# Obituaries: Joseph B. Keller

One of Keller’s most outstanding contributions is the Geometrical Theory of Diffraction (GTD), which he originated for solving problems of wave propagation. He began thinking about such problems in work during World War II, on problems of sonar. GTD is an important extension of the Geometrical Theory of Optics (GTO), where wave propagation is described by rays. The extension overcomes difficulties which cannot account for phenomena such as diffraction, or the occurrence of signals where GTO predicts none. Keller developed a systematic way to treat high-frequency waves, and thus derived and solved the equations determining the rays, or paths along which signals propagate, as well as those governing how signals propagate along the rays. These include predictions of what happens as rays encounter obstacles or inhomogeneities of the medium in which they travel. Prior to Keller’s work, only a few isolated problems were solved and understood, and there was no general theory for the solution of more complex and technologically important problems. Now there exist books devoted to Keller’s theory, as engineers and scientists employ his theory to this day. Indeed, it is an indispensable tool for engineers and scientists working on radar, antenna design, and in general, on high-frequency systems in complicated environments. The impact of his work could be judged by attending a meeting of URSI, the international society devoted to radio science, where sessions were devoted to Keller’s methods. This theory has been and is still applied to a variety of other problems in which signals are transmitted by waves, including acoustics, as in problems of sonar, and elastodynamics, as in quantitative non-destructive testing, and seismic exploration for oil, to name but a few. It is commonplace in all these fields to see articles which read, “we employ Keller’s method to...”.

Keller also showed that the methods developed for wave propagation were extendable to other classes of problems. Among these is his fundamental and penetrating work on semi-classical mechanics. In his work, Keller generalized work by Planck, Bohr, Sommerfeld, Wilson, Einstein and Brillouin to derive the correct quantization rules for non-separable systems, thus yielding results valid in any coordinate system. His results, referred to as the Einstein-Brillouin-Keller (EBK) quantization rules, are employed by many scientists. In his work on semiclassical quantization he introduced an important measure, the number of times a closed curve passes through a caustic surface, later generalized by Maslov and called the Keller-Maslov index. This too was subsequently extended by Keller to eigenvalue problems in bounded domains, not necessarily associated with quantum mechanics, but governed by general systems of partial differential equations.

Keller’s work stimulated a vast literature in both the U.S. and abroad, not only in areas of science and engineering, where his methods and results are routinely employed, but in the mathematics community as well, where his work was taken up by pure mathematicians. For example, his work was the impetus for developments in the theory of Fourier Integral Operators and Lagrangian Manifolds.

In addition, he opened up directions of investigation by considering problem areas which were enthusiastically taken up by the research community. For example, his pioneering work on the evolution of singularities of nonlinear wave equations, as well as on bifurcation theory and nonlinear eigenvalue problems, to which scant attention was paid until the notes of his seminar appeared, is now one of the hottest topics of investigation by both pure and applied mathematicians alike.

No stranger to national service, Keller worked on many problems related to national security and served on various advisory boards, national panels, and committees. After his work on sonar for the Columbia University Division of War Research, he worked on problems of underwater explosions in order to predict the shock wave and water waves to be expected at the Bikini atomic bomb tests. At the time, there was concern of producing a tsunami which might devastate Japan and other Pacific countries. His analysis showed there was no danger. He also spent time at Argonne and Los Alamos National Laboratories, studying hydrogen bomb explosions. In the early 1950s he served, with Von Neumann, on a Committee on Underwater Atomic Bombs for the Air Force to consider the effects of A-bomb explosions on ships and submarines, and headed another project on A-bomb explosions. During the late 1960s he was a member of JASON, a group of high-level consultants to the Defense Department and other governmental agencies on scientific and technical matters. Furthermore, he served as consultant to the Air Force Special Weapons Project, U.S. Naval Air Development Center, U.S. Army Chemical Corps, and Argonne, Brookhaven, and Los Alamos National Laboratories.

In addition to his important and prolific research, Keller was a teacher and expositor par excellence. He twice received the MAA’s Lester R. Ford Award for outstanding expository writing. He received awards from all three major U.S. mathematical societies, from various engineering societies, as well as from national honorary societies, both in the U.S. and abroad. The 60 Ph.D. students and numerous postdoctoral associates whom he has trained, now successful applied mathematicians in their own right, further attest to the impact that Keller had. In short, Keller was one of the most prolific and important investigators and educators of our time.

Finally, there is Joe Keller the man. Countless numbers of mathematicians, engineers and scientists have come to him through the years to benefit from his acumen and understanding. To each he listened patiently, contributed helpful insights, and offered words of advice and encouragement. For each he was simply “Joe,” teacher, colleague, and friend. The world has lost a giant. He will be sorely missed, though his legacy endures.