Abstract: A stack is, roughly speaking, the set of equivalence classes of an equivalence relation, together with information about multiple ways in which equivalent points may be related. For example, the orbit space of a non-free group action is best understood as a stack. Thus, isomorphism classes of geometric structures are frequently treated as moduli stacks, which incorporate information about automorphisms of objects.

The precise definition of stack requires a long excursion into category theory, but there is a more concrete way of "presenting" stacks by groupoids, which are in many ways like group actions.

A notion of "cardinality" for discrete groupoids was introduced recently by Baez and Dolan, recapitulating a concept which appeared first in Lefschetz formulas in arithmetic geometry and topology, and as the Euler characteristic of an orbifold. Observing first that this notion is really an invariant of stacks rather than just of the groupoids which present them, we extend it to the differentiable case by defining a concept of "volume" for differentiable stacks. Since there is no natural measure in the smooth case like the counting measure in the discrete case, we need extra data to define the volume. This data has the form of an invariant section of a natural line bundle over the stack.

In my talk, which will not require any prior knowledge of groupoids or stacks, I will explain the notion of volume for stacks, describe basic properties and examples, and pose some open problems.