Recent days have seen a rapid developments on shifted Poisson and symplectic structures on (derived) differentiable or algebraic stacks. A differentiable stack is, roughly speaking, a Lie groupoid up to Morita equivalence, and the stack represented by a symplectic groupoid of a Poisson manifold naturally has a 1- shifted symplectic structure. There have also been remarkable recent advances in other geometries, such as Dirac geometry and generalized complex geometry, that generalize Poisson geometry and have Lie groupoids and Lie algebroids at their cores. Many basic concepts and constructions in these geometries can be rephrased using the language of differential stacks, and such reformulations put these geometric structures in vastly new perspectives and establish further connections with other fields of mathematics such as algebraic geometry, deformation theory and high category theory.
Specific topics covered :
- Integrations of Poisson and Dirac structures
- Generalized complex geometry and mirror symmetry
- Multiplicative structures on Lie groupoids and stacks
- Shifted symplectic geometry
- Higher Lie groupoids and higher gauge theory