We study finite and infiite entangled graphs in the bond percolation process in three dimensions with density p. After a discussion of the relevant defi®nitions, the entanglement critical probabilities are defined. The size of the maximal entangled graph at the origin is studied for small p, and it is shown that this graph has radius whose tail decays at least as fast as exp … an = log n†; indeed, the logarithm may be replaced by any iterate of logarithm for an appropriate positive constant a. We explore the question of almost sure uniqueness of the infinite maximal open entangled graph when p is large, and we establish two relevant theorems. We make several conjectures concerning the properties of entangled graphs in percolation.

The first part of this paper extends the Doplicher-Haag-Roberts theory of superselection sectors to quantum field theory on arbitrary globally hyperbolic spacetimes. The statistics of a superselection sector may be defined as in flat spacetime and each charge has a conjugate charge when the spacetime possesses non-compact Cauchy surfaces. In this case, the field net and the gauge group can be constructed as in Minkowski spacetime. The second part of this paper derives spin-statistics theorems on spacetimes with appropriate symmetries. Two situations are considered: First, if the spacetime has a bifurcate Killing horizon, as is the case in the presence of black holes, then restricting the observables to the Killing horizon together with “modular covariance” for the Killing flow yields a conformally covariant quantum field theory on the circle and a conformal spin-statistics theorem for charged sectors localizable on the Killing horizon. Secondly, if the spacetime has a rotation and PT symmetry like the Schwarzschild-Kruskal black holes, “geometric modular action” of the rotational symmetry leads to a spin-statistics theorem for charged covariant sectors where the spin is defined via the SU(2)-covering of the spatial rotation group SO(3).

Abstract. We present a new direct proof of a topological representation theorem for oriented matroids in the general rank case. Our proof is based on an earlier rank 3 version. It uses hyperline sequences and the generalized Schönflies theorem. As an application, we show that one can read off oriented matroids from arrangements of embedded spheres of codimension one, even if wild spheres are involved.

. Prime end theory is essentially a compactification theory for simply connected, bounded domains, U , in E 2 , or simply connected domains in S 2 with nondegenerate complement. The planar case was originally due to Caratheodory and was later generalized to the sphere by Ursell and Young, and to arbitrary two manifolds by Mather. There are many applications of the two dimensional theory, including applications to fixed point problems, embedding problems, and homeomorphism (group) actions. Several constructions of a three dimensional topological prime end theory appear in the literature, including work by Kaufmann, Mazurkiewicz, and Epstein. In this paper, the authors develop a simple three dimensional prime end theory for certain open subsets of Euclidean three space. It includes conditions focusing on an "Induced Homeomorphism Theorem", which, the authors believe, provides the necessary ingredient for applications. 1. Introduction Prime end theory is essentially a compactificat...

There are several approaches to summarizing a mathematician’s research accomplishments, and each has its advantages and disadvantages. This article is based upon a talk given at Tulane that was aimed at a fairly general audience, including faculty members in other areas and graduate students who had taken the usual entry level courses. As such, it is meant to be relatively nontechnical and to emphasize qualitative rather than quantitative issues; in keeping with this aim, references will be given for some standard topological notions that are not normally treated in entry level graduate courses. Since this was an hour talk, it was also not feasible to describe every single piece of published mathematical work that Terry Lawson has ever written; in particular, some papers like [42] and [50] would require lengthy digressions that are not easily related to the central themes in his main lines of research. Instead, we shall focus on some ways in which Terry’s work relates to an important thread in geometric topology; namely, the passage from studying problems in a given dimension to studying problems in the next dimensions. Qualitatively speaking, there are fairly well-developed theories for very low dimensions and for all sufficiently large dimensions, but between these ranges there are some dimensions in which the answers to many fundamental

submanifolds of a smooth manifold

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Abstract. Taking the ℓ 1-completion and the topological dual of the singular chain complex gives rise to ℓ 1-homology and bounded cohomology respectively. In contrast to ℓ 1-homology, major structural properties of bounded cohomology are well understood by the work of Gromov and Ivanov. Based on an observation by Matsumoto and Morita, we derive a mechanism linking isomorphisms on the level of homology of Banach chain complexes to isomorphisms on the level of cohomology of the dual Banach cochain complexes and vice versa. Therefore, certain results on bounded cohomology can be transferred to ℓ 1-homology. For example, we obtain a new proof of the fact that ℓ 1-homology depends only on the fundamental group and that ℓ 1-homology with twisted coefficients admits a description in terms of projective resolutions. The latter one in particular fills a gap in Park’s approach. In the second part, we demonstrate how ℓ 1-homology can be used to get a better understanding of simplicial volume of non-compact manifolds. 1.

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