Recent developments (Alesker's theory of convex valuations) have revealed a beautiful algebraic structure underlying this array. We will describe this theory in general, and sketch the structure in the particular cases of the orthogonal and unitary groups.

At the same time these ideas offer an approach to the study of singular spaces, as the integrals involved tend to be rather insensitive to the smoothness of the objects in question. The basic construction is the normal cycle (also known as the characteristic cycle), which is a Lagrangian integral current canonically associated to a singular subspace X. In many respects this association is still very poorly understood.