Since at least the time when it was understood that the circumference of a circle is pi multiplied by its diameter, the applications of mathematics have raced on far ahead of the foundations of the subject itself. By considering a variety of examples, principally from the 19th century, we will explore the tension between mathematics and its applications, and reasons why it remains a valuable and rewarding occupation to develop the necessary framework for existing and “well understood” theories. If time permits, I will give a short discussion of recent work I have done on the mathematical foundations for diffusion models in population genetics.

Charles L. Epstein received his Ph.D. from New York University in 1983. After three years as a postdoctoral fellow at Princeton University, he joined the faculty of the Department of Mathematics at the University of Pennsylvania, where he currently holds the Thomas A. Scott Chair in Mathematics. He has worked in spectral theory, hyperbolic geometry, univalent function theory, microlocal analysis, several complex variables and index theory. For more than a decade, he has also worked on a range of problems in medical imaging, image analysis, computational electromagnetics and mathematical aspects of population genetics. He was a Sloan Foundation Fellow in 1988–90. In 2007, he founded the Graduate Group in Applied Mathematics and Computational Science at the University of Pennsylvania, which he continues to chair.