Book Review A History in Sum Reviewed by Steve Batterson mathematics.” Spoiler alert: If you would like to A History in Sum: 150 Years of Mathematics at make your own selections, the names are listed two Harvard (1825–1975) paragraphs below. In fleshing out these lives, the Steve Nadis and Shing-Tung Yau authors rely on interviews and published material Harvard University Press, October 2013 280 pages, $39.95 rather than archival sources. As in their first book, ISBN-13: 978-06747-250-03 The Shape of Inner Space, Nadis and Yau set out to make deep mathematics accessible. This time, instead of string theory, the topics range over the Mathematicians may be surprised by 1982 Fields various breakthroughs of their stars. Medalist Shing-Tung Yau’s collaboration with sci- Some readers will have opinions on the merits ence writer Steve Nadis on a history of the Harvard of a comprehensive departmental history versus mathematics department. Perhaps anticipating singling out its greatest men (at Harvard they are some bewilderment, Yau begins his portion of all men). I welcome both approaches as valuable the preface with a justification of why history is additions to the literature, particularly in view important to mathematics. For the purposes of of the distinction of the Harvard and Berkeley this review, I regard historical value as an axiom. A History in Sum joins Cal Moore’s Mathematics departments. In the interest of full disclosure, at Berkeley in the genre of book-length histories I need to state that I received honoraria from of American mathematics departments. Although Harvard University Press for commenting on the both volumes are dominated by biographies of manuscript at two stages of its development. university faculty, their underlying methodologies The division of labor between the two authors and objectives are very different. Moore wrote is not discussed beyond that the initiative arose what might be classified as a traditional history. He from Yau. One assumes that Yau selected the excavated the Berkeley archives and produced a names. His bona fides confer special interest on detailed record of the scholarly advancement of the what, in itself, is an intriguing list: Benjamin Peirce, department since the founding of the University Osgood, Bôcher, G. D. Birkhoff, Morse, Whitney, of California in 1868. His narrative includes the Mac Lane, Ahlfors, Mackey, Gleason, Zariski, Brauer, basic vitae of every faculty member and much Bott, and Tate. Yau acknowledges an element of more, analyzing changes in the department over subjectivity in making difficult decisions about the years. whom to include. While he understandably does not Nadis and Yau focus on the stories of Harvard discuss specific omissions, consider some of the personnel making pioneering mathematical dis- Harvard mathematicians who are not featured in coveries. A History in Sum features biographies of the book. Fields Medalists Mumford and Hironaka, fourteen Harvard faculty, from the period 1825– whose careers may have been regarded as too 1975, “that made the greatest contributions to late, get some attention as students of Zariski. Joseph Walsh and Marshall Stone receive a mere Steve Batterson is professor of mathematics and computer science at Emory University. His email address is sb@ paragraph apiece, comparable to Moore’s coverage mathcs.emory.edu. of Annie Dale Biddle Andrews, an obscure Berkeley DOI: http://dx.doi.org/10.1090/noti1133 instructor terminated in 1933. Dunham Jackson, June/July 2014 Notices of the AMS 603 who served on the Harvard agreed to provide full support for Zariski to visit faculty from 1911 until the institute for 1934–1935. In that same year leaving for Minnesota in Emmy Noether commuted to Princeton from Bryn 1919, is not mentioned. Mawr, delivering lectures on some of the algebraic The fourteen biogra- structures that Zariski needed. phies average about a In 1945 Zariski made the most of a posting dozen pages each, touch- as an exchange professor in São Paulo. There he ing on both personal and engaged in stimulating discussions with another mathematical lives. My visitor, André Weil. Two years after his return from prior knowledge of the Brazil, Zariski became the first tenured Jewish individual subjects var- mathematician on the Harvard faculty. By attracting ied substantially. To my strong students and bringing in distinguished surprise, I was most fasci- visitors, he soon made Harvard into an international nated by the story of Oscar center for algebraic geometry. In the late 1950s Zariski, about whom I knew the Zariski milieu included his students Heisuke Photo: AMS Archives. Oscar Zariski little. Hironaka, Michael Artin, and David Mumford, as Zariski was born at the well as the groundbreaking Europeans Jean-Pierre end of