Mathematical applications, computation, and complexity

Abstract

In this lecture I want to discuss some matters which are not themselves applied mathematics but are about the profession and practice of applied mathematics. In the trade, the measures of an article's quality are the sparsity of the prose, the swiftness of the mathematics, and the sweet connection to the real world. This one will be rather thick in prose and sparse in mathematics. I would have liked to have been able to respond to Prof. Davis's invitation to be philosophical, but that has always seemed to be a dangerous way to behave publicly. I offer these comments from the point of view of an industrial mathematician and as one of the Old Browns.When I arrived at Brown nearly twenty-five years ago, the world looked a little different to a young applied mathematician than it does to a middle-aged applied mathematician now. It was clear that what was to be learned at Brown was continuum mechanics and the complex function and differential equation theory that was a part of it. Fluid mechanics, elasticity, plasticity-we were beginning to grapple with those problems that von Karman had said we were to in his 1940 Gibbs lecture [1]. We were really not yet "tooled up" for nonlinearity, as he urged in the first issue of the Quarterly of Applied Mathematics [2], but there was a realization of its importance. There was an atmosphere of the beginning of something at Brown in the late 1940s and, in fact, it was a beginning-a feeling that there were many, many problems to be solved in aerodynamics, flutter and aero-elasticity, viscous flow and boundary layers, plastic design, vibration problems. These were obviously good problems to solve because someone, somewhere needed the solutions, or they were hard, or they were the next problem in their field, or just because they were there. We were (and, of course, I'm aware of that benevolent distortion that occurs in looking backward this way) an optimistic crew who had a great deal of confidence in the results, techniques, and styles of thought of several generations of European applied mathematicians. One can observe that for American science, applied mathematics was a rather new field then, not very populated or popular but with an apparent good potential for growth. Furthermore, when scientists thought of the uses of mathematics, they were used to thinking of the applications of classical analysis to the problems of the physical sciences.In a corner of the house at 27 Brown Street there was a large room full of girls and machines--our computers. It was a small but busy and necessary enterprise. Other important locations were the softball field and the house on Benevolent Street where the Friday afternoon post-colloquium sessions were held.Before nostalgia overcomes, what happened to all of that? Where are we? What I think has happened is that the needs for and uses of mathematics have