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