Richard Hepworth (University of Sheffield)
Suppose that a compact Lie group G acts smoothly on a manifold M. Then the quotient M/G is a perfectly good topological space, but it might not be a manifold. This problem can be resolved by extending our viewpoint from the world of manifolds to the world of differentiable stacks, where the quotient always exists. We must then ask whether the usual tools of differential geometry can be extended to the world of stacks.
I will begin by introducing stacks and explaining why they are useful, and then go on to explain how the notions of tangent bundle, vector field and flow can be extended to this new setting.