Physical asymptotics. Lecture 1

The singularities of light: intensity, phase, polarization
Intensity singularities are caustics, classified by catastrophe theory, which extends to detailed descriptions of the interference patterns that decorate them on fine scales. Phase singularities are optical vortices; in three dimensions, they can be linked and knotted. Polarization singularities, with half-integer index, describe the pattern of polarized daylight, conical refraction and give new insights into crystal optics. Asymptotic connections between different levels of theoretical description will be emphasized, and the talks will be strongly visual.
Variations on a theme of Aharonov and Bohm
The partial anticipation of the AB effect by Ehrenberg and Siday was an approximation whose wavefunction was not singlevalued; its connection with the singlevalued AB wave involves topology: ‘whirling waves’ winding round the flux. AB is a fine illustration of idealization in physics. There are four AB effects, depending on whether the waves and the flux are classical or quantum; in the classical-classical case, fine details of the AB wavefunction have been explored experimentally in ripples scattered by a water vortex. The AB wave possesses a phase singularity, and there is a similar phenomenon in general interferometers. There are connections between the AB wave and the Cornu spiral describing edge diffraction. For bound systems, the interplay of AB and geometric phases exemplifies general aspects of degeneracies induced by varying parameters.
Riemann and quantum
The Riemann hypothesis can be interpreted as stating that the prime numbers contain ‘music’, whose component frequencies are the Riemann zeros. The question “Frequencies of what?” leads to tantalizing connections with the energy levels of quantum systems whose corresponding classical motion is chaotic, and the analogy suggests directions for seeking the elusive dynamical system whose corresponding quantum eigenvalues are the zeros. At the level of statistics, predictions for the Riemann zeros based on semiclassical quantum asymptotics (with primes as periods of classical trajectories) have reached a high degree of accuracy and refinement. For the zeros themselves, the Riemann-Siegel formula and its improvements lead to new ways of calculating quantum levels.