Disciplina Discipline MAT5704
Teoria dos Modelos em Lógica Contínua
Model Theory in Continuous Logic

Área de Concentração: 45131

Concentration area: 45131

Criação: 03/09/2019

Creation: 03/09/2019

Ativação: 03/09/2019

Activation: 03/09/2019

Nr. de Créditos: 8

Credits: 8

Carga Horária:

Workload:

Docente Responsável:

Professor:

Ricardo Bianconi

Objetivos:

Desenvolver a Teoria dos Modelos em Lógica Contínua de modo sistemático, unificando a literatura existente e vindoura, e aplicá-la a problemas de Análise Funcional e Espaços Métricos

Objectives:

To develop Model Theory in Continuous Logic in a systematic way unifying the exsting and forthcoming literature, and apply it to problems in Functional Analysis and Metric Spaces.

Justificativa:

A Lógica Contínua tem sido aplicada a vários problemas de análise funcional atualmente, principalmente os voltados aos espaços de Banach e espaços vetoriais topológicos. Esta disciplina justifica-se pela sua aplicabilidade, não só na área de Lógica Matemática, mas também na Análise.

Rationale:

Continuous Logic has been applied recently to several problems in Functional Analysis, mainly those about Banach spaces and topological vector spaces. This course is justified by its applicability, not only on Mathematical Logic but also in Analysis.

Conteúdo:

Todos os itens devem conter exemplos de suas aplicações. 1. Estruturas métricas calibradas, sintaxe e semântica. 2. Teoremas de Completitude e Compacidade. 3. Ultraprodutos de estruturas métricas e o Teorema de Los. 4. Forcing na Lógica Contínua. 5. Saturação e categoricidade. 6. Espaços de tipos e suas topologias. 7. Estabilidade. 8. Tópicos extras escolhidos na literatura.

Content:

All items must contain examples of their applications. 1. Metric gauged structures, syntax and semantics. 2. Theorems of Completeness and Compactness. 3. Ultraproducts and Los Theorem. 4. Forcing in Continuous Logic. 5. Saturation and Categoricity. 6. Spaces of types and their topologies. 7. Stability. 8. Extra topics chosen from the literature.

Forma de Avaliação:

A avaliação dos estudantes faz-se por uma média ponderada de provas, exercícios, seminários, etc, a critério da(o) ministrante.

Type of Assessment:

The evaluation of the students may be done by a weighted average of exams, exercises, seminars, etc, at the discretion of the lecturer.

Observação:

A disciplina poderá ser ministrada também em inglês se for conveniente.

Notes/Remarks:

The subject can be taught in English if convenient.

Bibliografia:

1. I. Ben Yaacov, A. Berenstein, C. Ward Henson, A. Usvyatsov. Model Theory for Metric Structures. Em Model Theory with applications to Algebra and Analysis, vol 2, p. 315-427, Z. Chatzidakis, D. Macpherson, A. Pillay, A. Wilkie eds., London Mathematical Society Lecture Note Series: 350, Cambridge University Press, Cambridge, Inglaterra, 2008. 2. I. Ben Yaacov, Continuous first order logic for unbounded metric structures, Journal of Mathematical Logic 8 (2008), no. 2, 197-223. 3. I. Ben Yaacov and Arthur Paul Pedersen, A proof of completeness for continuous first-order logic, Journal of Symbolic Logic 75 (2010), no. 1, 168-190. 4. C. Ward Henson, J. Iovino. Ultraproducts in Analysis. Em Logic and Analysis, p.1-113, C. Ward Henson, J. Iovino A. Kechris, E. Odell eds. London Mathematical Society Lecture Note Series: 262, Cambridge University Press, Cambridge, Inglaterra, 2002. 5. I. Ben Yaacov, J. Iovino, Model theoretic forcing in analysis, Annals of Pure and Applied Logic 158 (2009), no. 3, 163-174. 6. I. Ben Yaacov and Alexander Usvyatsov, Continuous first order logic and local stability, Transactions of the American Mathematical Society 362 (2010), no. 10, 5213-5259. 7. I. Ben Yaacov, Model theoretic stability and definability of types, after A. Grothendieck, Bull. Symb. Log. 20 (2014), no. 4, 491-496. 8. J. Iovino. Applications of Model Theory to Functional Analysis. Dover Publications, Mineola, NY, EUA, 2002.

Bibliography:

1. I. Ben Yaacov, A. Berenstein, C. Ward Henson, A. Usvyatsov. Model Theory for Metric Structures. Em Model Theory with applications to Algebra and Analysis, vol 2, p. 315-427, Z. Chatzidakis, D. Macpherson, A. Pillay, A. Wilkie eds., London Mathematical Society Lecture Note Series: 350, Cambridge University Press, Cambridge, Inglaterra, 2008. 2. I. Ben Yaacov, Continuous first order logic for unbounded metric structures, Journal of Mathematical Logic 8 (2008), no. 2, 197-223. 3. I. Ben Yaacov and Arthur Paul Pedersen, A proof of completeness for continuous first-order logic, Journal of Symbolic Logic 75 (2010), no. 1, 168-190. 4. C. Ward Henson, J. Iovino. Ultraproducts in Analysis. Em Logic and Analysis, p.1-113, C. Ward Henson, J. Iovino A. Kechris, E. Odell eds. London Mathematical Society Lecture Note Series: 262, Cambridge University Press, Cambridge, Inglaterra, 2002. 5. I. Ben Yaacov, J. Iovino, Model theoretic forcing in analysis, Annals of Pure and Applied Logic 158 (2009), no. 3, 163-174. 6. I. Ben Yaacov and Alexander Usvyatsov, Continuous first order logic and local stability, Transactions of the American Mathematical Society 362 (2010), no. 10, 5213-5259. 7. I. Ben Yaacov, Model theoretic stability and definability of types, after A. Grothendieck, Bull. Symb. Log. 20 (2014), no. 4, 491-496. 8. J. Iovino. Applications of Model Theory to Functional Analysis. Dover Publications, Mineola, NY, EUA, 2002.

Tipo de oferecimento da disciplina:

Presencial

Class type:

Presencial