4,821 research outputs found
Topological Signals of Singularities in Ricci Flow
We implement methods from computational homology to obtain a topological
signal of singularity formation in a selection of geometries evolved
numerically by Ricci flow. Our approach, based on persistent homology, produces
precise, quantitative measures describing the behavior of an entire collection
of data across a discrete sample of times. We analyze the topological signals
of geometric criticality obtained numerically from the application of
persistent homology to models manifesting singularities under Ricci flow. The
results we obtain for these numerical models suggest that the topological
signals distinguish global singularity formation (collapse to a round point)
from local singularity formation (neckpinch). Finally, we discuss the
interpretation and implication of these results and future applications.Comment: 24 pages, 14 figure
Generalized Sums over Histories for Quantum Gravity I. Smooth Conifolds
This paper proposes to generalize the histories included in Euclidean
functional integrals from manifolds to a more general set of compact
topological spaces. This new set of spaces, called conifolds, includes
nonmanifold stationary points that arise naturally in a semiclasssical
evaluation of such integrals; additionally, it can be proven that sequences of
approximately Einstein manifolds and sequences of approximately Einstein
conifolds both converge to Einstein conifolds. Consequently, generalized
Euclidean functional integrals based on these conifold histories yield
semiclassical amplitudes for sequences of both manifold and conifold histories
that approach a stationary point of the Einstein action. Therefore sums over
conifold histories provide a useful and self-consistent starting point for
further study of topological effects in quantum gravity. Postscript figures
available via anonymous ftp at black-hole.physics.ubc.ca (137.82.43.40) in file
gen1.ps.Comment: 81pp., plain TeX, To appear in Nucl. Phys.
Surgery on -manifolds
We show that although closed -manifolds
do not admit metrics of nonpositive sectional curvature, the arguments of
Farrell and Jones can be extended to show that such manifolds are topologically
rigid, if .Comment: 7 pages, AMS-LaTeX file, To appear in the Canadian Mathematical
Bulletin
In-plane magnetic anisotropy of Fe atoms on BiSe(111)
The robustness of the gapless topological surface state hosted by a 3D
topological insulator against perturbations of magnetic origin has been the
focus of recent investigations. We present a comprehensive study of the
magnetic properties of Fe impurities on a prototypical 3D topological insulator
BiSe using local low temperature scanning tunneling microscopy and
integral x-ray magnetic circular dichroism techniques. Single Fe adatoms on the
BiSe surface, in the coverage range are heavily relaxed
into the surface and exhibit a magnetic easy axis within the surface-plane,
contrary to what was assumed in recent investigations on the opening of a gap.
Using \textit{ab initio} approaches, we demonstrate that an in-plane easy axis
arises from the combination of the crystal field and dynamic hybridization
effects.Comment: 5 pages, 3 figures, typos correcte
Topological representations of matroid maps
The Topological Representation Theorem for (oriented) matroids states that
every (oriented) matroid can be realized as the intersection lattice of an
arrangement of codimension one homotopy spheres on a homotopy sphere. In this
paper, we use a construction of Engstr\"om to show that structure-preserving
maps between matroids induce topological mappings between their
representations; a result previously known only in the oriented case.
Specifically, we show that weak maps induce continuous maps and that the
process is a functor from the category of matroids with weak maps to the
homotopy category of topological spaces. We also give a new and conceptual
proof of a result regarding the Whitney numbers of the first kind of a matroid.Comment: Final version, 21 pages, 8 figures; Journal of Algebraic
Combinatorics, 201
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