This thesis concerns the use of perverse sheaves with coefficients in commutative rings and in particular, fields of positive characteristic, in the study of modular representation theory. We begin by giving a new geometric interpretation of classical connections between the representation theory of the general linear groups and symmetric groups. We then survey work, joint with D. Juteau and G. Williamson, in which we construct a class of objects, called parity sheaves. These objects share many properties with the intersection cohomology complexes in characteristic zero, including a decomposition theorem and a close relation to representation theory. The final part of this document consists of two computations of IC stalks in the nilpotent cones of sl₃and sl₄. These computations build upon our calculations in sections 3.5 and 3.6 of (31), but utilize slightly more sophisticated techniques and allow us to compute the stalks in the remaining characteristics. / text

Approved for public release, distribution is unlimited / Radar polarimetry is a recent development seeing active research only in the last few decades. The phenomenon that optimal (maximal power) reflected fields exist in both the co-polarized and cross polarized channels of the receiving radar antenna was first introduced by Kennaugh and Huynen. Current research efforts focus on target scattering matrices and relating them to physical attributes of the target. This thesis provides a comprehensive survey of the polarimetry theories that have been put forth by various researchers to characterize, manipulate and optimize target radar returns via polarization states. One such theory is the Target Decomposition (TD) theorem that seeks to decompose the target returns into individual scattering mechanisms. The topic of optimization of polarization states of the incident field for maximizing power return is also examined. Two models are implemented in Matlab to verify and demonstrate these polarimetry theories. The first model uses TD theorems to simulate foliage clutter and study its effect on the polarization of the incident electric field. A (simulated) static dihedral target is introduced and its effect on wave polarization is also simulated. The second model studies optimization of polarization states. Both models are able to produce the expected results for known canonical targets. / Civilian, Republic of Singapore

Given a random dynamical system T constructed from Jablonski transformations, consider its Perron-Frobenius operator P_T. We prove a weak form of the Lasota-Yorke inequality for P_T and thereby prove the existence of BV- invariant densities for T. Using the Spectral Decomposition Theorem we prove that the support of an invariant density is open a.e. and give conditions such that the invariant density for T is unique. We study the asymptotic behavior of the Markov operator P_T, especially when T has a unique absolutely continuous invariant measure (ACIM). Under the assumption of uniqueness, we obtain spectral stability in the sense of Keller. As an application, we can use Ulam's method to approximate the invariant density of P_T.

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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.

L’objectif de ce mémoire est d’introduire une nouvelle méthode smoothed particle hydrodynamics (SPH) implicite purement lagrangienne, pour la résolution des équations de Navier- Stokes incompressibles bidimensionnelles en présence d’une surface libre. Notre schéma de discrétisation est basé sur celui de Kéou Noutcheuwa et Owens [19]. Nous avons traité la surface libre en combinant la méthode multiple boundary tangent (MBT) de Yildiz et al. [43] et les conditions aux limites sur les champs auxiliaires de Yang et Prosperetti [42]. Ce faisant, nous obtenons un schéma de discrétisation d’ordre $\mathcal{O}(\Delta t ^2)$ et $\mathcal{O}(\Delta x ^2)$, selon certaines contraintes sur la longueur de lissage $h$. Dans un premier temps, nous avons testé notre schéma avec un écoulement de Poiseuille bidimensionnel à l’aide duquel nous analysons l’erreur de discrétisation de la méthode SPH. Ensuite, nous avons tenté de simuler un problème d’extrusion newtonien bidimensionnel. Malheureusement, bien que le comportement de la surface libre soit satisfaisant, nous avons rencontré des problèmes numériques sur la singularité à la sortie du moule. / The objective of this thesis is to introduce a new implicit purely lagrangian smoothed particle hydrodynamics (SPH) method, for the resolution of the two-dimensional incompressible Navier-Stokes equations in the presence of a free surface. Our discretization scheme is based on that of Kéou Noutcheuwa et Owens [19]. We have treated the free surface by combining Yildiz et al. [43] multiple boundary tangent (MBT) method and boundary conditions on the auxiliary fields of Yang et Prosperetti [42]. In this way, we obtain a discretization scheme of order $\mathcal{O}(\Delta t ^2)$ and $\mathcal{O}(\Delta x ^2)$, according to certain constraints on the smoothing length $h$. First, we tested our scheme with a two-dimensional Poiseuille flow by means of which we analyze the discretization error of the SPH method. Then, we tried to simulate a two-dimensional Newtonian extrusion problem. Unfortunately, although the behavior of the free surface is satisfactory, we have encountered numerical problems on the singularity at the output of the die.