Research / Presentations
This page contains material from various presentations that I have given.
This section describes some survey talks I have given.
A Brief Introduction to Mathematical Relativity
The following slides are from a talk introducing some of the mathematics behind Einstein's theory of relativity. This talk was aimed at advanced undergraduate and beginning postgraduate students.
The following slides are from a similar talk, but aimed at interested Year 11 students in the UK.
The following are some of my past research talks.
Bulk-Boundary Correspondence for Vacuum Asympotically Anti-de Sitter Spacetimes
These slides discuss the second phase of my research program regarding unique continuation from the conformal boundary of asymptotically AdS spacetimes and its connections to the AdS/CFT correspondence from theoretical physics. The focus here is on unique continuation properties for the Einstein-vacuum equations from the conformal boundary.
On Controllability of Waves and Geometric Carleman Estimates
These slides discuss my results on the controllability of a general class of (non-analytic) wave equations on non-cylindrical domains with moving boundaries.
A longer article, which discusses these results in more detail, can be found in the proceedings for the Séminaire Laurent Schwartz:
Correspondence and Rigidity Results on Asymptotically Anti-de Sitter Spacetimes
These slides discuss the first phase of my research program regarding unique continuation from the conformal boundary of asymptotically AdS spacetimes and its connections to the AdS/CFT correspondence from theoretical physics. The focus here is on unique continuation properties for wave equations from the conformal boundary.
Unique Continuation for Waves, Carleman Estimates, and Applications
The slides below are from various talks summarizing some of my past results (with various collaborators) regarding unique continuation results for linear and nonlinear wave equations, the methods used in establishing them, and further applications of these methods.
Unique Continuation, Carleman Estimates, and Blow-up for Nonlinear Waves
These slides discuss two rather different problems for nonlinear wave equations: unique continuation from infininity, and singularity formation. The talk describes how the same methods—novel global Carleman estimates—can be applied to study both problems.
Unique Continuation from Infinity for Linear Waves
This talk describes work (joint with Spyros Alexakis and Volker Schlue) describing when solutions to wave equations on various asymptotically flat spacetimes can be uniquely continued from parts of infinity. In particular, we ask when solutions to wave equations are determined by their radiation data.
Null Cones to Infinity, Curvature Flux, and Bondi Mass
These slides summarize a sequence of results (partially joint with Spyros Alexakis) on controlling the geometry of null cones extending to infinity by its weighted curvature flux. In particular, one controls the Bondi mass associated with the cone by the curvature flux.
A Generalized Representation Formula for Tensor Wave Equations on Curved Spacetimes
This talk describes a new representation formula for solutions of tensor wave equations on general Lorentzian manifolds. This result further generalized an earlier formula of Sergiu Klainerman and Igor Rodnianski and simplified its derivation.
Breakdown Criteria for Nonvacuum Einstein Equations
This talk describes the research conducted for my PhD thesis, in which I developed an improved criterion for continuing solutions of the Einstein-scalar and Einstein-Maxwell equations. This extended an earlier result of Sergiu Klainerman and Igor Rodnianski for solutions of the Einstein-vacuum equations.