Interview with Joseph Keller he moved to Stanford University, where he is now an emeritus professor. Keller has received many awards and honors throughout his career, including the von Karman Prize of the Society for Industrial and Applied Mathematics (1979), the Timoshenko Medal of the American Society of Mechanical Engineers (1984), the National Medal of Science (1988), the National Academy of Sciences Award in Applied Mathe- matics and Numerical Analysis (1995), the Nemmers Joe Keller at home in Stanford, CA. Prize from Northwestern University (1996), and Joseph B. Keller is one of the premier applied math- the Wolf Prize (1997). ematicians of recent times. The problems he has What follows is the edited text of an interview worked on span a wide range of areas, including with Keller conducted in March 2004 by Notices se- wave propagation, semiclassical mechanics, geo- nior writer and deputy editor Allyn Jackson. physical fluid dynamics, epidemiology, biome- chanics, operations research, finance, and the math- Early Years ematics of sports. His best-known work includes Notices: How did you get interested in mathemat- the Geometrical Theory of Diffraction, which is an ics? extension of the classical theory of optics and in- Keller: I was always good at mathematics as a spired the introduction of geometric methods in child. My father, who was not educated but was a the study of partial differential equations, and the smart guy, used to give my Einstein-Brillouin-Keller Method, which provided brother and me mathematical new ways to compute eigenvalues in quantum me- puzzles when we were kids. chanics. Keller has had about fifty doctoral students Here’s an example. A goose met and many collaborators. With a publication list of a flock of geese, and the goose over four hundred articles, he remains active in re- said, “Hello, 100 geese.” The search; his most recent paper appeared this year leader of the flock said, “We are in the Proceedings of the National Academy of Sci- not 100 geese. But if we were ences. twice as many as we are, plus Keller was born in Paterson, New Jersey, in 1923. half of that, and you, then we He received his bachelor’s degree from New York would be 100. How many were University in 1943. He was an instructor in physics there?” That was a typical ex- at Princeton during 1943–44 and then became a re- ample, which I happen to re- search assistant in the Columbia University Divi- member. sion of War Research during 1944–45. After re- In my high school, which was ceiving his Ph.D. from NYU in 1948, he joined the East Side High School in Paterson, faculty there and participated in the building of the New Jersey, I had several good Courant Institute of Mathematical Sciences. In 1979 mathematics teachers. One of Keller at age 4. AUGUST 2004 NOTICES OF THE AMS 751 them was Mr. Dougherty. After I left high school, Chevalley. One course I had was with Pauli. Of I didn’t hear from him until maybe twenty years course many of the people were away during the later. I was a faculty member at the Courant Insti- war, some at Los Alamos. tute. I got a letter from him saying that he had just I was able to continue studying at Princeton by purchased a copy of one of Morris Kline’s books. teaching in programs in which the armed services On the dust jacket there was a curve, and he had put people to be trained as engineers before the end tried to figure out the equation of that curve, and of the war. After I was at Princeton for one year, he didn’t succeed. He wrote 1943–44, it was recognized that the war would end that he knows that Kline is a before the students ever came out as engineers. The very busy man, but perhaps I program was terminated, so I had to find a new po- could ask him about the equa- sition. I went to the American Institute of Physics, tion of that curve. So I took which sent me to the Columbia University Divi- the letter to Kline, and he re- sion of War Research. The position was working for membered Dougherty because the Office of Scientific Research and Development, Doughtery used to bring a in offices on the fiftieth floor of the Empire State team from our high school for Building. The work was to analyze the use of sonar the Pi Mu Epsilon examination, in submarine detection. That’s what I worked on which was held at NYU every during the rest of the war. year. I had been a member of I had very good colleagues there, primarily physi- that team. When Kline read cists. My boss was a physicist named Henry Pri- Dougherty’s letter, he said, makoff, and I shared an office with another grad- “Joe, there is something funny uate student, Martin Klein, who is now a professor about that figure. I had given in the history of science at Yale. I got to meet a lot the draftsman a portion of it in of people there who were working in various as- the first quadrant, and he was pects of submarine detection. For example, I met supposed to flip it over to fill Conyers Herring, now an emeritus professor of out the whole picture. But he Joe Keller, top right, with his physics here at Stanford. did it incorrectly, and conse- brother Herbert B. Keller and Incidentally, during the time I was working there, quently there is no simple their parents, early 1950s. an airplane hit the Empire State Building. It hap- equation that would give that pened on a Saturday, and I was late to work. By the figure.” Kline was amazed that time I arrived, the building was cordoned off and Dougherty had found this, and he sent him a copy I couldn’t get in. That was a precursor of the World of the book as a reward. Dougherty was then teach- Trade Center disaster, although this one was an ac- ing at a state college in Pennsylvania. He was no cident. It was due to the fact that the cloud level longer at the high school—he had “graduated”. was very low and the pilot couldn’t see. I was always interested in mathematics. When I We worked there on the following kind of ques- went to college, I thought I would major in math- tion: How much sound is reflected back from a sub- ematics, but the first year I took a course in physics. marine? We had devices, called projectors, which I found that so attractive that I switched my major are like underwater loudspeakers and which sent to physics. By the end of my stay I had majored in out sound waves. The waves would hit whatever was both mathematics and physics. Then I got an in- out there, and some of the waves would be re- structorship in physics at Princeton—that was dur- flected back and picked up by a device called a re- ing World War II. ceiver. The sonar officer would listen, oftentimes Notices: Somebody told me that you were in- with earphones, to the reflected sound, if there volved in a course that Einstein was teaching. Is that was any. When he heard something, then he con- true? cluded there was an object out there, and he would Keller: No. When I was at Princeton, I had two try to locate it precisely. The problem was to cal- contacts with Einstein. Once I walked past him on culate how the waves traveled through the water, the street. Another time I went to a lecture by spread out, became weaker, hit the object, and Bertrand Russell, and Einstein was in the audience. then how much was scattered back, and what the I fell asleep, but Einstein didn’t. So I figured it was strength of the signal would be when it got back easy for me, and hard for Einstein. to the receiving sonar. We wanted to know, for ex- Notices: Did you hang around much with the ample, at what range would it be possible to de- mathematicians in Princeton? tect a submarine, and what was the best frequency Keller: Oh sure, because during the war there to use? The projector sent out a beam, and the beam were very few graduate students, so I hung around spread out, and if you didn’t detect anything, then with all of them. I was taking half mathematics and you would change direction and do it again. How half physics courses. I had courses from Lefschetz much should you move each time, through what and Tucker and Church and Bohnenblust and angle? How long should you wait for the signal to 752 NOTICES OF THE AMS VOLUME 51, NUMBER 7 come back? How deep is the submarine, and should you look down and up? There were all kinds of questions like that. We had a laboratory in Mountain Lakes, New Jersey. Occasionally I was sent out there to test the projectors and receivers. One of the nicest jobs I had was testing an “underwater flashlight”. It was an underwater, handheld sonar device the size of a large flashlight, and it was to be used by troops who were coming up on a beach to locate land mines and other obstructions underwater. I would have to go out swimming in the lake, which was great in nice weather, to test this device. It sent out a signal, and when the signal bounced back, I would hear it on earphones.