Functoriality of Lie groupoid convolution algebras

I will discuss the following guiding slogan: The noncommutative differential geometry of a differentiable stack is encoded in the convolution algebra of smooth functions on a Lie groupoid presentation. I will introduce the bornological convolution algebra associated to a Lie groupoid and show how this construction assembles into a 2-functor. In particular, this functorial perspective implies Morita invariance. I will also describe several examples illustrating how aspects of the transverse geometry of a Lie groupoid manifest algebraically in the convolution algebra. This is joint work with Christian Blohmann.