Nonlinear stability of supersymmetric spacetimes

Spacetimes with compact directions which have special holonomy, such as Calabi-Yau spaces, play an important role in supergravity and string theory. In this talk I will discuss two recent works showing the nonlinear stability of spacetimes which are the cartesian product of Minkowski space with a compact special holonomy space. I will not assume any prior knowledge of the Cauchy formulation of general relativity, and I will also explain how these results are related to claims of Penrose and Witten.