Unique Continuation for Geometric Wave Equations, and Applications to Relativity, Holography, and Controllability

A wide variety of phenomena in science, economics, and engineering are mathematically modelled by partial differential equations, or PDEs. Wave equations form an important subclass of PDEs and are found within many fundamental equations of physics, such as the Maxwell equations (electromagnetics), Yang-Mills equations (particle physics), Einstein field equations (gravitation), and Euler equations (fluid dynamics). In particular, the wave structure hidden within the Einstein equations led to the prediction of gravitational waves a century before their experimental detection in 2016.

A basic problem in PDEs, and for wave equations in particular, is to find a unique solution given appropriate data. This can be interpreted as being able to "predict the future" given initial conditions for a system. On the other hand, in settings where the equation may not always be solved, it remains pertinent to ask whether solutions, if they exist, remain unique; this is the problem of unique continuation. Intuitively, this question asks whether there is a one-to-one correspondence between initial data and solutions.

There is now significant literature surrounding the theory of unique continuation. Modern developments began with the work of Carleman in 1939 and continued with breakthroughs by Calderón, Hörmander, Tataru, and many others. The main analytic technique is a class of weighted inequalities now known as Carleman estimates.

My recent contributions in this direction revolve around unique continuation properties for linear and nonlinear waves; this also forms the backbone of my proposed research. The focus is on studying "degenerate" settings, for which the classical theory fails to apply. Another essential goal is the development of robust geometric techniques that apply to a wide variety of curved settings.

The bulk of the proposed research programme deals with applying the results and techniques of this unique continuation theory toward other problems. In the past decades, there have been numerous connections between unique continuation and other aspects of PDEs and physics. For instance, unique continuation results for wave equations have been recently applied in relativity toward symmetry extension and rigidity results. Furthermore, Carleman estimates, in particular for wave equations, have been an invaluable tool for studying inverse and control theory problems in the context of PDEs and differential geometry.

One ongoing research project, a major component of the proposed programme, is to apply unique continuation techniques to study holographic principles in theoretical physics. This is largely motivated by the AdS/CFT correspondence, which roughly posits a correspondence between gravitational dynamics in Anti-de Sitter spacetime and conformal field theories on its boundary. While this idea has been tremendously influential in theoretical physics, there has been scant progress in terms of rigorous mathematical formulations and results. This project ultimately aims to develop mathematical underpinnings to these physical ideas. A first major step in this endeavour, in the context of classical relativity, is to formulate and prove a correspondence statement as a unique continuation problem for the Einstein field equations.

Another major aspect of my research is to apply these techniques, novel Carleman estimates in particular, toward other problems for geometric PDEs. One example is the question of controllability: this asks whether one can drive a system (say, modelled by PDEs) to a preferred state using limited controls (for instance, through boundary data). Another area of interest is inverse problems, which asks whether one can determine a system (mathematically, a PDE) by making only limited measurements (for instance, the boundary values of solutions). Given the prevalence of waves in physics, both control and inverse problems for waves are well-connected to important questions in science and engineering.

Wave equations arise in many equations of physics describing aspects of our universe, including those of electromagnetics (Maxwell equations), gravity (Einstein equations), fluid dynamics (Euler equations), particle physics (Dirac equation), acoustics, and many others. As a result, much physical understanding of the above systems has been impacted by the study of wave equations. Recent notable examples include gravitational waves and radiation in general relativity.

As mentioned before, the proposed research would benefit multiple academic communities, including within linear and nonlinear PDEs, analysis, differential geometry, mathematical relativity, control theory, inverse problems, and theoretical physics. Because of its ties to a wide number of areas, the project would be particularly impactful in terms of building connections among these groups within and beyond mathematics. For example, the research on holography would involve ideas from differential geometry, analysis, and theoretical physics. Moreover, the novel Carleman estimates developed in this programme, obtained via ideas in differential geometry and geometric PDEs, would be connected to scientific and engineering problems in inverse problems and control theory.

For several decades, results from mathematical relativity have impacted physics by illuminating various aspects of Einstein's theory. Along these lines, my work with Spyros Alexakis and Volker Schlue on unique continuation for waves from infinity has been applied to rule out the existence of many time-periodic spacetimes. Moreover, various phenomena in physics can be modelled by nonlinear PDEs for which singularities can form. In some recent results with Spyros Alexakis, I introduced Carleman estimates as an innovative method for studying and describing singularities formed by nonlinear wave equations.

A fundamental open question in theoretical physics is that of reconciling gravity, via general relativity, with quantum theories. The AdS/CFT correspondence has been an influential direction of investigation toward this question in the past twenty years, finding applications in string theory, condensed matter physics, and nuclear physics. However, there is currently very little rigorous mathematical knowledge in this area, especially in dynamical settings. My proposed work on AdS and holography would contribute toward filling this gap. My research with Gustav Holzegel on a linearised model problem has already yielded novel observations of interest to physicists: the importance of considering sufficiently large portions of the AdS boundary.

Inverse problems is another fast-growing area, in particular in the UK and Europe, with many applications in science and engineering. Examples related to wave equations include seismology and tomography. There is now considerable literature regarding inverse problems for wave equations, as well as recent interest in inverse problems in the context of general relativity. As a result, advancements in the proposed research would have positive impacts on the areas mentioned above.

Control theory is yet another field that has been impactful in science and engineering, with applications in robotics, chemistry, material sciences, economics, among many others. There has also been a tremendous number of mathematical advances in the past three decades. Like for inverse problems, the proposed research would find impact in studying the controllability of wave equations, as well as the multitude of equations in physics that contain waves.

Finally, I intend to achieve wider impact through outreach efforts promoting aspects of my research programme and PDEs in general. For instance, this would involve presenting and speaking to students of all levels. For example, in spring 2016, I gave a presentation introducing mathematical relativity at the Warwick Imperial Spring Meeting, an inter-university conference for advanced undergraduate and early postgraduate students.