Abstract: Topological and differentiable stacks are like spaces whose points possess their own intrinsic groups of symmetry. These include several natural objects of study: quotients of manifolds by group actions, leaf spaces of foliated manifolds, and orbifolds. They have many applications in foliation theory, twisted K-theory, string topology, and symplectic and Poisson geometry, as well as applications in theoretical physics, particularly in string theory and higher gauge theory. I will give an intuitive introduction to the concept of topological and differentiable stacks. I will then provide a brief summary of some results proven in my thesis. In particular, I will speak about compactly generated stacks, which are to topological stacks what compactly generated spaces are to topological spaces. Time permitting, I then hope explain how ineffective data of an étale stack is the same as a gerbe over its effective part, and what this means intuitively.
