Variedades Diferenciáveis

Código:

M505

Sigla:

M505

Áreas Científicas
Classificação	Área Científica
OFICIAL	Matemática

Ocorrência: 2023/2024 - 1S

Ativa?	Sim
Unidade Responsável:	Departamento de Matemática
Curso/CE Responsável:	Programa Doutoral em Matemática - Interuniversitário

Ciclos de Estudo/Cursos

Sigla	Nº de Estudantes	Plano de Estudos	Anos Curriculares	Créditos UCN	Créditos ECTS	Horas de Contacto	Horas Totais
IUD-M	3	PE do Prog Inter-Univ Dout Mat	1	-	9	60	243

Língua de trabalho

Inglês

Objetivos

To treat the basic theory of differential manifolds.

Resultados de aprendizagem e competências

The student should acquire a thorough knowledge of the theory of differential manifolds and be able to use its tools in mathematical problem solving and research.

Modo de trabalho

Presencial

Pré-requisitos (conhecimentos prévios) e co-requisitos (conhecimentos simultâneos)

Elements of general topology. Calculus of functions of several variables. Basic algebra.

Programa

Topological and differentiable manifolds, differentiable maps; partitions of unity. Tangent and cotangent bundles; the differential of a smooth map. Immersions, submersions, and submanifolds; inverse images of regular values, transversality, Sard's Theorem; Whitney embedding.

Vector fields and flows; the Lie bracket of vector fields; Foliations, distributions and the Frobenius theorem; Lie groups and Lie algebras; Lie group and Lie algebra actions and representations; quotients, principal bundles, and homogenous spaces. Fibre bundles and Vector bundles.

Differential forms, exterior derivative. Integration on manifolds. Stokes' theorem. Elements of homological algebra; de Rham cohomology. The Poincaré Lemma. Homotopy and homotopy invariance of de Rham cohomology. Euler characteristic. The Mayer-Vietoris sequence.

Some additional topics may be treated, such as: Degree of a map; the index of a vector field with isolated singularities and the Poincaré-Hopf Theorem. Constructions with vector and fiber bundles; vector valued forms; classification of bundles. Symplectic, Riemannian, complex, or other geometric structures.

Bibliografia Obrigatória

Jaques Lafontaine; An Introduction to Differential Manifolds, Springer, 2015
Barden, D. and Thomas, C.; An introduction to differential manifolds, Imperial College Press, 2003

Bibliografia Complementar

Ib Madsen; From calculus to cohomology. ISBN: 0-521-58956-8
Raoult Bott; Differential forms in algebraic topology. ISBN: 0-387-90613-4
Tu, L.W.; An Introduction to Manifolds, Springer, 2008
Fulton, W.; Algebraic Topology - A First Course, Springer, 1997
Sutherland, W.A. ; Introduction to Metric and Topological Spaces, Oxford University Press, 1975

Métodos de ensino e atividades de aprendizagem

Lectures, problem sessions, student presentations.

Palavras Chave

Ciências Físicas > Matemática > Geometria

Tipo de avaliação

Avaliação por exame final

Componentes de Avaliação

Designação	Peso (%)
Apresentação/discussão de um trabalho científico	30,00
Trabalho escrito	70,00
Total:	100,00

Componentes de Ocupação

Designação	Tempo (Horas)
Estudo autónomo	187,00
Frequência das aulas	56,00
Total:	243,00

Obtenção de frequência

Attendance is not compulsory.

Fórmula de cálculo da classificação final

H+P

The number H (ranging from 0 to 14) is the total mark of the Homework problems, assigned approximately every 4 lectures.

The number P (ranging from 0 to 6) is the mark of the final component of the evaluation. This can be either:

Option a) A presentation (1 - hour lecture) given by the student on an advanced topic of their interest related with the course. The written notes for the presentation are handed in as well.
Oprion b) A final set of problems to solve.

Each student must decide whether they prefer option a) or b) before the end of December.

 

Avaliação especial (TE, DA, ...)

By written and/or oral exam.

Melhoria de classificação

By final exam.