Differentiable Manifolds

Code:

M505

Acronym:

M505

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2011/2012 - 1S

Active?	Yes
Web Page:	http://cmup.fc.up.pt/cmup/pbgothen/manifolds/
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	5	PE do Prog Inter-Univ Dout Mat	1	-	9	60	243

Teaching language

English

Objectives

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.

Program

Elements of general topology: metric and topological spaces, connectedness, compactness, quotient spaces. Topological groups. Brief reference to surfaces. Homotopy, covering spaces, fundamental group. Differentiable manifolds: local structure, Sard Lemma, transversality, vector fields and flows. Fiber bundles, tangent and cotangent bundles of a manifold. Lie derivative of vector fields. Lie algebras. Lie groups (classical). Homogeneous spaces. Differential forms, exterior derivative. Integration on manifolds. Stokes' theorem. Degree of a map. Index of a vector field. Singular and Cech cohomology. De Rham cohomology and de Rham's theorem.

Mandatory literature

Barden, D. and Thomas, C.; An introduction to differential manifolds, Imperial College Press, 2003

Complementary Bibliography

Sutherland, W.A. ; Introduction to Metric and Topological Spaces, Oxford University Press, 1975
Tu, L.W.; An Introduction to Manifolds, Springer, 2008
Conlon, L.; Differentiable Manifolds, 2nd ed., Birkhäuser, 2008
Fulton, W.; Algebraic Topology - A First Course, Springer, 1997

Teaching methods and learning activities

Lectures, problem sessions, student presentations.

Evaluation Type

Distributed evaluation without final exam

Assessment Components

Description	Type	Time (hours)	Weight (%)	End date
Attendance (estimated)	Participação presencial	44,00
Second test	Exame	2,00		2011-11-16
Third test	Exame	2,00		2011-12-15
First test	Exame	2,50		2011-10-17
	Total:	-	0,00

Eligibility for exams

The following compulsory requirements must be satisfied:
- Presence in 75% of classes;
- Approval in 75% of the (approximately) weekly homework sets.

Calculation formula of final grade

There will be three tests during the semester. The final mark is the average of the two highest marks obtained in the three tests. The mark in each individual test must be at least 7 (seven) out of 20 (twenty). Provisional dates: 13/10/2011, 16/11/2011, 15/12/2011.