Searcharxiv⌕ Search

arXiv subjects

Jeffrey R. Weeks

Publications and source records attributed to Jeffrey R. Weeks.

7 recordsLinked to original sources
arXiv2006

Exact Polynomial Eigenmodes for Homogeneous Spherical 3-Manifolds

Observational data hints at a finite universe, with spherical manifolds such as the Poincare dodecahedral space tentatively providing the best fit. Simulating the physics of a model universe requires knowing the eigenmodes of the Laplace operator on the space. The present article provides explicit polynomial eigenmodes for all globally homogeneous 3-manifolds: the Poincare dodecahedral space S3/I*, the binary octahedral space S3/O*, the binary tetrahedral space S3/T*, the prism manifolds S3/D_m* and the lens spaces L(p,1).

math.SP↗
arXiv2004

Maxwell's Multipole Vectors and the CMB

The recently re-discovered multipole vector approach to understanding the harmonic decomposition of the cosmic microwave background traces its roots to Maxwell's Treatise on Electricity and Magnetism. Taking Maxwell's directional derivative approach as a starting point, the present article develops a fast algorithm for computing multipole vectors, with an exposition that is both simpler and better motivated than in the author's previous work. Tests show the resulting algorithm, coded up as a Mathematica notebook, to be both fast and robust. This article serves to announce the free availability of the code. The algorithm is then applied to the much discussed anomalies in the low-ell CMB modes, with sobering results. Simulations show the quadrupole-octopole alignment to be unusual at the 98.7% level, corroborating earlier estimates of a 1-in-60 alignment while showing recent reports of 1-in-1000 (using non-unit-length normal vectors) to have unintentionally relied on the near orthogonality of the quadrupole vectors in one particular data set. The alignment of the quadrupole and octopole vectors with the ecliptic plane is confirmed at better than the 2sigma level.

astro-ph↗
arXiv2003

Computation of Hyperbolic Structures in Knot Theory

This chapter from the upcoming Handbook of Knot Theory (eds. Menasco and Thistlethwaite) shows how to construct hyperbolic structures on link complements and perform hyperbolic Dehn filling. Along with a new elementary exposition of the standard ideas from Thurston's work, the article includes never-before-published explanations of SnapPea's algorithms for triangulating a link complement efficiently and for converging quickly to the hyperbolic structure while avoiding singularities in the parameter space.

math.GT↗
arXiv2002

Detecting topology in a nearly flat hyperbolic universe

Cosmic microwave background data shows the observable universe to be nearly flat, but leaves open the question of whether it is simply or multiply connected. Several authors have investigated whether the topology of a multiply connect hyperbolic universe would be detectable when 0.9 < Omega < 1. However, the possibility of detecting a given topology varies depending on the location of the observer within the space. Recent studies have assumed the observer sits at a favorable location. The present paper extends that work to consider observers at all points in the space, and (for given values of Omega_m and Omega_Lambda and a given topology) computes the probability that a randomly placed observer could detect the topology. The computations show that when Omega = 0.98 a randomly placed observer has a reasonable chance (~50%) of detecting a hyperbolic topology, but when Omega = 0.99 the chances are low (<10%) and decrease still further as Omega approaches one.

astro-ph↗
arXiv1999

Cosmetic surgery on knots

This paper concerns the Dehn surgery construction, especially those Dehn surgeries leaving the manifold unchanged. In particular, we describe an oriented 1-cusped hyperbolic 3-manifold X with a pair of slopes r_1, r_2 such that the Dehn filled manifolds X(r_1), X(r_2) are oppositely oriented copies of the lens space L(49,18), and there is no homeomorphism h of X such that h(r_1)=h(r_2).

math.GT↗
arXiv1998

Measuring the shape of the Universe

Since the dawn of civilization, humanity has grappled with the big questions of existence and creation. Modern cosmology seeks to answer some of these questions using a combination of mathematics and measurement. The questions people hope to answer include ``how did the universe begin?''; ``how will the universe end?''; ``is space finite or infinite?''. After a century of remarkable progress, cosmologists may be on the verge of answering at least one of these questions -- is space finite? Using some simple geometry and a NASA satellite set for launch in the year 2000, the authors and their colleagues hope to measure the size and shape of space. This review article explains the mathematics behind the measurement, and the cosmology behind the observations.

astro-ph↗