“Recountings tells of the influential US mathematics department at the Massachusetts Institute of Technology (MIT) through interviews with a dozen faculty members . The interest in teaching among these senior faculty members is broad and deep. The professors share their strategies for achieving research success, from working on prize problems to developing an intuitive feel for proofs. They explain how new research directions have come from interactions with students and colleagues or from writing a review article. The insights in [this book] will inspire mathematicians and scientists to come.” —Eric Altschuler, NATURE “Students currently contemplating or pursuing a mathematics-related career should find MIT’s oral history illuminating and thought provoking. It is a remarkable institutional story, recalled and superbly narrated by those much concerned.” —Mathematics Teacher “Recountings provides a history of the MIT Mathematics Department, as told through interviews conducted with 12 of its current and former faculty members, plus Zipporah (Fagi) Levinson, widow of Norman Levinson (and one-time ‘den mother’ of the department). What emerges is a piecemeal yet compelling portrait of the department’s rapid post-WW II transformation from a program largely focused on offering mathematical instruction to engineering students, to a world-class research enterprise. Highly recommended.” —CHOICE “Though never in the eye of popular culture, these men kept society advancing with their minds. Recountings is a collection of interviews and anecdotes from the geniuses of MIT who have pursued mathematics as their life’s careers and obsessions. These men have been responsible for major scientific advances throughout history . Recountings is an intriguing look at mathematics and the men behind it.” —Library Bookwatch “It’s full of really good stuff. these offerings are in the form of interviews, much along the lines of the way Mathematical People and More Mathematical People were set Recountings up—and equally successfully. It all reads very well, and is incredibly fascinating . It’s a terrifically interesting book, and just plain fun. I’ll read it again and again.” —MAA Reviews EDITED BY JOEL SEGEL Recountings Recountings Conversations with MIT Mathematicians Edited by Joel Segel A K Peters Ltd. Natick, Massachusetts CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2010 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20140703 International Standard Book Number-13: 978-1-4398-6541-5 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reason- able efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. 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Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com Dedicated to William “Ted” Martin (1911–2004) & Norman Levinson (1912–1975) Contents Introduction ix From the Department Head xv Acknowledgments xvii Photo Credits xix Zipporah (Fagi) Levinson 1 Isadore M. Singer 25 Arthur P. Mattuck 43 Hartley Rogers 105 Gilbert Strang 159 Kenneth M. Hoffman 183 Alar Toomre 231 Steven L. Kleiman 271 Harvey P. Greenspan 307 Bertram Kostant 329 Michael Artin 351 Daniel J. Kleitman 375 Sigurdur Helgason 411 vii Introduction This book is not about mathematics but about mathematicians, indi- viduals who have made mathematics the pursuit of a lifetime. It also recounts the history of an academic department that wanted to learn about its most recent and successful half century from the faculty members with the longest memories. The project began with a thought. Professor Gilbert Strang was regretting the loss of Gian-Carlo Rota, who died suddenly in his sleep at age 66. Rota had single-handedly put combinatorics on the map of mathematical respectability. He held the unique title of Professor of Applied Mathematics and Philosophy, had published a collection of thought- provoking essays called Indiscrete Thoughts, and was one of MIT’s most popular teachers. What a pity his lectures were not preserved, Strang thought. His wife, Jillian Strang, brought a historian’s perspective to the problem. Why not interview some of the older generation? she suggested. Get their recollections down on paper before these memories, too, were lost. Strang mentioned the idea to I. M. (Is) Singer, who discovered the index theorem with Michael Atiyah and whose opinion is key to department enterprise. Singer liked the idea, as did department head Michael Sipser. The core mis- sion: ask surviving members of today’s senior generation what they remem- ber of the department’s history. By any account, after all, the department that these mathematicians built is a success story. In the fi rst half of the twentieth century, astonishing as this may seem for the home of Norbert Wiener, the department’s primary mis- sion was teaching MIT’s engineering students the mathematics they needed to know. The generation interviewed here was hired as part of an ongoing effort to transform MIT mathematics from a service department to one focused on both teaching and research. Today MIT is ranked in the top handful of any- one’s list of the best mathematics departments in the country. The collective memory of the department’s oldest representatives—twelve men, 65 and older, all but two still working, as well as the widow of department godfather Norman Levinson—provides an important chapter in the story of that success. ix xIntroduction The core idea of the project, once articulated, soon expanded to include not only memories of the department but also a focus on the individuals themselves. Anyone hired to a tenure-track position at MIT is exceptional. It therefore made sense to ask about the years before they came on board, beginning in childhood. How did bright youngsters fi rst become attracted to mathematics, and how was their initial interest nurtured? Any educated person today will have a sense of mathematics as the fabric underlying our understanding of the physical world, a crucial element of the language we use to think about science and engineering. This realization was dawning only gradually in the years following World War II: most of the interviewees were only dimly aware, if at all, of a research mathematician as something real people became. In the 1930s this was to be expected: Norman Levinson (1912–1975), a formidable mathematician whose judgment and behind-the- scenes leadership profoundly shaped the mathematics department at MIT, assumed he would train as an actuary and get a job in insurance. Levinson knew jobs were scarce in those Depression-era days, even more so for the son of Jewish immigrants. Yet an entire generation later, an Estonian immi- grant of practical bent like Alar Toomre still saw mathematics as “too much like opera. You got paid only if you were terrifi c.” Even Michael Artin, a future leader in modern algebraic geometry, assumed he would get a teach- ing position at a small-town college somewhere. The mathematics community has made great strides in identifying and nurturing mathematical talent and inclination. Fifty years ago, the process was much more haphazard. The paths these young men took to a career in mathematics were varied, and few had the fi nal goal in mind. Some ambled up to the front door, set on their path by an early delight in number manipu- lation that deepened over time. Others clambered in through the back win- dow, trying to understand the mathematics that undergirded an initial love of chemistry or physics. Bertram Kostant remembers being fascinated by the formulas of chemical reactions, as well as by “the magic and power of chemistry”—a dangerous attraction in the hands of a brilliant but rebel- lious youngster. But he was always good at mathematics as well, and he was entranced by the notion of statements that could be proved, in such contrast to the argumentative, fast-talking culture of Brooklyn where he’d grown up. Kenneth Hoffman attended Occidental College because of their track-and- fi eld team. When told by his mathematics teacher, Mabel Barnes, that he should consider graduate school, his response was, “What’s that?” Then, as now, exceptional students will often gravitate toward—or attract—exceptional teachers, thesis supervisors, mentors. Fagi Levinson Recountings xi remembers how her husband’s talent got the attention of Norbert Wiener, not a mathematician known for his attentiveness to the faces in front of him but a generous mentor once he realized that the young Levinson was worthy of his attention.