Introduction to Smootheology

Smooth manifolds form a successful theory in geometry that uses topology to generalize real analysis from Euclidean space to much more general shapes. For example, think of how the 2-torus locally looks like a disk in the plane at every point. This is exactly what a manifold is: a space that locally looks like a disk/ball in Euclidean space. However, in practice, mathematics is full of singular spaces, where singularities spoil the local structure, but we still need to do analysis at this locations. Think of the apex of the double cone z 2 = x 2 + y 2 in 3-space. In this talk, I will introduce you to a few attempts to generalize smooth manifolds to bigger categories in order to include singular and function spaces, and mention some of their pros and cons.