Symplectic Geometry

Symplectic geometry is a branch of differential geometry with historical roots in the geometric formulation of classical mechanics of the XIX century , known as “Hamiltonian formalism”. Its recent developments, however, have deep connections with a wide range of areas in modern mathematics (including topology, dynamics, complex geometry) as well as mathematical physics.

In the context of symplectic geometry, the so-called “Poisson brackets” (originated in the classical works of Poisson, Jacobi, Lie and Hamilton) play a key role and have led to the concept of “Poisson manifolds”, which generalize symplectic manifolds. Poisson geometry started as a field in the 1980s and has since then become an active area of research, combining techniques from symplectic geometry, foliation theory and Lie theory. Poisson structures also arise as the semi-classical limit of quantum systems and can be seen as intermediate objects between differential geometry and the world of noncommutative algebras.

The main areas of research at IMPA are currently the following:

• Equivariant symplectic geometry; hamiltonian actions and momentum maps;
• Poisson manifolds and related geometries: Dirac structures, Courant algebroids, generalized complex geometry;
• Lie groupoids and Lie algebroids;
• Poisson geometry and deformation quantization; connections with non-commutative geometry.