Rothschild Lecture: Extraordinary waves and math: from beaches to photonics

Waves are fascinating. They are of keen interest across the realm of science and mathematics and they often elicit excitement in the general public.  There is a class of extraordinary localized waves, called solitary waves, that were first observed 185 years ago. More recently certain solitary waves, often called solitons, have been found to possess numerous special properties. This lecture will discuss the history of these waves, and will indicate why mathematics played a crucial role in both historical and modern developments. Some applications will be described including as time permits: water waves, rogue waves, photonics and cellular automata. The discussion will be general; it will  `leave almost all equations behind'.