424 research outputs found
Geodesics in Heat
We introduce the heat method for computing the shortest geodesic distance to
a specified subset (e.g., point or curve) of a given domain. The heat method is
robust, efficient, and simple to implement since it is based on solving a pair
of standard linear elliptic problems. The method represents a significant
breakthrough in the practical computation of distance on a wide variety of
geometric domains, since the resulting linear systems can be prefactored once
and subsequently solved in near-linear time. In practice, distance can be
updated via the heat method an order of magnitude faster than with
state-of-the-art methods while maintaining a comparable level of accuracy. We
provide numerical evidence that the method converges to the exact geodesic
distance in the limit of refinement; we also explore smoothed approximations of
distance suitable for applications where more regularity is required
Discrete approaches to quantum gravity in four dimensions
The construction of a consistent theory of quantum gravity is a problem in
theoretical physics that has so far defied all attempts at resolution. One
ansatz to try to obtain a non-trivial quantum theory proceeds via a
discretization of space-time and the Einstein action. I review here three major
areas of research: gauge-theoretic approaches, both in a path-integral and a
Hamiltonian formulation, quantum Regge calculus, and the method of dynamical
triangulations, confining attention to work that is strictly four-dimensional,
strictly discrete, and strictly quantum in nature.Comment: 33 pages, invited contribution to Living Reviews in Relativity; the
author welcomes any comments and suggestion
Lattice Gravity and Random Surfaces
I review recent progress in simplicial quantum gravity in three and four
dimensions, in particular new results on the phase structure of modified models
of dynamical triangulations, the application of a strong-coupling expansion,
and the benefits provided by including degenerate triangulations. In addition,
I describe some recent numerical and analytical results on anisotropic
crystalline membranes.Comment: 22 pages, Latex2e, 13 eps-figures. Review talk presented at LATTICE98
(Boulder
Quantum Gravity via Causal Dynamical Triangulations
"Causal Dynamical Triangulations" (CDT) represent a lattice regularization of
the sum over spacetime histories, providing us with a non-perturbative
formulation of quantum gravity. The ultraviolet fixed points of the lattice
theory can be used to define a continuum quantum field theory, potentially
making contact with quantum gravity defined via asymptotic safety. We describe
the formalism of CDT, its phase diagram, and the quantum geometries emerging
from it. We also argue that the formalism should be able to describe a more
general class of quantum-gravitational models of Horava-Lifshitz type.Comment: To appear in "Handbook of Spacetime", Springer Verlag. 31 page
Diameter Bounds on the Complex of Minimal Genus Seifert Surfaces for Hyperbolic Knot
Given a link L in the 3-sphere, one can build simplicial complexes MS(L) and IS(L), called the Kakimizu complexes. These complexes have isotopy classes of minimal genus and incompressible Seifert surfaces for L as their vertex sets and have simplicial structures defined via a disjointness property. The Kakimizu complexes enjoy many topological properties and are conjectured to be contractible. Following the work of Gabai on sutured manifolds and Murasugi sums, MS(L) and IS(L) have been classified for various classes of links. This thesis focuses on hyperbolic knots; using minimal surface representatives and Kakimizu's formulation of the path-metric on MS(K), we are able to bound the diameter of this complex in terms of only the genus of the knot. The techniques of this paper are also generalized to one-cusped manifolds with a preferred relative homology class
Lattice Quantum Gravity: Review and Recent Developments
We review the status of different approaches to lattice quantum gravity
indicating the successes and problems of each. Recent developments within the
dynamical triangulation formulation are then described. Plenary talk at LATTICE
95 July 11-15, Melbourne, Australia.Comment: 12 pages, 8 figure
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