Differentiable Manifolds

Code:

M505

Acronym:

M505

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2021/2022 - 1S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Doctoral Program in Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
IUD-M	7	PE do Prog Inter-Univ Dout Mat	1	-	9	60	243

Teaching language

English

Objectives

To treat the basic theory of differential manifolds.

Learning outcomes and competences

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.

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

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

Program

Elements of general topology (revision): topological spaces, connectedness, compactness, quotient spaces. Differentiable manifolds, differentiable maps. Inverse images of regular values. Transversality. Sard's Theorem. Fiber bundles, tangent and cotangent bundles of a manifold. Vector fields and flows.The Lie bracket of vector fields. Lie groups and Lie algebras. 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. Degree of a map. The index of a vector field with isolated singularities and the Poincaré-Hopf Theorem. Some additional topics may be treated.

Mandatory literature

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

Complementary Bibliography

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

Teaching methods and learning activities

Lectures, problem sessions, student presentations.

keywords

Physical sciences > Mathematics > Geometry

Evaluation Type

Evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	100,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	187,00
Frequência das aulas	56,00
Total:	243,00

Eligibility for exams

Attendance is not compulsory.

Calculation formula of final grade

The final mark is the mark obtained in the exam.

Special assessment (TE, DA, ...)

By written and/or oral exam.

Classification improvement

By final exam.