Complete Calabi-Yau metrics: recent progress and open problems

Complete Calabi-Yau metrics are fundamental objects in Kahler geometry arising as singularity models or “bubbles” in degenerations of compact Calabi-Yau manifolds.  The existence of these metrics and their relationship with algebraic geometry are the subjects of several long standing conjectures due to Yau and Tian-Yau.  I will describe some recent progress towards the question of existence, and explain some future directions, highlighting connections with notions of algebro-geometric stability.