# Crossing borders, by mathematical means

14 April 2011

A large number joined Professor Sir Michael Berry at the BRLSI in February for a fascinating journey of discovery across the varied territories of physical theories and models.

The “crossing of borders” – with its challenges and perils – was in fact the focal point of this talk, the indispensible tool for safe passage being the branch of maths called *asymptotics*, the study of limits of infinite series – divergent series in particular.

Professor Berry showed how progress in the understanding of infinite series used to represent physical quantities can shed light onto the apparent unsettling existence of singularities – discontinuities or infinities – when attempting to reduce a physical theory to another, usually less general, model of description.

He showed that connections between the different models – “patches of description” – can thus be established, and that these in fact lead to the discovery of novel – emergent – phenomena, or to the explanation of previously puzzling observations.

He looked at several examples, from the unusually “smooth” case of reduction of special relativity to Newtonian mechanics, to the case of wave optics “merging” into geometrical ray optics in the limit of infinitesimal wavelengths, to the reduction of quantum mechanics to classical mechanics in the limit of Planck constant becoming infinitesimally small – the smallness of such quantities to be intended with respect to the typical size of the system under consideration.

Professor Berry described in detail the beautiful case of the Airy integral in optics, and the interference oscillations that lead to the so-called supernumerary rainbows.

He explained the transition between microscopic statistical mechanics and thermodynamics, with the singularity describing the critical state of a system and its fractal nature.

He also explained how singular behaviour in the transition from viscous to inviscid flow is linked to the emergent feature of turbulence.

Mathematics is arguably more than just a powerful tool: it is the language we have developed to effectively decipher Nature’s mysteries, in the words of the greatest – from Galileo to Feynman – it is the very language Nature speaks in.

Professor Berry has employed his fluency in that language to achieve its most effective translation into metaphors and analogies of great communicative impact, aided in this by awesome images and experimental observations.