Smooth asymptotics for collapsing Calabi-Yau metrics

Yau's solution of the Calabi conjecture provided the first nontrivial examples of Ricci-flat Riemannian metrics on compact manifolds. Attempts to understand the degeneration patterns of these metrics in families have revealed many remarkable phenomena over the years. I will review some aspects of this story and present recent joint work with Valentino Tosatti where we obtain a complete asymptotic expansion of the degenerating metrics in a natural class of examples. This relies on a new analytic method where each additional term of the expansion arises as the obstruction to proving a uniform bound on one additional derivative of the remainder.