Nonlinear inverse problems in holography and particle kinematics

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By placing electrodes on a patient's chest, a medical technician may use an electrocardiogram to monitor a patient's heart. Could we use the electrical information on the skin of the patient to build a 3-dimensional image of the patient's heart? This imaging question can be mathematically formulated as an inverse problem. The goal of this project is to study inverse problems which arise in the physical fields of holography, condensed matter physics, and atmospheric physics. In these fields, we will seek to build mathematical objects describing the physical situation in a bulk region (analogous to the patient's chest) from measurement data (analogous to the electrical measurements on the patient's skin). The main postulate of holography is that our 4-dimensional universe is described by a 3-dimensional space, kind of how a 3-dimensional object can be represented in a 2-dimensional hologram image. Broadly, some of the questions this project will consider are (1) can the bulk spacetime modelling gravity be completely described by conformal field theory objects defined on a lower dimensional subspacetime? In the areas of condensed matter physics and atmospheric physics, we will address questions such as (2) Is it possible to determine the kinetic properties of a system of particles in a region from measurement data such as particle concentrations or estimated counts? The answers found to these and related questions over the course of this project aim to provide researchers in the fields of holography theory, condensed matter physics, and atmospheric physics analytical tools for the physical models in their fields. It also aims to equip researchers with mathematical techniques for the development of algorithms which compute key properties, such as the spacetime geometry describing quantum gravity or the operator describing how particles collide. Both geometric and analytic mathematical viewpoints will be employed towards the solutions of such problems. From this dual approach, novel combinations of theoretical techniques from across the disciplines of minimal surface theory, partial differential equations theory, microlocal analysis, and inverse problems theory will be created to develop new approaches for solving key problems in holography theory and condensed matter physics, as well as theoretical problems related to the Boundary Rigidity Problem in geometry.