James Eells

Innovative mathematician

James Eells, mathematician: born Cleveland, Ohio 25 October 1926; Professor of Analysis, Warwick University 1969-92; married 1950 Nan Munsell (one son, three daughters); died Cambridge 14 February 2007.

The mathematician James Eells moved to Britain from the United States in the late Sixties, becoming Professor of Analysis at Warwick University, and had an important, and continuing, effect on the development of mathematics in his adopted country.

His appreciation of the difficulties faced by scientists, and mathematicians in particular, working in Third World countries led to his becoming in 1986 the first director of a new mathematics section at the International Centre for Theoretical Physics at Trieste: one of the centre's main aims is to give Third World scientists the opportunity to keep up to date with rapidly moving areas of science and mathematics.

Born in 1926 in Cleveland, Ohio, Eells went to school at Western Reserve Academy until his exuberance led to his expulsion. However, he was admitted to Bowdoin College, a distinguished liberal arts college in Maine, graduating in 1947. After Bowdoin he had a "gap" year teaching mathematics at Robert College (now Bogaziçi University) in Istanbul. He returned to the US to be an instructor in mathematics at Amherst, during which time he met his future wife, Nan.

After two years he decided that his career really was to be in mathematics and enrolled for a PhD at Harvard under the topologist and analyst Hassler Whitney. In 1954 he, with Nan, two young daughters and a PhD, set off on his academic career. This started at the Institute for Advanced Study at Princeton, went on to the University of California at Berkeley, returned to the East Coast for a position at Columbia University, New York; and in 1964, after a year at Churchill College, Cambridge, he took a full professorship at Cornell University.

He returned to Cambridge for 1966-67; two events that year gave a foretaste of the future. In December he visited Ghana, in connection with Churchill College, to give a mathematics course, having persuaded a Cornell colleague, Cliff Earle, to make a follow-up visit early in 1967. Between the two visits they began fruitful joint research, and Jim Eells was invited to run a small symposium in the summer of 1967 in Warwick, with Earle as one of the principal participants.

The innovative mathematics department of the new university of Warwick had already gained an international reputation and in 1969, excited by the freedoms and potential in Warwick, Eells joined the Mathematics Department there, becoming its first Professor of Analysis.

It was an appointment which fitted perfectly with the philosophy of the department at that time, which was to feature research in global, rather than traditional, analysis; and it was already becoming known as a centre for the global approach to dynamical systems theory. It is tempting to describe global analysis as a holistic approach to mathematics. In it the whole geometry or topology of the spaces involved play a role, rather than just the equations describing the behaviour or motion in small areas. Non-linearity, especially that caused by curvature, is a prevalent aspect. A prime example is Eells's most famous article, "Harmonic Mappings of Riemannian Manifolds", published in the American Journal of Mathematics in 1964. Written with J.H. Sampson of Johns Hopkins University, it founded the theory of "harmonic maps" and the "non-linear heat flow".

The latter describes the behaviour of one shape stretched over another. For example, think of an uninflated balloon wrapped around the surface of a sphere, anchor ring or of some other shape, and allowed to settle down, relaxing, but only moving on the surface. Strange-shaped balloons are also allowed, for example water rings or just a piece of balloon with ends fixed. The final position of minimal stretching would depend on the shape of the balloon, the shape of the surface, and the way it was wrapped. For example, you might force the balloon over the sphere before sealing the end, or you might just lie the balloon slightly stretched over part of the surface. The end result is the "harmonic map".

If you did it with a rubber band rather than a balloon, the harmonic maps would be along geodesics, lines of shortest length (possibly just one point). These generalisations of geodesics were found to have profound importance in various parts of geometry, and in some aspects of theoretical physics such as string theory. The "settling down" motion, governed by the non-linear heat equation, was essentially the first example of a "geometric evolution", the study of which has been of increasing importance and finally appears to have led to a proof of the famous "Poincaré conjecture", one of the classic problems, for which there is a $1m prize. In fact, in lectures and articles in the early Sixties, Eells gave the Poincaré conjecture as a potential application of the general approach he was following.

Infinite dimensional objects such as the space of all paths on a surface often occur in global analysis, and one of Eells's early works was to show that many such spaces have a "smooth" structure which enables calculus to be applied on them. He also played a leading role in the classification of such infinite dimensional manifolds. Global analysis and probability theory were combined in foundational work in the analysis of Brownian Motion on curved spaces, such as the surface of spheres.

One result was that, if you roll a surface along the trajectory of a typical Brownian path on an ink-covered plane, the mark the ink makes on the surface will be the trajectory of a "Brownian particle on the surface". A difficulty here is that Brownian particles move so irregularly that standard, Newtonian, calculus cannot be applied, so even what it means to "roll" is unclear. It has to be replaced by Ito calculus, the calculus which is also now in regular use in mathematical finance. From the mid-Seventies, however, the main part of Eells's work was related to harmonic map theory.

The highly successful, year-long, Warwick symposia he organised, "Global Analysis" in 1971-72 and "Geometry of the Laplace Operator" in 1976-77, to a large extent introduced novel mathematics to Britain, and together with Eells's many and energetically pursued connections abroad, attracted fine PhD students and research fellows to study with him. After the first of these, in the summer of 1972, he took the symposium to the International Centre for Theoretical Physics at Trieste for the first session held there on mathematics. This was an early move in the development of a Mathematics Division at ICTP, of which he became the first director, from 1986 to 1992, on partial secondment from Warwick. His research activities in Warwick were barely diminished, however, and if anything there were more visitors and visiting scholars, together with another year-long symposium, "Partial Differential Equations in Differential Geometry", in 1989-90.

Eells's mathematical influence internationally became even more substantial, both personally, as a mentor, and scientifically, through the spread of his work and that which was built on it. He had at least 38 graduate students, most of whom became academics. However there are many more throughout the world who owe their careers to him.

Many, together with their families, will also especially remember Nan Eells for her friendship and excellent dinner parties, with only mild attempts to control the exuberance of her husband. Jim Eells was a man of irrepressible enthusiasm for mathematics, and for most other things; especially people, irreverent fun, lots of wine, and music. His interest in the latter ranged across most styles and he would delight the younger children of his colleagues with lively performances of scatological songs.

He retired from Warwick in 1992, moving to Cambridge. He continued working on harmonic maps and travelling abroad to discuss mathematics or for conferences. He also helped to set up a regular UK-Japan Winter School in Mathematics. His final monograph, Harmonic Maps between Riemannian Polyhedra, co- authored with the Danish mathematician B. Fuglede, was published in 2001.

David Elworthy

Published: 17 April 2007 The Independent