Groups acting on veering pairs and Kleinian groups

We show that some laminar group which has an invariant veering pair of laminations is a hyperbolic 3-orbifold group. On the way, we show that from a veering pair of laminations, one can construct a loom space (in the sense of Schleimer-Segerman) as a quotient. Our approach does not assume the existence of any 3-manifold to begin with so this is a geometrization-type result, and supersedes some of the results regarding the relation among veering triangulations, pseudo-Anosov flows, taut foliations in the literature.Comment: 60 pages, 3 figure

Black hole boundaries

Classical black holes and event horizons are highly non-local objects, defined in relation to the causal past of future null infinity. Alternative, quasilocal characterizations of black holes are often used in mathematical, quantum, and numerical relativity. These include apparent, killing, trapping, isolated, dynamical, and slowly evolving horizons. All of these are closely associated with two-surfaces of zero outward null expansion. This paper reviews the traditional definition of black holes and provides an overview of some of the more recent work on alternative horizons.Comment: 27 pages, 8 figures, invited Einstein Centennial Review Article for CJP, final version to appear in journal - glossary of terms added, typos correcte

Gravitoelectromagnetic knot fields

We construct a class of knot solutions of the gravitoelectromagnetic (GEM) equations in vacuum in the linearized gravity approximation by analogy with the Ra\~{n}ada-Hopf fields. For these solutions, the dual metric tensors of the bi-metric geometry of the gravitational vacuum with knot perturbations are given and the geodesic equation as a function of two complex parameters of the GEM knots are calculated. Finally, the Landau--Lifshitz pseudo-tensor and a scalar invariant of the GEM knots are computed.Comment: 22 pages. Published in the Special Issue "Frame-Dragging and Gravitomagnetism

Low-dimensional Topology and Number Theory

The workshop brought together topologists and number theorists with the intent of exploring the many tantalizing connections between these areas

Stability-aware simplification of curve networks

La conception de réseaux de courbes nécessite la considération de plusieurs facteurs: la stabilité de la structure, l'efficience matérielle, et l'aspect esthétique - des objectifs complexes et interdépendants rendant la conception manuelle difficile. Nous présentons une nouvelle méthode permettant de simplifier des réseaux de courbes destinés à la fabrication. Pour un ensemble de courbes 3D donné, notre algorithme en sélectionne un sous-ensemble stable. Bien que la stabilité soit traditionnellement mesurée par l'ordre de grandeur des déformations entraînées par des charges prédéfinies, une telle approche peut s'avérer limitante. Elle ne tient ni compte des effets de vibration pour les structures de grandes tailles, ni des multiples possibilités de forces appliquées pour les structures et objets de plus petite taille. Ainsi, nous optimisons directement pour une déformation minimale avec la charge dans le pire des cas (de l'anglais "worst-case"). Notre contribution technique est une nouvelle formulation de la simplification de réseaux de courbes pour la stabilité dans le pire des cas. Celle-ci mène à un problème d'optimisation semi-définie positive en nombres entiers (MI-SDP). Malgré que résoudre ce problème MI-SDP directement est irréaliste dans la plupart des cas, une intuition physique nous mène à un algorithme vorace efficace. Enfin, nous démontrons le potentiel de notre approache à l'aide plusieurs réseaux de courbes et validons l'efficacité de notre méthode en la comparant de façon quantitative à des approaches plus simples.Designing curve networks for fabrication requires simultaneous consideration of structural stability, cost effectiveness, and visual appeal - complex, interrelated objectives that make manual design a difficult and tedious task. We present a novel method for fabrication-aware simplification of curve networks, algorithmically selecting a stable subset of given 3D curves. While traditionally, stability is measured as the magnitude of deformation induced by a set of predefined loads, predicting applied forces for common day objects can be challenging. Instead, we directly optimize for minimal deformation under the worst-case load. Our technical contribution is a novel formulation of 3D curve network simplification for worst-case stability, leading to a mixed-integer semi-definite programming problem (MI-SDP). We show that while solving MI-SDP directly is impractical, a physical insight suggests an efficient greedy heuristic algorithm. We demonstrate the potential of our approach on a variety of curve network designs and validate its effectiveness compared to simpler alternatives using numerical experiments

