arrays by formulating it as one of ending the right storage partitioning hyperplanes: each storage partition corresponds to a single storage location. Statement-wise storage partitioning hyperplanes are determined that partition a unit end global array space so that values with overlapping live ranges are not mapped to the same partition. Our integrated heuristic for exploiting intra-array as well as inter-array reuse opportunities is driven by a fourfold objective function that not only minimizes the dimensionality and storage requirements of arrays required for each high-level statement, but also maximizes inter-statement storage reuse. We built an automatic polyhedral storage optimizer called SMO using our storage partitioning approach. Storage reduction factors and other results we report from SMO demon-strate the e activeness of our approach on several benchmarks drawn from the domains of image processing, stencil computations, high-performance computing, and the class of tiled codes in general. The reductions in storage requirement over previous approaches range from a constant factor to asymptotic in the loop blocking factor or array extents { the latter being a dramatic improvement for practical purposes. As an incidental and related topic, we also studied the problem of polyhedral compilation of graphical data programs. While polyhedral techniques for program transformation are now used in several proprietary and open source compilers, most of the research on poly-herald compilation has focused on imperative languages such as C, where the computation is species in terms of statements with zero or more nested loops and other control structures around them. Graphical data ow languages, where there is no notion of statements or a schedule specifying their relative execution order, have so far not been studied using a powerful transformation or optimization approach. The execution semantics and ref-eventual transparency of data ow languages impose a di errant set of challenges. In this work, we attempt to bridge this gap by presenting techniques that can be used to extract polyhedral representation from data ow programs and to synthesize them from their equivalent polyhedral representation. We then describe Polyglot, a framework for automatic transformation of data ow programs that we built using our techniques and other popular research tools such as Clan and Pluto. For the purpose of experimental evaluation, we used our tools to compile LabVIEW, one of the most widely used data ow programming languages. Results show that data ow programs transformed using our framework are able to outperform those compiled otherwise by up to a factor of seventeen, with a mean speed-up of 2.30 while running on an 8-core Intel system.

We consider the distribution of the zero and typical cells of a (homogeneous) Poisson hyperplane tessellation. We give a direct proof adapted to our setting of the well known Complementary Theorem. We provide sharp bounds for the tail distribution of the number of facets. We also improve existing bounds for the tail distribution of size measurements of the cells, such as the volume or the mean width. We improve known results about the generalised D.G. Kendall's problem, which asks about the shape of large cells. We also show that cells with many facets cannot be close to a lower dimensional convex body. We tacle the much less study problem of the number of facets and the shape of small cells. In order to obtain the results above we also develop some purely geometric tools, in particular we give new results concerning the polytopal approximation of an elongated convex body.