Simplified Geometry is a branch of Differential Geometry with historical roots in the geometric formulation of classical mechanics in the 19th century, known as the “Hamiltonian formalism”. Its recent developments are the fruit of its close relationship with diverse areas of contemporary mathematics (including topology, dynamics and complex geometry) and mathematical physics.

In the simplicial field, the so-called “Poisson brackets” (originated in the classical works of Poisson, Jacobi, Lie and Hamilton) play a prominent role and lead to the concept of “Poisson varieties”, which generalize simplicial varieties. Poisson geometry has become an active field of research since the 1980s, combining techniques from simplical geometry, foliation theory and Lie theory. On the other hand, Poisson structures arise naturally as semiclassical limits of quantum systems and can be seen as intermediate objects between Differential Geometry and the world of non-commutative algebras.

The main lines of research currently being carried out at IMPA are:

  • Equivariant simplicial geometry: Hamiltonian actions and moment applications;
  • Poisson varieties and related geometries: Dirac structures and Courant algebroids, generalized complex geometry;
  • Lie groupoids and algebroids;
  • Poisson geometry and deformation quantization; relations with non-commutative geometry.