4. An Intellectual Crossroads (2005)

Europe has nothing like the MRC, where scientists are given complete freedom to think, with no lectures or administrative responsibilities—no interruptions. —Alexander Ostrowski, Basel, Switzerland, 1961 Introduction At a time when the ascendant view of mathematics was that of pure mathematics (much of it in the Bourbaki tradition), the Mathematics Research Center presented a more complete view of the mathematical world. MRC provided a rich intellectual environment where resident mathematicians continuously interacted with a stream of visitors both domestic and foreign. In a very real sense it was an intellectual crossroads in the world of applied mathematics, and one could argue that for a considerable time it was preeminent in the world in that respect. Many senior members of the center had a rich history of working on applied problems, generated either by other disciplines or by real-life applications. Several bright young scholars who visited the center or were postdoctoral fellows there benefited immensely in the formulation of their perspective on the world of mathematics and future directions and opportunities for creative mathematical research. A mathematician may obtain great benefit by solving a practical problem, and the center provided ample opportunities for such solution through its interactions with the Army and other government agencies. However, MRC provided more than that: namely, an opportunity to pursue related questions that were, at the core, of purely mathematical interest. This allowance to deviate on mathematical tangents was at the crux of the center's intellectual achievements. In this chapter, we describe a few of these outstanding scientific accomplishments, which have had a profound impact on application domains as well as on the evolution of applied mathematics during these decades. We also mention other aspects of the MRC environment that were important in promoting and sustaining this scientific productivity. Spline functions Background Spline functions are a class of piecewise polynomial functions satisfying continuity properties only slightly less stringent than those of polynomials. They are a natural generalization of polynomials, and they have highly desirable characteristics for use as approximating, interpolating, and curve-fitting functions. It was Newton who pioneered the use of polynomials for obtaining numerical approximations for interpolation, integration, differentiation, and other such operations on functions of one variable. In these methods, interpolation plays a preferred and fundamental role. In 1949, Arthur Sard introduced a new point of view leading to his best approximation formulae. In this new approach to Newton's old problems, it seemed as if interpolation was losing the fundamental role that it played in Newton's work. It turns out that interpolation can be restored to its central role, provided that we replace the Newtonian polynomial interpolation of the data by interpolation using the so-called natural spline functions. Early history Although they were not referred to by this term, spline functions had been used before mid-century in a few isolated instances. In 1904, Carl Runge used periodic quadratic splines.