Book Review

A History in SumReviewed by Steve Batterson

A History in Sum: 150 Years of Mathematics atHarvard (1825–1975)Steve Nadis and Shing-Tung YauHarvard University Press, October 2013280 pages, $39.95ISBN-13: 978-06747-250-03

Mathematicians may be surprised by 1982 FieldsMedalist Shing-Tung Yau’s collaboration with sci-ence writer Steve Nadis on a history of the Harvardmathematics department. Perhaps anticipatingsome bewilderment, Yau begins his portion ofthe preface with a justification of why history isimportant to mathematics. For the purposes ofthis review, I regard historical value as an axiom.

A History in Sum joins Cal Moore’s Mathematicsat Berkeley in the genre of book-length historiesof American mathematics departments. Althoughboth volumes are dominated by biographies ofuniversity faculty, their underlying methodologiesand objectives are very different. Moore wrotewhat might be classified as a traditional history. Heexcavated the Berkeley archives and produced adetailed record of the scholarly advancement of thedepartment since the founding of the Universityof California in 1868. His narrative includes thebasic vitae of every faculty member and muchmore, analyzing changes in the department overthe years.

Nadis and Yau focus on the stories of Harvardpersonnel making pioneering mathematical dis-coveries. A History in Sum features biographies offourteen Harvard faculty, from the period 1825–1975, “that made the greatest contributions to

Steve Batterson is professor of mathematics and computerscience at Emory University. His email address is sb@mathcs.emory.edu.

DOI: http://dx.doi.org/10.1090/noti1133

mathematics.” Spoiler alert: If you would like tomake your own selections, the names are listed twoparagraphs below. In fleshing out these lives, theauthors rely on interviews and published materialrather than archival sources. As in their first book,The Shape of Inner Space, Nadis and Yau set outto make deep mathematics accessible. This time,instead of string theory, the topics range over thevarious breakthroughs of their stars.

Some readers will have opinions on the meritsof a comprehensive departmental history versussingling out its greatest men (at Harvard they areall men). I welcome both approaches as valuableadditions to the literature, particularly in viewof the distinction of the Harvard and Berkeleydepartments. In the interest of full disclosure,I need to state that I received honoraria fromHarvard University Press for commenting on themanuscript at two stages of its development.

The division of labor between the two authorsis not discussed beyond that the initiative arosefrom Yau. One assumes that Yau selected thenames. His bona fides confer special interest onwhat, in itself, is an intriguing list: Benjamin Peirce,Osgood, Bôcher, G. D. Birkhoff, Morse, Whitney,Mac Lane, Ahlfors, Mackey, Gleason, Zariski, Brauer,Bott, and Tate. Yau acknowledges an element ofsubjectivity in making difficult decisions aboutwhom to include. While he understandably does notdiscuss specific omissions, consider some of theHarvard mathematicians who are not featured inthe book. Fields Medalists Mumford and Hironaka,whose careers may have been regarded as toolate, get some attention as students of Zariski.Joseph Walsh and Marshall Stone receive a mereparagraph apiece, comparable to Moore’s coverageof Annie Dale Biddle Andrews, an obscure Berkeleyinstructor terminated in 1933. Dunham Jackson,

June/July 2014 Notices of the AMS 603

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Oscar Zariski

who served on the Harvardfaculty from 1911 untilleaving for Minnesota in1919, is not mentioned.

The fourteen biogra-phies average about adozen pages each, touch-ing on both personal andmathematical lives. Myprior knowledge of theindividual subjects var-ied substantially. To mysurprise, I was most fasci-nated by the story of OscarZariski, about whom I knewlittle.

Zariski was born at theend of the nineteenth century in a Russian citythat is now part of Belarus. For his education hemoved to the Ukraine, where World War I and thenthe Russian Revolution unfolded around him. In1919 Zariski was wounded by shrapnel when hehappened into a skirmish between Bolshevik andUkrainian forces. Two years later he left embattledKiev to continue his mathematical education inItaly.

In Rome, Zariski came under the influence of thepioneering algebraic geometers Guido Castelnuovo,Federigo Enriques, and Francesco Severi. As wascharacteristic of his life, Zariski made the mostof the opportunities in an environment withmonumental barriers. Despite being a CommunistJew in a time and place where Mussolini wasadvancing his fascist agenda, Zariski absorbedthe classical techniques of his Italian teachers. Hecompleted his Ph.D. in 1924 under Castelnuovo.

Zariski was fortunate in that, of the Italiangeometers, Castelnuovo recognized the limitationsof the Italian school. He encouraged Zariski tostudy the topological techniques being introducedby Solomon Lefschetz. Lefschetz, himself a RussianJew, had just moved from the University of Kansasto Princeton. Lefschetz used his influence to assistZariski in obtaining a research fellowship for1927–1928 at nearby Johns Hopkins.

