In this talk we describe some very recent leaps in our understanding of singularities for the surface diffusion flow, including finite-time pinchoff of embedded tori and self-similar immersed tori. This is joint work with Madeline Mcrae.
We prove uniqueness in the Lorentzian Calderon problem in a semiglobal
setting, comparing a potentially large perturbation of the Minkowski metric
against a small one. Our proof uses a reduced amount of data, solving a
formally determined version of the problem. In contrast to traditional
approaches, it employs neither control-theoretic arguments nor a reduction to a
geometric inverse problem via microlocal methods or high-frequency solutions.
Instead, it relies on distorted plane waves and weighted energy estimates.