Twistor actions for gauge theory and gravity

This is a review of recent developments in the study of perturbative gauge theory and gravity using action functionals on twistor space. It is intended to provide a user-friendly introduction to twistor actions, geared towards researchers or graduate students interested in learning something about the utility, prospects, and shortcomings of this approach. For those already familiar with the twistor approach, it should provide a condensed overview of the literature as well as several novel results of potential interest. This work is based primarily upon the author's D.Phil. thesis. We first consider four-dimensional, maximally supersymmetric Yang-Mills theory as a gauge theory in twistor space. We focus on the perturbation theory associated to this action, which in an axial gauge leads to the MHV formalism. This allows us to efficiently compute scattering amplitudes at tree-level (and beyond) in twistor space. Other gauge theory observables such as local operators and null polygonal Wilson loops can also be formulated twistorially, leading to proofs for several correspondences between correlation functions and Wilson loops, as well as a recursive formula for computing mixed Wilson loop / local operator correlators. We then apply the twistor action approach to general relativity, using the on-shell equivalence between conformal and Einstein gravity. This can be extended to N=4 supersymmetry. The perturbation theory of the twistor action leads to formulae for the MHV amplitude with and without cosmological constant, yields a candidate for the Einstein twistor action, and induces a MHV formalism on twistor space. Appendices include discussion of super-connections and Coulomb branch regularization on twistor space.Comment: 178 pages, 30 figures. Review based on the author's D.Phil. thesis. v2: references adde

Generalizable automated pixel-level structural segmentation of medical and biological data

Over the years, the rapid expansion in imaging techniques and equipments has driven the demand for more automation in handling large medical and biological data sets. A wealth of approaches have been suggested as optimal solutions for their respective imaging types. These solutions span various image resolutions, modalities and contrast (staining) mechanisms. Few approaches generalise well across multiple image types, contrasts or resolution. This thesis proposes an automated pixel-level framework that addresses 2D, 2D+t and 3D structural segmentation in a more generalizable manner, yet has enough adaptability to address a number of specific image modalities, spanning retinal funduscopy, sequential fluorescein angiography and two-photon microscopy. The pixel-level segmentation scheme involves: i ) constructing a phase-invariant orientation field of the local spatial neighbourhood; ii ) combining local feature maps with intensity-based measures in a structural patch context; iii ) using a complex supervised learning process to interpret the combination of all the elements in the patch in order to reach a classification decision. This has the advantage of transferability from retinal blood vessels in 2D to neural structures in 3D. To process the temporal components in non-standard 2D+t retinal angiography sequences, we first introduce a co-registration procedure: at the pairwise level, we combine projective RANSAC with a quadratic homography transformation to map the coordinate systems between any two frames. At the joint level, we construct a hierarchical approach in order for each individual frame to be registered to the global reference intra- and inter- sequence(s). We then take a non-training approach that searches in both the spatial neighbourhood of each pixel and the filter output across varying scales to locate and link microvascular centrelines to (sub-) pixel accuracy. In essence, this \link while extract" piece-wise segmentation approach combines the local phase-invariant orientation field information with additional local phase estimates to obtain a soft classification of the centreline (sub-) pixel locations. Unlike retinal segmentation problems where vasculature is the main focus, 3D neural segmentation requires additional exibility, allowing a variety of structures of anatomical importance yet with different geometric properties to be differentiated both from the background and against other structures. Notably, cellular structures, such as Purkinje cells, neural dendrites and interneurons, all display certain elongation along their medial axes, yet each class has a characteristic shape captured by an orientation field that distinguishes it from other structures. To take this into consideration, we introduce a 5D orientation mapping to capture these orientation properties. This mapping is incorporated into the local feature map description prior to a learning machine. Extensive performance evaluations and validation of each of the techniques presented in this thesis is carried out. For retinal fundus images, we compute Receiver Operating Characteristic (ROC) curves on existing public databases (DRIVE & STARE) to assess and compare our algorithms with other benchmark methods. For 2D+t retinal angiography sequences, we compute the error metrics ("Centreline Error") of our scheme with other benchmark methods. For microscopic cortical data stacks, we present segmentation results on both surrogate data with known ground-truth and experimental rat cerebellar cortex two-photon microscopic tissue stacks.Open Acces

