COMMUNICATION Joseph B. Keller (1923–2016) Communicated by Stephen Kennedy assistant in the Columbia University Division of War Re- Editor’s Note: Alice S. Whittemore, George Papanico- search during 1944–1945. After receiving his PhD from laou, Donald S. Cohen, L. Mahadevan, and Bernard J. NYU in 1948, he joined the faculty there and participated Matkowsky have kindly contributed to this memorial in the building of the Courant Institute of Mathematical article. Sciences. In 1979, he joined Stanford University where he served as an active member of the departments of math- ematics and mechanical engineering. Alice S. Whittemore In 1974, when Joe was based at the Courant Institute, Joseph Bishop Keller was an ap- he was asked by Donald Thomsen, the founder of the SIAM plied mathematician of world Institute for Mathematics and Society (SIMS), to oversee renown, whose research inter- a SIMS transplant fellowship at NYU Medical Center. The ests spanned a wide range of mission of SIMS was to help research mathematicians topics, including wave propa- apply their training to societal problems by temporarily gation, semi-classical mechan- transplanting them from their theoretical academic en- ics, geophysical fluid dynamics, vironments to settings engaged with applied problems. operations research, finance, Joe agreed to Thomsen’s request and promptly arranged biomechanics, epidemiology, an initial meeting with the prospective transplant, me. biostatistics, and the math- My research goals were to explore biomedical problems ematics of sports. His work involving epidemiology and biostatistics, rather than combined a love of physics, the problems in group theory with which I had been mathematics, and natural phe- struggling. During this two-year fellowship, Joe provided nomena with an irrepressible characteristically supportive and inspiring mentoring at curiosity to pursue explana- weekly blackboard sessions. Joe Keller (right) tions of practical and often We were intrigued by the with his younger playful enigmas. unknown biological mecha- brother Herbert, Keller was born in Paterson, nisms underlying the forma- who also became a New Jersey in 1923. His father tion of cancers and the role of mathematician, in their emigrated from Russia in flight environmental exposures (like childhood home of cigarette smoke) in causing Paterson, NJ, 1930. from anti-Semitic pogroms and sold liquor wholesale during them. Several investigators had Prohibition, later opening a bar. His mother, who emi- proposed quantitative theories grated from England where her family had fled Russia, did of carcinogenesis in attempts the bookkeeping for the bar. Joe’s father challenged his to explain the temporal behav- two sons (Joe and Herbert) with math puzzles at dinner. ior of cancer occurrence in hu- Both boys became mathematicians. mans and laboratory rodents Joe received his bachelor’s degree from New York exposed to carcinogens. The Keller on his 1943 University in 1943. He was an instructor in physics at theories involved the transfor- graduation from New Princeton during 1943–1944 and then became a research mation of normal cells to ma- York University. lignancy and the subsequent Alice S. Whittemore is Professor of Epidemiology and Biostatistics proliferation of malignant cells to form a detectable tumor. at Stanford University School of Medicine. Her e-mail address is A major puzzle was why the incidence of many can- [email protected]. cers increases with the fifth or sixth power of an indi- For permission to reprint this article, please contact: vidual’s age. For example, if I’m twice as old as you, my [email protected]. cancer risks are 32 or 64 times yours. To account for DOI: http://dx.doi.org/10.1090/noti1538 this puzzle, investigators proposed that a normal cell 606 NOTICES OF THE AMS VOLUME 64, NUMBER 6 COMMUNICATION becomes malignant after suffering multiple dence rates are proportional to the fourth sequential mutations. Once transformed, it power of smoking duration but only to the proliferates more rapidly than normal cells [Problems] must square of smoking rate (packs per day) until its progeny form a detectable tumor. be neither too and why the lung cancer incidence rates Exposures to carcinogens increase cancer for ex-smokers never drop to the rates of incidence by increasing the common rate at difficult nor too lifelong nonsmokers. which the mutations occur. This theory ex- Joe’s catholic curiosity about all natural plains the steep rise in incidence with age, mathematically phenomena included a broad range of bio- but it conflicts with the linear or quadratic logical, biomechanical, and bio-mathemati- dependence of incidence on dose of carcin- trivial. cal enigmas. In addition to his work on lung ogens. To avoid this conflict, Armitage and cancer in smokers and skin tumors in mice, Doll instead assumed that the sequential he wrote about the mechanisms underlying breathing at- mutations occur at different rates, only some of which are affected by a given carcinogen. Although this multi- tacks in