Picture "The Thinker" in a heavy rain: this gives us a visualization of the Morse-Smale flow on a two dimensional manifold (which is the surface of the sculpture, of course). Generally speaking, a Morse-Smale flow on a compact Riemannian manifold is the negative gradient flow of a Morse function. The collection of Morse flows induced by a given Morse function, compactified with so-called broken flow lines, is named the (compactified) moduli space of Morse flows, and it is well-studied as a topological manifold with corners itself. However, understanding the differential structure of the compactified moduli space is surprisingly difficult, as the moduli space fails to be a differentiable manifold with corners in general. In this talk, we will see that a generic moduli space of Morse-Smale flows resembles a smooth manifold with analytical corners, which is a strictly weaker type of corners than the classical model; and we will give an outline of the proof.

This work is supervised by Prof. Dominic Joyce.