Geometry & Analysis Seminar Winter 2018

The Ricci flow applied to metrics of the form $dx^2 + u(x)^2 g_{S^p}$ has been a source of interesting dynamics of the Ricci flow that include the formation of slow blow-up degenerate neckpinch singularities and the forward continuation of the flow through neckpinch singularities. A natural next source of examples is then the Ricci flow of doubly-warped product metrics $dx^2 + u(x)^2 g_{S^p} + v(x)^2 g_{S^q}$. In this case, the Ricci flow is equivalent to a degenerate parabolic system or, by fixing a certain choice of gauge, a parabolic system with a nonlocal term. The doubly-warped product structure also allows for a larger collection of singularity models to appear compared to the singly-warped case. Indeed, formal matched asymptotic expansions suggest that some non-generic set of initial metrics on a compact manifold form finite-time Type II singularities modeled on a Ricci-flat cone at an appropriate scale. I will outline the formal matched asymptotics of this singularity formation and discuss the topological argument used to make such a formal construction rigorous. Along the way, we’ll draw comparisons to similar constructions for the mean curvature flow and a class of semilinear heat equations.

The Ends of the Hitchin Moduli Space

Hitchin's equations are a system of gauge theoretic equations on a Riemann surface that are of interest in many areas including representation theory, Teichm\"uller theory, and the geometric Langlands correspondence. In this talk, I'll describe what solutions of $SL(n,\mathbb{C})$-Hitchin's equations ``near the ends'' of the moduli space look like, and the resulting compactification of the Hitchin moduli space. Wild Hitchin moduli spaces are an important ingredient in this construction. This construction generalizes Mazzeo-Swoboda-Weiss-Witt's construction of $SL(2,\mathbb{C})$-solutions of Hitchin's equations where the Higgs field is "simple.''

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In this talk, we start with the inverse problem for semilinear wave equations on a 4-dimensional Lorentzian manifold. We present the results of determining the conformal and isometry class of the Lorentzian metric, and some information about the nonlinear term. A key component of the method is the (microlocal) analysis of the nonlinear interaction of (distorted) plane waves, and to show that new waves are generated from such interactions.  Next, we discuss the inverse problems for the Einstein equations in general relativity, that is the determination of space-time structures using gravitational or electromagnetic waves.