https://repositorio.ufpb.br/jspui/handle/tede/6873 PPGMAT 2026-10-31T18:44:28Z https://repositorio.ufpb.br/jspui/handle/123456789/39600 Título: Aut(P^3) - Órbitas do esquema de Hilbert, Hilb^{at+b}P^3 com a, b em {1, 2} Autor(es): Pereira, Antonio Fred Castro Orientador: Arancibia, Jacqueline Fabiola Rojas Abstract: The Hilbert scheme, introduced by Alexander Grothendieck in the 1960s, is a funda- mental object in algebraic geometry whose closed points parametrize closed subschemes of a projective scheme with a fixed Hilbert polynomial. This dissertation investigates the stratification of the Hilbert scheme of curves in P3, denoted by Hilb at+b P3 with a, b ∈ {1, 2}, by analyzing the orbits generated under the action of the automorphism group Aut(P3). The study focuses on the Hilbert polynomials: t + 1, t + 2, 2t + 1, and 2t + 2, which correspond geometrically to lines, the schematic union of a line and a point, conics, and the schematic union of disjoint lines or of a conic with a point in P3. Using techniques from commutative algebra, geometric invariant theory, and com- putational support from the software Macaulay2, we present an explicit classification of these orbits, the computation of their dimensions, and the description of specializa- tion relations (degenerations) among them. The results demonstrate that these spaces decompose into a finite number of orbits, revealing a well-defined geometric hierar- chy that reflects the relative position of irreducible components and the non-reduced structures of the studied curves. Editor: Universidade Federal da Paraíba Tipo: Dissertação 2026-07-17T00:00:00Z https://repositorio.ufpb.br/jspui/handle/123456789/39329 Título: Controllability and observability of coupled PDE Systems: boussinesq and thermoelasticity Autor(es): Silva, Hudson Cavalcante da Orientador: Santos, Maurício Cardoso Abstract: In this thesis, we examine two distinct problems in the context of distributed parameter systems: one concerns system control, more precisely, the issue of local null controllability, and the other concerns observability and inverse problems. First, we establish the local null controllability of the 3D Boussinesq system using a single distributed scalar control acting solely on the temperature equation. To achieve this, we first linearize the system around a suitable trajectory. Inspired by the classical work of M. Gromov and the recent contributions of J. M. Coron and P. Lissy on the Navier- Stokes equation, we employ an algebraic method to systematically reduce the number of required controls, leveraging known controllability results for systems with more controls. Finally, using an inverse function theorem, we prove the local null controllability of the nonlinear system. These results demonstrate that the Boussinesq system is locally nullcontrollable without direct action on the velocity field, highlighting the strength of the coupling between the thermal and fluid dynamic equations. Second, we investigate the optimal sensor placement problem for a linear thermoelasticity system on a bounded domain. Adopting a randomized approach to the initial data, we reduce the problem to the maximization of a spectral functional defined by the eigenfunctions of the Laplacian and a sequence of weights corresponding to the inverse of the maximum eigenvalues of a particular sequence of matrices. From a theoretical perspective, we analyze the existence of optimal sets by studying the relaxation gap between characteristic functions and density functions. We prove that if the spectral weights converge to their limit from below, the existence of a classical optimal set is guaranteed under Quantum Ergodicity (QUE) assumptions or weaker asymptotic separation conditions. We also address cases where QUE fails, such as the n-dimensional orthotope, proving existence results under specific geometric constraints. Applying this framework to the thermoelasticity system, we perform a detailed highfrequency asymptotic analysis of the spectral weights. We identify two distinct physical regimes depending on the wave speed and coupling parameters. We derive explicit algebraic sufficient conditions on the physical parameters (coupling strength, viscosity, wave speed, and time horizon) that ensure the spectral weights converge from below, thereby guaranteeing the existence of an optimal sensor domain. Editor: Universidade Federal da Paraíba Tipo: Tese 2026-01-20T00:00:00Z https://repositorio.ufpb.br/jspui/handle/123456789/38569 Título: Contributions to lineability theory and the Separable Quotient Problem in Banach Spaces Autor(es): Barbosa, Anderson Felipe de Souza Orientador: Pellegrino, Daniel Marinho Abstract: This work is structured in two parts. In the rst, we generalize, in a certain sense, two classical 2004 results of V. I. Gurariy and L. Quarta and develop relevant contributions to some recent notions of lineability and spaceability. In particular, we establish several general criteria in the context of complements of unions of vector subspaces, thereby broadening the range of applications in this framework. In the second part, we examine the Separable Quotient Problem and analyze its interaction with the theory of strongly continuous semigroups of linear operators. Moreover, we present new equivalent or suficient conditions for the problem in the context of operator ranges, providing approaches that connect geometric and analytic properties of these structures. Editor: Universidade Federal da Paraíba Tipo: Tese 2026-02-23T00:00:00Z https://repositorio.ufpb.br/jspui/handle/123456789/38051 Título: Limites de curvas planas duais via séries de Puiseux Autor(es): Custódio, Rony Héron Silva Orientador: Sousa, Wállace Mangueira de Abstract: This work addresses the limit of plane dual curves, a topic in algebraic geome- try that seeks to understand what happens to curves when they undergo degeneracy processes. In such situations, the dual of the limiting curve does not always coincide with the limit of the duals, and therefore a more careful examination is required. First, we introduce some fundamental concepts, such as Puiseux series, discri- minants of univariate polynomials, and intersections in the projective plane. Next, we address the duality between points and lines in this setting and apply this concept to the study of the dual curve of a smooth projective plane curve. We then examine how this duality behaves within flat families of curves. The main result provides a precise formula for describing the limit of such dual curves, relating its components to the duals of each component of the special fiber of the family, as well as to certain discriminants. Finally, examples illustrate the theorem and demonstrate the practical utility of the theory developed. The research thus seeks to bring the reader closer to a more intuitive understanding of how curves and their duals interact in limiting situations. Editor: Universidade Federal da Paraíba Tipo: Dissertação 2025-10-31T00:00:00Z