Modern Applications of Complex Variables: Modeling, Theory and Computation

All professional mathematicians have taken an undergraduate class in complex analysis, covering analytic functions, singularities, residue calculus, conformal mapping, and so on. Indeed, one could argue that an early appreciation of the beauty of complex analysis, and of its power to solve a range of physically-arising problems, is a good indicator of a budding mathematician. Nonetheless, following a “golden age” after the Second World War, when complex variable theory (including integral transform methods) was used to solve a huge range of applied, mainly idealized, problems arising in fluid dynamics, elasticity and many other areas, complex analysis fell into disfavor among some applied mathematicians, who believed that all of the exciting discoveries had been made and that the subject had little new to offer. (Classic references from this period include the books of Noble, Sneddon, and Mittra & Lee [31, 37, 48].) In particular, the increasing prevalence of computers towards the end of the 20th century meant that many previously “in principle” calculations could now be done in practice: more complicated models could be quickly and cheaply solved numerically, obviating what some saw as the primary use for complex variable methods, the need for exact solutions to simplified models. The range of speakers and topics gathered together for the 5-day workshop 15w5052 provided resounding evidence that the field of applied complex analysis is moving into a new era, burgeoning with new discoveries. The broad spectrum of subject areas they represented implies that complex analysis continues to play a central role in many areas of mathematics ranging from harmonic analysis to applied scientific computing. Particularly encouraging was the large number of early to mid-career mathematicians among the participants, whose work indicates that the field will be healthy for decades to come. The following sections of this report summarize the cutting edges of the subject that our Workshop considered, the specific outcomes of our Workshop, and the outlook for the future.