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

Homotopy Quantum Field Theories and Related Ideas

In this short note we provide a review of some developments in the area of homotopy quantum field theories, loosely based on a talk given by the second author at the Xth Oporto Meeting on Geometry, Topology and Physics.Comment: 8 pages, 2 figures; correcte

Teaching White Papers Through Client Projects

White papers are increasingly prevalent in business and professional settings. Although textbook resources for white paper assignments are limited, a white paper assignment completed for a community client can provide a learning experience that students enjoy and that strengthens ties between the university and the community. This article describes a way to approach the white paper assignment in a communications-focused course and identifies resources to support white paper assignments

The magnitude of odd balls via Hankel determinants of reverse Bessel polynomials

Magnitude is an invariant of metric spaces with origins in category theory. Using potential theoretic methods, Barceló and Carbery gave an algorithm for calculating the magnitude of any odd dimensional ball in Euclidean space, and they proved that it was a rational function of the radius of the ball. In this paper an explicit formula is given for the magnitude of each odd dimensional ball in terms of a ratio of Hankel determinants of reverse Bessel polynomials. This is done by finding a distribution on the ball which solves the weight equations. Using Schröder paths and a continued fraction expansion for the generating function of the reverse Bessel polynomials, combinatorial formulae are given for the numerator and denominator of the magnitude of each odd dimensional ball. These formulae are then used to prove facts about the magnitude such as its asymptotic behaviour as the radius of the ball grows