Differentiable Manifolds

Code:

M4135

Acronym:

M4135

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2021/2022 - 2S

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

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
M:M	13	Plano Oficial do ano letivo 2021	1	-	9	84	243

Teaching language

Suitable for English-speaking students

Objectives

Introduction to the theory of differentiable manifolds.

Learning outcomes and competences

Students should dominate basic concepts in the theory of differentiable manifolds as well as become autonomous in the theory, using the vast literature available.

Working method

Presencial

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

First degree with a strong mathematics component, namely, two semester of linear algebra, three semesters of real analysis (including differential equations).

Program

Differentiable manifolds. Derivative of differentiable maps between manifolds, submersions, immersions and embeddings. Tangent and cotangent buncle of a manifold. 
Transversality, homotopy and stability, Sard's Theorem and Morse functions, Whitney's Theorem, partitions of unity, tubular neighbourhoods, genericity and its relation to transversality.
Vector fields on a manifold: flow, derivations and one-parameter group of diffeomorphisms, Lie groups. 
Exterior algebra, differential forms , integration of differential forms, Stokes theorem.

Mandatory literature

Victor Guillemin; Differential topology. ISBN: 0-13-212605-2
Dennis Barden; An introduction to differential manifolds. ISBN: 1-86094-355-1

Complementary Bibliography

Serge Lang; Differential manifolds. ISBN: 0-387-96113-5
John M. Lee; Introduction to smooth manifolds. ISBN: 0-387-95448-1

Teaching methods and learning activities

Lectures will include time for theoretical exposition of concepts, examples of application and also for resolution (by the students) of exercises. 
Assessments will take place either in class time or on dates pre-arranged with the students.

Evaluation Type

Distributed evaluation without final exam

Assessment Components

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

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	165,00
Frequência das aulas	78,00
Total:	243,00

Eligibility for exams

Students should be present in 75% of the classes and also in the dates where assessments and resolution of exercises for evaluation purposes take place.

Calculation formula of final grade

In the first call the final score will be the sum of the scores obtained in the following components: 
1. first assessment (worth 10 points)
2. second assessment (worth 10 points). 

In the second call the final score is the one obtained in the exam (worth 20 points).

Examinations or Special Assignments

n.a.

Internship work/project

n.a.

Special assessment (TE, DA, ...)

Students with special conditions which exempt them from exercises and assessments will have an exam in the conditions described for the second call.

Classification improvement

Improvement of classification will be allowed only in the second call.