37 research outputs found
Spread: a measure of the size of metric spaces
Motivated by Leinster-Cobbold measures of biodiversity, the notion of the
spread of a finite metric space is introduced. This is related to Leinster's
magnitude of a metric space. Spread is generalized to infinite metric spaces
equipped with a measure and is calculated for spheres and straight lines. For
Riemannian manifolds the spread is related to the volume and total scalar
curvature. A notion of scale-dependent dimension is introduced and seen,
numerically, to be close to the Hausdorff dimension for approximations to
certain fractals.Comment: 18 page
An almost-integral universal Vassiliev invariant of knots
A `total Chern class' invariant of knots is defined. This is a universal
Vassiliev invariant which is integral `on the level of Lie algebras' but it is
not expressible as an integer sum of diagrams. The construction is motivated by
similarities between the Kontsevich integral and the topological Chern
character.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-29.abs.htm
The twisted Drinfeld double of a finite group via gerbes and finite groupoids
The twisted Drinfeld double (or quasi-quantum double) of a finite group with
a 3-cocycle is identified with a certain twisted groupoid algebra. The groupoid
is the loop (or inertia) groupoid of the original group and the twisting is
shown geometrically to be the loop transgression of the 3-cocycle. The twisted
representation theory of finite groupoids is developed and used to derive
properties of the Drinfeld double, such as representations being classified by
their characters.
This is all motivated by gerbes and 3-dimensional topological quantum field
theory. In particular the representation category of the twisted Drinfeld
double is viewed as the `space of sections' associated to a transgressed gerbe
over the loop groupoid.Comment: 25 pages, 10 picture
On the magnitude of spheres, surfaces and other homogeneous spaces
In this paper we define the magnitude of metric spaces using measures rather
than finite subsets as had been done previously and show that this agrees with
earlier work with Leinster in arXiv:0908.1582. An explicit formula for the
magnitude of an n-sphere with its intrinsic metric is given. For an arbitrary
homogeneous Riemannian manifold the leading terms of the asymptotic expansion
of the magnitude are calculated and expressed in terms of the volume and total
scalar curvature of the manifold. In the particular case of a homogeneous
surface the form of the asymptotics can be given exactly up to vanishing terms
and this involves just the area and Euler characteristic in the way conjectured
for subsets of Euclidean space in previous work.Comment: 21 pages. Main change from v1: details added to proof of Theorem