Symplectic Geometry

Code:

M425

Acronym:

M425

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2011/2012 - 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	3	PE do Mestrado em Matemática	1	-	7,5	-
			2

Teaching language

Portuguese

Objectives

The aims of this course are the study of the geometry in symplectic vector spaces and in symplectic manifolds.
Students should recognize the main similarities (and differences) between symplectic and Riemannian geometry, as well as the importance of symplectic geometry in the study of conservative mechanical systems.

Program

1. Symplectic geometry in vector spaces: (a) symplectic vector spaces, isotropic, coisotropic, symplectic and Lagrangian subspaces; (b) the group of linear symplectomorphisms; (c) existence of a symplectic basis; (d) Gromov's nonsqueezing theorem (affine case), linear symplectic width.

2. Differential calculus on differentiable manifolds: (a) differential forms, exterior product, pullback, exterior derivative; (b) vector fields, derivations, Lie bracket, flow; (c) chains in manifolds and their boundary, integration of differential forms, Stokes' theorem; (d) Cartan's differential calculus; (e) De Rham's cohomology, homotopy operator and Poincaré's lemma, orientable manifolds.

3. Symplectic geometry in differentiable manifolds: (a) symplectic manifolds, symplectomorphisms, symplectic and Hamiltonian vector fields, Hamilton's equations; (b) Poisson bracket, the Hamiltonian version of Noether's theorem; (c) Moser's method and Darboux-Weinstein's theorem; (d) Isotropic, coisotropic, symplectic and Lagrangian submanifolds, product symplectic structure, graph of a symplectomorphism.

4. Symplectic invariants
or, alternatively
4. Hamiltonian actions and moment maps.

Mandatory literature

000071863. ISBN: 0-387-96890-3
000052721. ISBN: 0-19-851177-9
000056783

Complementary Bibliography

000045791. ISBN: 3-7643-5066-0
000047233

Teaching methods and learning activities

Lectures of exposition of the theoretical topics, complemented with examples of application and assignments to be solved by the students.

Evaluation Type

Distributed evaluation without final exam

Assessment Components

Description	Type	Time (hours)	Weight (%)	End date
Attendance (estimated)	Participação presencial	64,00
	Total:	-	0,00

Eligibility for exams

To be eligible for exam, students should
1. attend at least two thirds of the estimated number of lectures;
2. submit the assignments given for evaluation purposes.

Calculation formula of final grade

Final grade is obtained by adding up the marks in the assignments with the mark obtained in the final exam.

Examinations or Special Assignments

Nor applicable.

Special assessment (TE, DA, ...)

Students with special status can be skip the rules for eligibility for exams.
The calculation of the final grade will follow the above described rules.

Classification improvement

Improvement of the final grade will apply only to the mark obtained in the final exam.