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.
The talk will focus on critical points of some isoperimetric-type problems, or, more precisely, volume-constrained shape optimisation problems. The celebrated Alexandrov’s Soap Bubble
Theorem and Serrin’s symmetry result provide characterisations of critical points for the isoperimetric problem associated with the area functional (the classical isoperimetric problem) and the isoperimetric problem associated with the torsional rigidity functional (Saint Venant’s problem), respectively. We present various generalisations of these celebrated symmetry theorems and discuss related quantitative stability results.
We'll describe a shrinking soliton of the mean curvature flow in R4 which is an embedded Lagrangian Möbius bundle. This shrinker was originally found by Y.-I. Lee and M.-T. Wang, and may be regarded as analogous to the so-called FIK shrinker in Ricci flow. In work with K. Braxton and T.-K. Lee, we show that the Möbius shrinker is linearly stable. This is the first non-round example of a stable shrinker. We’ll then discuss potential further parallels in mean curvature flow of recent progress on Ricci flow in 4 (real) dimensions.
Asymptotically hyperbolic (AH) manifolds are a class of complete, non-compact manifolds generalizing hyperbolic space. They have attracted considerable interest in the mathematical community over the past few decades, in part due to their connections with conformal geometry and the AdS/CFT correspondence in theoretical physics. In this talk, I will discuss geometric inverse problems on two-dimensional AH manifolds, focusing on tensor tomography and the non-abelian X-ray transform. The talk is based on joint works with F. Monard and J. Zou, and with H. Grebnev.
In AdS/CFT, reconstructing the bulk asymptotically hyperbolic metric from boundary entanglement entropy forms a geometric inverse problem via the Ryu-Takayanagi minimal surface prescription. I will frame this bulk reconstruction as a boundary rigidity problem, clarifying the nature of boundary "data." Specifically, I will discuss how DtN and NtD maps manifest within the holographic dictionary, touching upon recent developments connected to the inverse problem work by Uhlmann et al. I will also present ongoing work demonstrating that for effectively one-dimensional slab geometries, the linearized minimal surface variations reduce to a classical integral geometry problem that can be exactly inverted using the standard Abel transform.
In this talk we discuss recent progress on the bulk reconstruction inverse problem arising in the AdS/CFT correspondence, where the aim is to determine the geometry of an asymptotically anti-de Sitter spacetime from field theory data on its conformal boundary. The correspondence turns this into a geometric inverse problem closely related to boundary rigidity: by the Ryu-Takayanagi formula the data are areas of minimal surfaces anchored at the conformal boundary, and the unknown is the bulk metric. We study the linearized problem for minimal surfaces anchored on round spheres, obtaining an explicit inversion formula and recovering known bulk solutions to first order. We also discuss further directions, including uniqueness for the linearized problem and recovery of the fully nonlinear geometry.