Bulk-boundary correspondences and unique continuation in asymptotically Anti-de Sitter spacetimes

This article surveys the research presented by the author at the MATRIX Institute workshop "Hyperbolic Differential Equations in Geometry and Physics" in April 2022. The work is centered about establishing rigorous mathematical statements toward the AdS/CFT correspondence in theoretical physics, in particular in dynamical settings. The contents are mainly based on the recent paper with G. Holzegel that proved a unique continuation result for the Einstein-vacuum equations from asymptotically Anti-de Sitter (aAdS) conformal boundaries. We also discuss some preceding results, in particular novel Carleman estimates for wave equations on aAdS spacetimes, which laid the foundations toward the main correspondence theorems.Comment: To be published in "MATRIX Annals

Carleman estimates with sharp weights and boundary observability for wave operators with critically singular potentials

We establish a new family of Carleman inequalities for wave operators on cylindrical spacetime domains containing a potential that is critically singular, diverging as an inverse square on all the boundary of the domain. These estimates are sharp in the sense that they capture both the natural boundary conditions and the natural $H^1$-energy. The proof is based around three key ingredients: the choice of a novel Carleman weight with rather singular derivatives on the boundary, a generalization of the classical Morawetz inequality that allows for inverse-square singularities, and the systematic use of derivative operations adapted to the potential. As an application of these estimates, we prove a boundary observability property for the associated wave equations.Comment: 31 pages; accepted versio

Global stability of traveling waves for (1+1)(1+1)-dimensional systems of quasilinear wave equations

A key feature of $(1+1)$-dimensional nonlinear wave equations is that they admit left or right traveling waves, under appropriate algebraic conditions on the nonlinearities. In this paper, we prove global stability of such traveling wave solutions for $(1+1)$-dimensional systems of nonlinear wave equations, given a certain asymptotic null condition and sufficient decay for the traveling wave. We first consider semilinear systems as a simpler model problem; we then proceed to treat more general quasilinear systems.Comment: Comments are welcome

New tensorial estimates in Besov spaces for time-dependent (2+1)(2 + 1)-dimensional problems

In this paper, we consider various tensorial estimates in geometric Besov-type norms on a one-parameter foliation of surfaces with evolving geometries. Moreover, we wish to do this with only very weak control on these geometries. Several of these estimates were established in previous works by S. Klainerman and I. Rodnianski, but in very specific settings. A primary objective of this paper is to significantly simplify and make more robust the proofs of the estimates. Another goal is to generalize these estimates to more abstract settings. In upcoming papers (joint with S. Alexakis), we will apply these estimates in order to study truncated null cones in an Einstein-vacuum spacetime extending to infinity. This analysis will then be used to study and to control the Bondi mass and the angular momentum under minimal conditions.Comment: Corrected typo

A Generalized Representation Formula for Systems of Tensor Wave Equations

In this paper, we generalize the Kirchhoff-Sobolev parametrix of Klainerman and Rodnianski to systems of tensor wave equations with additional first-order terms. We also present a different derivation, which better highlights that such representation formulas are supported entirely on past null cones. This generalization is a key component for extending Klainerman and Rodnianski's breakdown criterion result for Einstein-vacuum spacetimes to Einstein-Maxwell and Einstein-Yang-Mills spacetimes.Comment: 29 page

On Breakdown Criteria for Nonvacuum Einstein Equations

The recent "breakdown criterion" result of S. Klainerman and I. Rodnianski stated roughly that an Einstein-vacuum spacetime, given as a CMC foliation, can be further extended in time if the second fundamental form and the derivative of the lapse of the foliation are uniformly bounded. This theorem and its proof were extended to Einstein-scalar and Einstein-Maxwell spacetimes in the author's Ph.D. thesis. In this paper, we state the main results of the thesis, and we summarize and discuss their proofs. In particular, we will discuss the various issues resulting from nontrivial Ricci curvature and the coupling between the Einstein and the field equations.Comment: 62 pages This version: corrected minor typos, expanded Section 6 (geometry of null cones

