| Abstract: |
| Minimal surfaces are fundamental objects in geometry, defined as critical points of the area functional. They also play a central role in the AdS/CFT correspondence in theoretical physics, where the Ryu-Takayanagi formula equates entanglement entropy on the boundary to the area of a minimal surface in the bulk. Bulk reconstruction---a central problem in physics---is precisely the inverse problem of recovering geometric information from areas of minimal surfaces.
I will first briefly recall results on this inverse problem based on a PDE approach for the minimal surface equation. These results rely on methods for the Calder\`{o}n problem and are mostly restricted to two dimensional minimal surfaces and one can at best expect logarithmic stability.
I will then introduce an integral geometry approach, viewing the first variation of area as a generalized Radon transform---the minimal surface transform. Interpreting this transform as a Fourier Integral Operator yields injectivity results for analytic metrics in dimensions three and higher. This framework achieves two major advances: it extends to higher-dimensional minimal surfaces and bulks, and it provides H\{o}lder stability rather than logarithmic.
These mathematical developments have improved upon the best known results in the physics literature, representing a rare instance where techniques from mathematics advance theoretical physics. |
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