Dimension 4 is the next horizon for applications of Ricci flow to topology, where the main goal is to understand the topological operations that Ricci flow generically performs at singular times. Shrinking Ricci solitons model these topological operations, and only the stable ones should arise generically.

I will present recent joint works with Olivier Biquard and Keaton Naff that determine the stability of all of the currently known shrinking Ricci solitons in dimension 4. The arguments use structural features unique to dimension four, in particular self-duality.