Aim of the course
This course provides an introduction to the theory of Poisson structures, with a strong emphasis on the differential geometric aspects of the subject. In addition, the students becomes familiar with topics such as symplectic geometry, Lie theory, foliations theory and complex geometry, because all of these structures interact in the framework of Poisson geometry.
The following subjects are treated:
- Symplectic and Poisson structures;
- Symplectic foliations;
- Weinstein's splitting theorem;
- Submanifolds in Poisson geometry;
- Dirac structures, Generalized complex structures, Holomorphic Poisson structures;
- Poisson Lie groups;
- Symplectic realizations;
- Lie algebroids and Lie groupoids;
- Conn's theorem.
- An introductory course to smooth manifolds;
- Basics of the theory of Lie groups and Lie algebras (e.g. read Chapter 3 of "Foundations of differentiable manifolds and Lie groups", Frank W. Warner, Graduate Texts in Mathematics, 94. Springer-Verlag, New York-Berlin, 1983; or any other introductory book on smooth manifolds);
- Basic knowledge of symplectic and/or complex geometry will certainly be useful, but is not required.
Rules about Homework / Exam
There will be weekly homework assignments (40% of the final mark) and an oral exam (60% of the final mark).