At the interface between semiclassical analysis and numerical analysis of wave propagation problems

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Our understanding of wave propagation underpins several important technologies in everyday use. For example, WiFi and mobile phones use electromagnetic waves to transmit information, and the technologies in seismic and medical imaging use acoustic, elastic, and electromagnetic waves to obtain information about the ocean floor and the human body. Over the last few years, the investigator has worked with, on the one hand, researchers interested in wave phenomena from a purely mathematical point of view (without any thought of applications), and, on the other hand, researchers interested primarily in improving how wave phenomena are simulated in applications (with this understanding then feeding into new technologies). This experience has uncovered huge untapped potential in the relationship between the theoretical aspects of wave propagation and the more-practical aspects, and this fellowship seeks to exploit this.The overall goals are to (i) prove fundamental theoretical results about wave propagation, motivated by applications, and (ii) use these theoretical results to prove fundamental results about how wave propagation is simulated using computers, addressing long-standing open problems and developing new numerical methods that have the potential to change the technologies used in the huge variety of practical applications of wave propagation.