asthmatics, vision in kittens, running in athletes, stage theory accounts for some of the cancer data, it has crawling in worms, genetics in families, hearing in humans, some biological defects. The main defect is the lack of any and leukemia in children. This work, together with his direct experimental evidence that cancer occurs in more major contributions in mathematical physics and applied than two stages. This led Armitage and Doll to modify the mechanics, has earned him some of the world’s highest theory so that only two stages were needed, but cells in scientific honors, including the National Medal of Science the intermediate stage could multiply more rapidly than and foreign membership in the Royal Society of London. normal cells, providing an increasing supply of partially When asked how he selected problems to work on, Joe transformed cells awaiting a final transition to malignancy. replied that he needed to understand the phenomenon These and other theories had been presented in a wide underlying the problem and he needed to see that it had variety of medical journals with varying degrees of math- a mathematical aspect. Moreover he needed to see that its ematical rigor, and their predictions were compared to observed patterns of cancer incidence in epidemiological or experimental data. Joe quickly saw the utility of a re- view and synthesis of the many different theories, using a common framework of stochastic equations to describe the rates of cell transformation and tumor growth in our joint 1978 SIAM Review paper on “Quantitative Theories of Carcinogenesis.” The modified two-stage theory and its subsequent extensions form the basis for much of our current under- standing of how genetic and non-genetic factors cause human cancers. They explain certain enigmas regarding the roles of cigarette smoking in lung cancer, of mam- mographic density in breast cancer, and of genetics in colorectal cancer and retinoblastoma (a malignancy of the eye). For example, they explain why lung cancer inci- Mathematician and future wife Alice Whittemore with Whittemore and Keller on a long-distance hiking trail Joe Keller in NJ, 1975. (Grande Randonnée) in France, circa 1995. JUNE/JULY 2017 NOTICES OF THE AMS 607 COMMUNICATION solution could be enlightening and significant. And, like the careful use of the very Goldilock’s chair, it must be neither too difficult nor too few exact solutions of dif- mathematically trivial. Joe loved working with students fraction problems that were There was and colleagues, many of whom approached him for help available, and a consistent in solving problems they had encountered in their own geometric interpretation of something work, and these associations led to fruitful collaborations. the wave components that special about One evening soon after the start of my transplant contributed to the overall fellowship at NYU, Joe and I were both hungry at the wave field. The insight that Joe Keller and conclusion of our weekly blackboard session, so we de- his geometrical theory of cided to grab a bite at a nearby Chinese restaurant. My diffraction provided came seminars. mathematical education continued at dinner, but now from its elegant conceptual the subject was inverse problems. Joe explained that in a simplicity. From the location typical mathematical problem, you are given a question and geometry of corners, edges, and other features of the for which you must provide an answer. In an inverse scattering environment, one could in principle write down problem, however, you are given the answer and your job the high-frequency form of the field anywhere, but this is to provide a corresponding question. For example, you would require numerical computations except in relatively might be asked to provide a question to which the answer simple cases. Scattering phenomena in the high-frequency is “1 and -1,” and your question might be, “What are the regime even today are difficult to analyze with direct roots of the equation x2 – 1 = 0?” To further clarify the numerical computations because the region of interest concept, he then gave me several other answers in need of could extend over thousands or millions of wavelengths corresponding questions. When he asked me to provide a in a three-dimensional setting. question whose answer was “Dr. Livingston I presume,” I I learned diffraction theory and uniform asymptotic suggested a question based on Stanley’s search for Living- methods by going to the Friday afternoon applied mathe- ston near the Nile. He promptly informed me that, while matics seminar at Courant. This was a more general theory that question was acceptable, the optimal question was that could, in diffraction for example, express the field “And what is your full name, Dr. Presume?” across a shadow boundary in a way that was almost exact That evening, neither of us knew that we were starting near it. It was a lot more elaborate and less geometrical a life of work, fun, and happiness together that would last than Joe’s original geometrical theory of diffraction.