Hamilton's Diabolical Singularity (Shih-I Pai Lecture)
Hamilton’s first application of the concept of phase-space – later so fruitful in the transition to quantum physics - was a prediction in optics: conical refraction in biaxial crystals. This created a sensation at the time (1831), probably because it was one of the first successful uses of mathematics to predict a qualitatively new phenomenon. At the heart of conical refraction is a singularity, anticipating the simplest geometric phase and the conical intersections extensively studied in quantum chemistry and now resurrected in graphene. The light emerging from the crystal contains many subtle diffraction details, whose definitive understanding and observation have been achieved only recently. Generalizations of the phenomenon involve radically different mathematical structures, reflecting the different physics of real symmetric, complex hermitian, and nonhermitian two by two matrices, that are still being explored theoretically and experimentally.