the nineteenth century in a Russian city Serre and Alexander Grothendieck. that is now part of Belarus. For his education he The biographies in A History in Sum illus- moved to the Ukraine, where World War I and then trate contrasting approaches to doing mathe- the Russian Revolution unfolded around him. In matics. Whereas Zariski thrived on interaction with 1919 Zariski was wounded by shrapnel when he other great scholars, Hassler Whitney preferred happened into a skirmish between Bolshevik and “solitude”. According to Nadis and Yau, Saunders Ukrainian forces. Two years later he left embattled Mac Lane’s most important contributions came Kiev to continue his mathematical education in out of his long-term collaboration with Samuel Italy. Eilenberg. Andrew Gleason never wrote a paper In Rome, Zariski came under the influence of the with his Harvard colleague George Mackey, but pioneering algebraic geometers Guido Castelnuovo, found inspiration from their frequent discussions. Federigo Enriques, and Francesco Severi. As was Interspersed throughout the twenty-one-page characteristic of his life, Zariski made the most section on Zariski is a variety of mathematical of the opportunities in an environment with excursions, beginning with the basic idea of alge- monumental barriers. Despite being a Communist braic geometry. The authors discuss the motivation Jew in a time and place where Mussolini was behind Zariski’s development of algebraic tools as advancing his fascist agenda, Zariski absorbed well as provide an introduction to problems over the classical techniques of his Italian teachers. He finite fields. I liked the explanation of resolution completed his Ph.D. in 1924 under Castelnuovo. of singularities, taken largely from an interview of Zariski was fortunate in that, of the Italian Hironaka in the October 2005 Notices. geometers, Castelnuovo recognized the limitations Nadis and Yau draw heavily from Carol Parikh’s of the Italian school. He encouraged Zariski to biography The Unreal Life of Oscar Zariski. Their study the topological techniques being introduced narrative is enhanced by fresh recollections of by Solomon Lefschetz. Lefschetz, himself a Russian mathematicians from Zariski’s circle. Over sixty Jew, had just moved from the University of Kansas interviews were conducted for the book, including to Princeton. Lefschetz used his influence to assist Tate, the only featured subject who survives. The Zariski in obtaining a research fellowship for remembrances about Raoul Bott give the reader a 1927–1928 at nearby Johns Hopkins. genuine feeling of Bott’s jovial charm. On the other At Hopkins, Zariski came into his own as hand, the section on Marston Morse only hints at an independent scholar, earning a position on the magnitude of his ego. the faculty. In preparing his comprehensive text The authors turned up a variety of biographical Algebraic Surfaces, Zariski gradually realized that sources on their subjects. A minor criticism is the entire subject of algebraic geometry rested that, in some cases, they could have used more on a wobbly geometric foundation. As he began discretion in filtering biased perspectives. For to craft a more rigorous algebraic replacement, a example, Garrett Birkhoff should not shape the fortuitous opportunity arose. When the Institute impression of his father. The dogmatic Norbert for Advanced Study opened in 1933, his Johns Wiener is a less-than-objective source on Harvard Hopkins colleague Egbert van Kampen was part faculty. Memorial tributes have a tendency to of an experiment in which several promising airbrush personal qualities. mathematicians spent a year in residence. The trial Nevertheless, the featured subjects stand on was so successful that the president of Hopkins the merits of their theorems. Tying them together 604 Notices of the AMS Volume 61, Number 6 is their link to Harvard. The strength of the A M S Harvard mathematics department, going back to Ahlfors and Birkhoff, is well known. The careers of Benjamin Peirce, W. F. Osgood, and Maxime Bôcher demonstrate that, with the exception of the ten years from Peirce’s death (1880) to the appointment of Osgood (1890), the university Selected Papers of V. S. faculty has included leading mathematicians since Varadarajan 1831. Indeed, Harvard merits consideration with Johns Hopkins and the University of Chicago as the Volumes 2 and 3 first academic home for mathematical scholarship Donald G. Babbitt and Ramesh Gangolli, K. R. Parthasarathy, Indian Statistical Institute, New Delhi, India, Enrico G. Beltrametti in the United States. and Gianni Cassinelli, University of Genova, Italy, Rita Fioresi, Although the authors focus on mathematics Università di Bologna, Italy, and Anatoly N. Kochubei, National at Harvard, a connection to the