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Doctoral Thesis
Full name
Thiago Filipe da Silva
Knowledge Area
Date of Defense
São Carlos, 2018
Grulha Junior, Nivaldo de Góes (President)
Frühbis-krüger, Anne
Hernandes, Marcelo Escudeiro
Pereira, Miriam da Silva
Title in English
Bi-Lipschitz invariant geometry
Keywords in English
Bi-Lipschitz equisingularity
Determinantal varieties
Double of a module
Integral closure of ideals and modules
Lipschitz saturation of a module
Abstract in English
The study about bi-Lipschitz equisingularity has been a very important subject in Singularity Theory in last decades. Many different approach have cooperated for a better understanding about. One can see that the bi-Lipschitz geometry is able to detect large local changes in curvature more accurately than other kinds of equisingularity. The aim of this thesis is to investigate the bi-Lipschitz geometry in an algebraic viewpoint. We define some algebraic tools developing classical properties. From these tools, we obtain algebraic criterions for the bi-Lipschitz equisingularity of some families of analytic varieties. We present a categorical and homological viewpoints of these algebraic structure developed before. Finally, we approach algebraically the bi-Lipschitz equisingularity of a family of Essentially Isolated Determinantal Singularities.
Title in Portuguese
Geometria Bi-Lipschitz invariante
Keywords in Portuguese
Equisingularidade bi-Lipschitz
Fecho Integral de ideais e módulos
O double de um módulo
Saturação Lipschitz de um módulo
Variedades Determinantais
Abstract in Portuguese
O estudo da equisingularidade bi-Lipschitz tem sido amplamente investigado nas últimas décadas. Diversas abordagens têm contribuído para uma melhor compreensão a respeito. Observa-se que a geometria bi-Lipschitz é capaz de detectar grandes alterações locais de curvatura com maior precisão quando comparada a outros padrões de equisingularidade. O objetivo desta tese é investigar a geometria bi-Lipschitz do ponto de vista algébrico. Definimos algumas estruturas algébricas desenvolvendo algumas propriedades clássicas. A partir de tais estruturas obtemos critérios algébricos para a equisingularidade bi-Lipschitz de algumas classes de famílias de variedades analíticas. Apresentamos uma visão categórica e homológica dos elementos desenvol- vidos. Finalmente abordamos algebricamente a equisingularidade de famílias de Singularidades Determinantais Essencialmente Isoladas.
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