Moduli Theory and Singularities

Abstract: The construction of compact moduli spaces of higher dimensional varieties has been an open problem for a long time. It was finally completed about 2 years ago, mainly due to the work of János Kollár, but also utilizing the work of many other researchers. Kollár's work on the topic eclipsed more than 3 decades and included many obstacles along the way. 

I will talk about the main challenges presented by this problem and their solutions. Time permitting I will also mention a result, joined with Kollár, in which we prove that the conditions of KSB/A stability need only be checked in codimension 2 and the rest is automatic.