Report of the Second European Zebrafish Principal Investigator Meeting in Karlsruhe, Germany, March 21-24, 2012.

The second European Zebrafish Principal Investigator (PI) Meeting was held in March, 2012, in Karlsruhe, Germany. It brought together PIs from all over Europe who work with fish models such as zebrafish and medaka to discuss their latest results, as well as to resolve strategic issues faced by this research community. Scientific discussion ranged from the development of new technologies for working with fish models to progress in various fields of research such as injury and repair, disease models, and cell polarity and dynamics. This meeting also marked the establishment of the European Zebrafish Resource Centre (EZRC) at Karlsruhe that in the future will serve as an important focus and community resource for zebrafish- and medaka-based research.

Steiner’s formula in the Heisenberg group

Steiner’s tube formula states that the volume of an ϵ-neighborhood of a smooth regular domain in Rn is a polynomial of degree n in the variable ϵ whose coefficients are curvature integrals (also called quermassintegrals). We prove a similar result in the sub-Riemannian setting of the first Heisenberg group. In contrast to the Euclidean setting, we find that the volume of an ϵ-neighborhood with respect to the Heisenberg metric is an analytic function of ϵ that is generally not a polynomial. The coefficients of the series expansion can be explicitly written in terms of integrals of iteratively defined canonical polynomials of just five curvature terms.