Paco A. Lagerstrom Lecture

The need to solve a concrete problem of physical significance occasionally leads to the development of a new mathematical technique. It is often realized that this technique can actually be used for the solution of a plethora of other problems, and thus it becomes a mathematical method. Problems arising in fluid mechanics have historically played a crucial role in the emergence of several methods in Applied Mathematics. The birth at Caltech of the department of Applied Mathematics from the department of Aeronautics provides a clear illustration of the above fact. In this lecture, a review will be presented of how a problem posed by the late Julian Cole, who was the first (joint) student of the late Paco Lagerstrom, led to the development of the so-called "Unified Transform." This method has been acclaimed by Israel Gelfand as, "the most important development on the exact analysis of PDE since the work of the classics in the 18th century." Remarkable connections with the Riemann hypothesis, as well as the development of several effective algorithms for Medical Imaging will also be reviewed. Regarding the latter, it should be noted that Mathematics provides a marvelous illustration of the innate ability of the brain to learn and to create using associations and generalizations, which forms the basis of abstract thinking. The role of the above brain imaging algorithms for elucidating underlying neuronal mechanisms will also be briefly discussed.