Two amusing problems in geometry and topology
Seminar Room 1, Newton Institute
This lecture will provide two extended discussions of mathematical problems with strong physical content. The first concerns the dynamics of topological rearrangements of soap films under slow deformation of their boundaries. Through a combination of theory and experiment we illustrate some fascinating new phenomena involving reconnection of Plateau borders and finite-time singularities. Some conjectures are advance on the simplest dynamical models for such behaviour. The second problem of interest is the shape of a ponytail, which we show involves understanding the statistical physics of quenched random curvatures. A density functional theory of hair fibre bundles is constructed that leads to a 'Ponytail Shape Equation' whose solutions describe the envelope of a hair bundle in terms of an equivalent single fibre subject to a radial pressure arising from those random curvatures. Comparison with experimental observations allows for the extraction of the 'equation of state' of hair.