Systematic review of tools to measure outcomes for young children with autism spectrum disorder

Background: The needs of children with autism spectrum disorder (ASD) are complex and this is reflected in the number and diversity of outcomes assessed and measurement tools used to collect evidence about children's progress. Relevant outcomes include improvement in core ASD impairments, such as communication, social awareness, sensory sensitivities and repetitiveness, skills such as social functioning and play, participation outcomes such as social inclusion, and parent and family impact. Objectives: To examine the measurement properties of tools used to measure progress and outcomes in children with ASD up to the age of 6 years. To identify outcome areas regarded as important by people with ASD and parents. Methods: The MeASURe (Measurement in Autism Spectrum disorder Under Review) research collaboration included ASD experts and review methodologists. We undertook systematic review of tools used in ASD early intervention and observational studies from 1992 to 2013, systematic review, using the COSMIN checklist (Consensus-based Standards for the selection of health Measurement Instruments) of papers addressing the measurement properties of identified tools in children with ASD, and synthesis of evidence and gaps. The review design and process was informed throughout by consultation with stakeholders including parents, young people with ASD, clinicians and researchers. Results: The conceptual framework developed for the review was drawn from the International Classification of Functioning, Disability and Health, including the domains 'Impairments', 'Activity Level Indicators', 'Participation', and 'Family Measures'. In review 1, 10,154 papers were sifted - 3091 by full text - and data extracted from 184, in total, 131 tools were identified, excluding observational coding, study-specific measures and those not in English. In review 2, 2665 papers were sifted and data concerning measurement properties of 57 (43%) tools were extracted from 128 papers. Evidence for the measurement properties of the reviewed tools was combined with information about their accessibility and presentation. Twelve tools were identified as having the strongest supporting evidence, the majority measuring autism characteristics and problem behaviour. The patchy evidence and limited scope of outcomes measured mean these tools do not constitute a 'recommended battery' for use. In particular,there is little evidence that the identified tools would be good at detecting change in intervention studies. The obvious gaps in available outcome measurement include well-being and participation outcomes for children, and family quality-of-life outcomes, domains particularly valued by our informants (young people with ASD and parents). Conclusions: This is the first systematic review of the quality and appropriateness of tools designed to monitor progress and outcomes of young children with ASD. Although it was not possible to recommend fully robust tools at this stage, the review consolidates what is known about the field and will act as a benchmark for future developments. With input from parents and other stakeholders, recommendations are made about priority targets for research. Future work: Priorities include development of a tool to measure child quality of life in ASD, and validation of a potential primary outcome tool for trials of early social communication intervention. Study registration: This study is registered as PROSPERO CRD42012002223. Funding: The National Institute for Health Research Health Technology Assessment programme

Null cones to infinity, curvature flux, and Bondi mass

Non UBCUnreviewedAuthor affiliation: University of TorontoPostdoctora

Correspondence and Rigidity Results on Asymptotically Anti-de Sitter Spacetimes

In theoretical physics, it is often conjectured that a correspondence exists between the gravitational dynamics of asymptotically Anti-de Sitter (aAdS) spacetimes and a conformal field theory of their boundaries. In the context of classical relativity, one can attempt to rigorously formulate such a correspondence statement as a unique continuation problem for PDEs: Is an aAdS solution of the Einstein equations uniquely determined by its data on its conformal boundary In this talk, we report on recent progress in this direction, and we highlight the connections between correspondence conjectures in physics, unique continuation theory for wave equations, and the geometry of aAdS spacetimes. We discuss recent unique continuation theorems for waves on aAdS spacetimes that form the key step toward correspondence results, as well as novel geometric obstructions to these results. As an application, we provide an answer to the following question: when can a symmetry on the conformal boundary be extended into the interior This is mostly joint work with Gustav Holzegel (Imperial College London).Non UBCUnreviewedAuthor affiliation: Queen Mary University of LondonResearche