In this talk we will discuss deformations of Lie groupoids and explain the construction of the cohomology which controls them. We will see some of its properties, and look at relations with other cohomologies: deformation cohomologies for classical examples, such as Lie group actions and foliations, and deformation cohomology of Lie algebroids. The deformation cohomology is then used to prove rigidity and normal form results for compact and for proper Lie groupoids. The talk is based on joint work with Marius Crainic and Ivan Struchiner.
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