Higher derivative gravity and holography

In de afgelopen eeuw heeft de inzet van natuurkundigen geleid tot een theorie die drie fundamentele natuurkrachten beschrijft, de elektromagnetische kracht, de zwakke kernkracht en de sterke kernkracht. Elke poging om de vierde natuurkracht, zwaartekracht, hiermee te verenigen is tot nu toe onbevredigend geweest. Om de aard van de zwaartekracht beter te begrijpen is het holografisch principe ontwikkeld. Dit is geïnspireerd door de eigenschappen van zwarte gaten en laat een verband zien tussen de zwaartekrachttheorie en de veel beter begrepen kwantum velden theorie. Men kan zich dan afvragen of het mogelijk is om beperkingen of onbekende effecten te vinden als we buiten de bekende formulering van het holografisch principe kijken. Dit proefschrift probeert deze vraag te beantwoorden door wiskundige structuren en fysische grootheden te gebruiken als gereedschap om het principe mee te verkennen. Dit wordt gedaan in de context van een aangepaste zwaartekrachttheorie genaamd New Massive Gravity. We zien dat New Massive Gravity ons meer vrijheid geeft in deze verkenning omdat het een breder spectrum aan oplossingen biedt. Daarnaast laat het bestuderen van de fysische grootheid verstrengelingsentropy zien dat er een relatie is tussen de kwantum eigenschappen van materie en de wiskundige eigenschappen van een geometrisch object. Deze relatie kan alleen bestaan als we het holografisch principe beschouwen met een aangepaste zwaartekrachttheorie

Higher derivative gravity and holography

In de afgelopen eeuw heeft de inzet van natuurkundigen geleid tot een theorie die drie fundamentele natuurkrachten beschrijft, de elektromagnetische kracht, de zwakke kernkracht en de sterke kernkracht. Elke poging om de vierde natuurkracht, zwaartekracht, hiermee te verenigen is tot nu toe onbevredigend geweest. Om de aard van de zwaartekracht beter te begrijpen is het holografisch principe ontwikkeld. Dit is geïnspireerd door de eigenschappen van zwarte gaten en laat een verband zien tussen de zwaartekrachttheorie en de veel beter begrepen kwantum velden theorie. Men kan zich dan afvragen of het mogelijk is om beperkingen of onbekende effecten te vinden als we buiten de bekende formulering van het holografisch principe kijken. Dit proefschrift probeert deze vraag te beantwoorden door wiskundige structuren en fysische grootheden te gebruiken als gereedschap om het principe mee te verkennen. Dit wordt gedaan in de context van een aangepaste zwaartekrachttheorie genaamd New Massive Gravity. We zien dat New Massive Gravity ons meer vrijheid geeft in deze verkenning omdat het een breder spectrum aan oplossingen biedt. Daarnaast laat het bestuderen van de fysische grootheid verstrengelingsentropy zien dat er een relatie is tussen de kwantum eigenschappen van materie en de wiskundige eigenschappen van een geometrisch object. Deze relatie kan alleen bestaan als we het holografisch principe beschouwen met een aangepaste zwaartekrachttheorie

Numerical Algorithms for finding Black Hole solutions of Einstein's Equations

Einstein's Theory of General Relativity has proven remarkably successful at modelling a wide range of gravitational phenomena. Amongst some of the novel features in this description is the existence of black holes; regions of space-time where gravity is so strong that light cannot escape. The properties of black holes have been extensively studied within General Relativity, culminating in the result that the few known space-times are the only allowed stationary black hole solutions in four dimensions. In the past half century, research has focused on how to unify the distinct theories of gravity and quantum mechanics. A common theme amongst several strong candidates is that space-time, the backdrop for gravity, is fundamentally higher dimensional. In these theories, the structure of black hole solutions is relatively unknown and expected to be much richer; finding such solutions is, however, a very hard task. In this thesis, we introduce new numerical methods to study higher dimensional black holes. The methods, based on refinements of existing work and the novel application of standard techniques, are then used to study a number of black hole space-times. Namely the structure of black holes on a Kaluza-Klein background, and rotating Kerr black holes. We demonstrate that these algorithms can be applied in a wide class of situations and yield good quality results with comparative ease. New results are presented in both cases studied. We examine the predicted merger between non-uniform black strings and localised black holes on a Kaluza-Klein background. We find evidence for a new type of non-uniform black string with one Euclidean negative mode and lower entropy than the uniform strings. We discover a window of localised black holes with one Euclidean negative mode but positive specific heat. We also look at the local structure of the merger point and find consistency with Kol's cone prediction

