Shapes of Quantum States

The shape space of k labelled points on a plane can be identified with the space of pure quantum states of dimension k-2. Hence, the machinery of quantum mechanics can be applied to the statistical analysis of planar configurations of points. Various correspondences between point configurations and quantum states, such as linear superposition as well as unitary and stochastic evolution of shapes, are illustrated. In particular, a complete characterisation of shape eigenstates for an arbitrary number of points is given in terms of cyclotomic equations.Comment: Submitted to Proc. R. Statist. So

Moduli Spaces of Embedded Constant Mean Curvature Surfaces with Few Ends and Special Symmetry

We give necessary conditions on complete embedded \cmc surfaces with three or four ends subject to reflection symmetries. The respective submoduli spaces are two-dimensional varieties in the moduli spaces of general \cmc surfaces. We characterize fundamental domains of our \cmc surfaces by associated great circle polygons in the three-sphere.Comment: latex2e, AMS-latex, 24 page