The problem of finding and classifying constant curvature metrics with conical singularities has a long history bringing together several different areas of mathematics. This talk will focus on the particularly difficult spherical case where many new phenomena appear. When some of the cone angles are bigger than $2\pi$, uniqueness fails and existence is not guaranteed; smooth deformation is not always possible and the moduli space is expected to have singular strata. I will give a survey of several recent results regarding this singular uniformization problem, connecting microlocal techniques with complex analysis and synthetic geometry. Based on joint works with Rafe Mazzeo and Bin Xu.

This talk will be a (very gentle) introduction of real blow ups with applications in singular geometry. We will start with two motivating toy problems, including an eigenvalue resolution of self-adjoint matrices and a simple Fredholm operator on a manifold with boundary. Then if time permits we will mention some applications in singular uniformization problems, where the resolution or compactification via blow ups plays an important role in understanding the moduli spaces of constant curvature metrics with cusp and conical singularities.

The subject of this talk is the moduli space of Riemann surfaces, which is the set of isometry classes of constant curvature metrics on a surface. The cotangent space of the moduli space is given by holomorphic quadratic differentials, and there is a natural Weil-Petersson metric defined by an $L^2$-type pairing. I will discuss the behavior of the moduli space when approaching the boundary of the Deligne-Mumford compactification, and show how to use tools from microlocal analysis to understand the degeneration of the Weil-Petersson metric.