The concept of symmetry is ubiquitous in mathematics and leads naturally to different theoretical frameworks. For instance, the standard notion of a group is suitable to study discrete symmetries (e.g. permutations), while continuous symmetries are modelled by Lie groups. Lie groupoids, introduced by Ehresmann in the 1950's, are groupoids endowed with a compatible smooth structure, and they model "point-dependent" continuous symmetries. Lie algebroids, introduced by Pradines in 1967, are "linear approximations" of Lie groupoids: they consist in vector bundles equipped with a special Lie bracket on the space of global sections. In the last few decades, these objects led to countless applications in category theory, algebraic geometry, noncommutative geometry, pseudodifferential operators, PDEs and mathematical physics.

Poisson geometry, i.e. the study of manifolds endowed with a Poisson structure, is a young field as well: it emerged between the 1970s and 1980s with the works of Lichnerowicz and Weinstein. It originates within the classical formalism of Hamiltonian mechanics, and it has developed strong connections with various areas of differential geometry, including symplectic geometry, foliation theory and complex geometry. Additionally, it has applications in both theoretical (representation theory, algebraic and differential topology, etc.) and applied mathematics (deformation quantisation, fluid dynamics, symplectic integrators, etc.).

Since the end of the 1980s, Poisson geometry has greatly benefitted from the theory of Lie groupoids and algebroids, and, in turn, has significantly contributed to the development of Lie theory. Accordingly, the goal of this course is to explore the fundamental concepts and theorems related to Lie groupoids/algebroids and to Poisson geometry, as well as their interplay, culminating with the integrability of Poisson manifolds.

The only prerequisite is a standard master course in differential geometry, covering smooth manifolds, vector fields, differential forms, and vector bundles. Previous knowledge of Lie groups/Lie algebras is helpful but not required. Attending in parallel the master course "Geometric Mechanics" can be beneficial, due to the many synergies between symplectic and Poisson geometry.