- An introductory course to smooth manifolds is an absolute prerequisite, e.g., a course equivalent to the content of "An Introduction to Manifolds", Loring W. Tu, or equivalent to my lecture;
- Basics of the theory of Lie groups and Lie algebras: at least the definitions of Lie groups and Lie algebras, and the definition of the differentiation of a Lie group morphism. Read for example any other introductory text to smooth manifolds, as in the sources above, or Chapter 3 of "Foundations of differentiable manifolds and Lie groups", Frank W. Warner, Graduate Texts in Mathematics, 94. Springer-Verlag, New York-Berlin, 1983;
- Basic knowledge of symplectic geometry will be very useful, but will be not required. If needed, we will review some basics of symplectic geometry at the beginning of the course.
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, theory of Lie groupoids and Lie algebroids, 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;
- Linearization and Conn's theorem;
- Submanifolds in Poisson geometry;
- Dirac structures;
- Symplectic realizations;
- Poisson cohomology;
- Lie algebroids and Lie groupoids;
Ioan Marcut (RU)