Differentiable world programs

L'intelligence artificielle (IA) moderne a ouvert de nouvelles perspectives prometteuses pour la création de robots intelligents. En particulier, les architectures d'apprentissage basées sur le gradient (réseaux neuronaux profonds) ont considérablement amélioré la compréhension des scènes 3D en termes de perception, de raisonnement et d'action. Cependant, ces progrès ont affaibli l'attrait de nombreuses techniques ``classiques'' développées au cours des dernières décennies. Nous postulons qu'un mélange de méthodes ``classiques'' et ``apprises'' est la voie la plus prometteuse pour développer des modèles du monde flexibles, interprétables et exploitables : une nécessité pour les agents intelligents incorporés. La question centrale de cette thèse est : ``Quelle est la manière idéale de combiner les techniques classiques avec des architectures d'apprentissage basées sur le gradient pour une compréhension riche du monde 3D ?''. Cette vision ouvre la voie à une multitude d'applications qui ont un impact fondamental sur la façon dont les agents physiques perçoivent et interagissent avec leur environnement. Cette thèse, appelée ``programmes différentiables pour modèler l'environnement'', unifie les efforts de plusieurs domaines étroitement liés mais actuellement disjoints, notamment la robotique, la vision par ordinateur, l'infographie et l'IA. Ma première contribution---gradSLAM--- est un système de localisation et de cartographie simultanées (SLAM) dense et entièrement différentiable. En permettant le calcul du gradient à travers des composants autrement non différentiables tels que l'optimisation non linéaire par moindres carrés, le raycasting, l'odométrie visuelle et la cartographie dense, gradSLAM ouvre de nouvelles voies pour intégrer la reconstruction 3D classique et l'apprentissage profond. Ma deuxième contribution - taskography - propose une sparsification conditionnée par la tâche de grandes scènes 3D encodées sous forme de graphes de scènes 3D. Cela permet aux planificateurs classiques d'égaler (et de surpasser) les planificateurs de pointe basés sur l'apprentissage en concentrant le calcul sur les attributs de la scène pertinents pour la tâche. Ma troisième et dernière contribution---gradSim--- est un simulateur entièrement différentiable qui combine des moteurs physiques et graphiques différentiables pour permettre l'estimation des paramètres physiques et le contrôle visuomoteur, uniquement à partir de vidéos ou d'une image fixe.Modern artificial intelligence (AI) has created exciting new opportunities for building intelligent robots. In particular, gradient-based learning architectures (deep neural networks) have tremendously improved 3D scene understanding in terms of perception, reasoning, and action. However, these advancements have undermined many ``classical'' techniques developed over the last few decades. We postulate that a blend of ``classical'' and ``learned'' methods is the most promising path to developing flexible, interpretable, and actionable models of the world: a necessity for intelligent embodied agents. ``What is the ideal way to combine classical techniques with gradient-based learning architectures for a rich understanding of the 3D world?'' is the central question in this dissertation. This understanding enables a multitude of applications that fundamentally impact how embodied agents perceive and interact with their environment. This dissertation, dubbed ``differentiable world programs'', unifies efforts from multiple closely-related but currently-disjoint fields including robotics, computer vision, computer graphics, and AI. Our first contribution---gradSLAM---is a fully differentiable dense simultaneous localization and mapping (SLAM) system. By enabling gradient computation through otherwise non-differentiable components such as nonlinear least squares optimization, ray casting, visual odometry, and dense mapping, gradSLAM opens up new avenues for integrating classical 3D reconstruction and deep learning. Our second contribution---taskography---proposes a task-conditioned sparsification of large 3D scenes encoded as 3D scene graphs. This enables classical planners to match (and surpass) state-of-the-art learning-based planners by focusing computation on task-relevant scene attributes. Our third and final contribution---gradSim---is a fully differentiable simulator that composes differentiable physics and graphics engines to enable physical parameter estimation and visuomotor control, solely from videos or a still image

Computer Aided Verification

The open access two-volume set LNCS 11561 and 11562 constitutes the refereed proceedings of the 31st International Conference on Computer Aided Verification, CAV 2019, held in New York City, USA, in July 2019. The 52 full papers presented together with 13 tool papers and 2 case studies, were carefully reviewed and selected from 258 submissions. The papers were organized in the following topical sections: Part I: automata and timed systems; security and hyperproperties; synthesis; model checking; cyber-physical systems and machine learning; probabilistic systems, runtime techniques; dynamical, hybrid, and reactive systems; Part II: logics, decision procedures; and solvers; numerical programs; verification; distributed systems and networks; verification and invariants; and concurrency