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.In this lecture, a review will be presented of how a concrete problem in the area of integrability led to the development of a new method in mathematical physics for analyzing boundary value problems for linear and for integrable nonlinear PDEs,called the "Unified Transform". This method has been acclaimed by the late 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 development of several effective algorithms for Medical Imaging,and with the Riemann hypothesis will also be reviewed.