At Hopkins, Zariski came into his own asan independent scholar, earning a position onthe faculty. In preparing his comprehensive textAlgebraic Surfaces, Zariski gradually realized thatthe entire subject of algebraic geometry restedon a wobbly geometric foundation. As he beganto craft a more rigorous algebraic replacement, afortuitous opportunity arose. When the Institutefor Advanced Study opened in 1933, his JohnsHopkins colleague Egbert van Kampen was partof an experiment in which several promisingmathematicians spent a year in residence. The trialwas so successful that the president of Hopkins

agreed to provide full support for Zariski to visitthe institute for 1934–1935. In that same yearEmmy Noether commuted to Princeton from BrynMawr, delivering lectures on some of the algebraicstructures that Zariski needed.

In 1945 Zariski made the most of a postingas an exchange professor in São Paulo. There heengaged in stimulating discussions with anothervisitor, André Weil. Two years after his return fromBrazil, Zariski became the first tenured Jewishmathematician on the Harvard faculty. By attractingstrong students and bringing in distinguishedvisitors, he soon made Harvard into an internationalcenter for algebraic geometry. In the late 1950sthe Zariski milieu included his students HeisukeHironaka, Michael Artin, and David Mumford, aswell as the groundbreaking Europeans Jean-PierreSerre and Alexander Grothendieck.

The biographies in A History in Sum illus-trate contrasting approaches to doing mathe-matics. Whereas Zariski thrived on interaction withother great scholars, Hassler Whitney preferred“solitude”. According to Nadis and Yau, SaundersMac Lane’s most important contributions cameout of his long-term collaboration with SamuelEilenberg. Andrew Gleason never wrote a paperwith his Harvard colleague George Mackey, butfound inspiration from their frequent discussions.

Interspersed throughout the twenty-one-pagesection on Zariski is a variety of mathematicalexcursions, beginning with the basic idea of alge-braic geometry. The authors discuss the motivationbehind Zariski’s development of algebraic tools aswell as provide an introduction to problems overfinite fields. I liked the explanation of resolutionof singularities, taken largely from an interview ofHironaka in the October 2005 Notices.

Nadis and Yau draw heavily from Carol Parikh’sbiography The Unreal Life of Oscar Zariski. Theirnarrative is enhanced by fresh recollections ofmathematicians from Zariski’s circle. Over sixtyinterviews were conducted for the book, includingTate, the only featured subject who survives. Theremembrances about Raoul Bott give the reader agenuine feeling of Bott’s jovial charm. On the otherhand, the section on Marston Morse only hints atthe magnitude of his ego.

The authors turned up a variety of biographicalsources on their subjects. A minor criticism isthat, in some cases, they could have used morediscretion in filtering biased perspectives. Forexample, Garrett Birkhoff should not shape theimpression of his father. The dogmatic NorbertWiener is a less-than-objective source on Harvardfaculty. Memorial tributes have a tendency toairbrush personal qualities.

Nevertheless, the featured subjects stand onthe merits of their theorems. Tying them together

604 Notices of the AMS Volume 61, Number 6

is their link to Harvard. The strength of theHarvard mathematics department, going back toAhlfors and Birkhoff, is well known. The careersof Benjamin Peirce, W. F. Osgood, and MaximeBôcher demonstrate that, with the exception ofthe ten years from Peirce’s death (1880) to theappointment of Osgood (1890), the universityfaculty has included leading mathematicians since1831. Indeed, Harvard merits consideration withJohns Hopkins and the University of Chicago as thefirst academic home for mathematical scholarshipin the United States.

Although the authors focus on mathematicsat Harvard, a connection to the university isnot necessary for enjoyment of the book nor isany special knowledge of the areas explored. AHistory in Sum should find an audience amongmathematicians from two broad classes: those whoenjoy biography and those who would like to gaina nontechnical flavor of major developments intheir science.

Selected Papers of V. S. VaradarajanVolumes 2 and 3Donald G. Babbitt and Ramesh Gangolli, K. R. Parthasarathy, Indian Statistical Institute, New Delhi, India, Enrico G. Beltrametti and Gianni Cassinelli, University of Genova, Italy, Rita Fioresi, Università di Bologna, Italy, and Anatoly N. Kochubei, National Academy of Sciences of Ukraine, Kiev, Ukraine, Editors

The current volumes contain the papers on fundamental questions of individual and families of meromorphic differential equations that are treated by a new group theoretic and functional approach; papers on representation theory of Lie groups; papers on foundations of physics, supersymmetry, and P-adic aspects of quantum physical theories; papers on analysis, especially oscillatory integrals, on semi-simple lie groups, their conjugacy classes, and their flag manifolds; and finally, several review articles, both personal and mathematical, on a number of the above topics.

Hindustan Book Agency; 2013; 1366 pages; Hardcover; ISBN: 978-93-80250-57-1; List US$165; AMS members US$132; Order code HIN/65

Publications of Hindustan Book Agency are distributed within the Americas by the American Mathematical Society. Maximum discount of 20% for all commercial channels.

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