Dev2PQ: Planar Quadrilateral Strip Remeshing of Developable Surfaces

We introduce an algorithm to remesh triangle meshes representing developable surfaces to planar quad dominant meshes. The output of our algorithm consists of planar quadrilateral (PQ) strips that are aligned to principal curvature directions and closely approximate the curved parts of the input developable, and planar polygons representing the flat parts of the input. Developable PQ-strip meshes are useful in many areas of shape modeling, thanks to the simplicity of fabrication from flat sheet material. Unfortunately, they are difficult to model due to their restrictive combinatorics and locking issues. Other representations of developable surfaces, such as arbitrary triangle or quad meshes, are more suitable for interactive freeform modeling, but generally have non-planar faces or are not aligned to principal curvatures. Our method leverages the modeling flexibility of non-ruling based representations of developable surfaces, while still obtaining developable, curvature aligned PQ-strip meshes. Our algorithm optimizes for a scalar function on the input mesh, such that its level sets are extrinsically straight and align well to the locally estimated ruling directions. The condition that guarantees straight level sets is nonlinear of high order and numerically difficult to enforce in a straightforward manner. We devise an alternating optimization method that makes our problem tractable and practical to compute. Our method works automatically on any developable input, including multiple patches and curved folds, without explicit domain decomposition. We demonstrate the effectiveness of our approach on a variety of developable surfaces and show how our remeshing can be used alongside handle based interactive freeform modeling of developable shapes

Analysis of a Reduced-Order Model for the Simulation of Elastic Geometric Zigzag-Spring Meta-Materials

We analyze the performance of a reduced-order simulation of geometric meta-materials based on zigzag patterns using a simplified representation. As geometric meta-materials we denote planar cellular structures which can be fabricated in 2d and bent elastically such that they approximate doubly-curved 2-manifold surfaces in 3d space. They obtain their elasticity attributes mainly from the geometry of their cellular elements and their connections. In this paper we focus on cells build from so-called zigzag springs. The physical properties of the base material (i.e., the physical substance) influence the behavior as well, but we essentially factor them out by keeping them constant. The simulation of such complex geometric structures comes with a high computational cost, thus we propose an approach to reduce it by abstracting the zigzag cells by a simpler model and by learning the properties of their elastic deformation behavior. In particular, we analyze the influence of the sampling of the full parameter space and the expressiveness of the reduced model compared to the full model. Based on these observations, we draw conclusions on how to simulate such complex meso-structures with simpler models.Comment: 14 pages, 12 figures, published in Computers & Graphics, extended version of arXiv:2010.0807

Non-abelian 4-d black holes, wrapped 5-branes, and their dual descriptions

We study extremal and non-extremal generalizations of the regular non-abelian monopole solution of hep-th/9707176, interpreted in hep-th/0007018 as 5-branes wrapped on a shrinking S^2. Naively, the low energy dynamics is pure N=1 supersymmetric Yang-Mills. However, our results suggest that the scale of confinement and chiral symmetry breaking in the Yang-Mills theory actually coincides with the Hagedorn temperature of the little string theory. We find solutions with regular horizons and arbitrarily high Hawking temperature. Chiral symmetry is restored at high energy density, corresponding to large black holes. But the entropy of the black hole solutions decreases as one proceeds to higher temperatures, indicating that there is a thermodynamic instability and that the canonical ensemble is ill-defined. For certain limits of the black hole solutions, we exhibit explicit non-linear sigma models involving a linear dilaton. In other limits we find extremal non-BPS solutions which may have some relevance to string cosmology.Comment: 53 pages, 21 figures, latex. v2: slightly improved figure

Approaching the Planck Scale from a Generally Relativistic Point of View: A Philosophical Appraisal of Loop Quantum Gravity

My dissertation studies the foundations of loop quantum gravity (LQG), a candidate for a quantum theory of gravity based on classical general relativity. At the outset, I discuss two---and I claim separate---questions: first, do we need a quantum theory of gravity at all; and second, if we do, does it follow that gravity should or even must be quantized? My evaluation of different arguments either way suggests that while no argument can be considered conclusive, there are strong indications that gravity should be quantized.LQG attempts a canonical quantization of general relativity and thereby provokes a foundational interest as it must take a stance on many technical issues tightly linked to the interpretation of general relativity. Most importantly, it codifies general relativity's main innovation, the so-called background independence, in a formalism suitable for quantization. This codification pulls asunder what has been joined together in general relativity: space and time. It is thus a central issue whether or not general relativity's four-dimensional structure can be retrieved in the alternative formalism and how it fares through the quantization process. I argue that the rightful four-dimensional spacetime structure can only be partially retrieved at the classical level. What happens at the quantum level is an entirely open issue.Known examples of classically singular behaviour which gets regularized by quantization evoke an admittedly pious hope that the singularities which notoriously plague the classical theory may be washed away by quantization. This work scrutinizes pronouncements claiming that the initial singularity of classical cosmological models vanishes in quantum cosmology based on LQG and concludes that these claims must be severely qualified. In particular, I explicate why casting the quantum cosmological models in terms of a deterministic temporal evolution fails to capture the concepts at work adequately. Finally, a scheme is developed of how the re-emergence of the smooth spacetime from the underlying discrete quantum structure could be understood