Mathematics Colloquium

Many problems in low-dimensional geometry can be approached from a number of different perspectives, and the general question about identifying canonical metrics on surfaces with prescribed singularities is no exception. The history of the problem of finding and classifying constant curvature metrics with prescribed conic singularities includes a long program using the calculus of variations by Malchiodi and his collaborators, and other approaches using topology and synthetic geometry. However, a complete solution has remained elusive. I will describe recent work, joint with Xuwen Zhu, which gives a new and broader existence theorem and shows that the nature of the moduli space of solutions has a different nature than had perhaps been anticipated.