Serge Lang, 1927â€“2005

Serge Lang, 1927–2005Jay Jorgenson and Steven G. Krantz

536 NOTICES OF THE AMS VOLUME 53, NUMBER 5

On September 12, 2005, the mathemat-ics community lost Serge Lang, whopassed away in his apartment in Berke-ley, California. Lang was well known asa mathematician, and also as an edu-

cator and political activist. The main force in Serge’slife was his enthusiasm for mathematics. In a worldof vagaries and irrational passions, he saw math-ematics as a noble pursuit that represented hon-esty and goodness. Within mathematics alone,Serge had many facets—a researcher, an expositor,a popularizer, and a teacher. Generations of math-ematicians around the world know the name SergeLang through his numerous books and articles.

For those individuals who knew Serge, one strik-ing feature most everyone noted was the com-partmentalized manner in which he showed him-self to anyone: His mathematical colleagues weretold virtually nothing about his personal life, hisfamily knew very little abut his mathematical re-search, his political allies were only slightly in-formed of his mathematical interests, and evenhis closest friends were unaware of each other’spresence in his life.

As we prepared this article discussing the manyaspects of Serge’s life, we chose to follow Serge’smethod of “file-making”, where the reader is

informed through the presentation of original doc-umentation. We have sought to bring out a fullpicture of Serge’s life by inviting contributionsfrom a large number of individuals who knew himwell. For the editors, it was fascinating to witnessthe diversity of these reminiscences; they representa broad range of interests and achievements. It isclear that, with Lang’s passing, we have lost some-one unique and irreplaceable.

After Lang’s passing, Yale University presidentRichard C. Levin wrote about Serge, “While havingsomeone like this in the community is not alwayseasy, it is salubrious.” It is entirely possible Sergewould have agreed with this assessment, perhapseven assigning a letter grade for President Levin’ssummary.

To repeat, our article is an attempt to followLang’s insistence for an honest and complete rep-resentation, allowing readers to draw their ownconclusions. With this said, we have no doubt thata common judgment will be drawn by everyone:With Lang’s death, the mathematical world, and be-yond, has lost someone without equal, and in timewe will better understand the significance of Lang’slife.

On Serge Lang’s retirement from Yale Universityin the spring of 2005, Yale president Richard C.Levin honored him with these words:

Serge Lang, A.B., California Institute ofTechnology, Ph.D. Princeton University,faculty member at Yale since 1972: Yourprimary love has always been numbertheory and you have written, by onecolleague’s estimate, over 50 books and

Jay Jorgenson is professor of mathematics at City Collegeof New York and Graduate Center. His email address [email protected].

Steven G. Krantz is professor of mathematics at WashingtonUniversity, St. Louis. His email address is [email protected].

With the assistance of numerous contributors.

Editor’s Note: This is the first part of a two-part article. In part two, which will ap-pear in a later issue, the authors discuss the mathematical accomplishments of SergeLang and the impact of those achievements.

MAY 2006 NOTICES OF THE AMS 537

monographs, many of them concernedwith this topic. Several of your mono-graphs are the only, or nearly the only,book treatments of their important sub-jects. Your famous theorem in Dio-phantine equations earned you the dis-tinguished Cole Prize of the AmericanMathematical Society. Your textbooksalso have garnered accolades. Your cal-culus for undergraduates went throughmany editions in the seventies andeighties, and your algebra textbook is astandard reference in the field. So prodi-gious are you as a scholar that there areactual jokes in your profession aboutyou. One joke goes: “Someone calls theYale Mathematics Department, and asksfor Serge Lang. The assistant who an-swers says, ‘He can’t talk now, he iswriting a book. I will put you on hold.’”

In your character, you are uncompro-mising in your insistence on what youperceive as logical consistency andrhetorical honesty, and you have ques-tioned much received wisdom and manyauthorities in the external world as wellas here at Yale. You are an excellentand deeply caring teacher, and in honorof this several years ago you receivedthe Dylon Hixon Prize for teaching inYale College. Your students keep intouch with you years after they gradu-ate and one has created an endowedfund in your honor. Among your manymonographs there is one called TheBeauty of Doing Mathematics, a collec-tion of three dialogues you gave in Parisin the ‘80s. Yale is grateful to you forthe passion with which you understand,practice and profess the mathematicalarts, and wishes you well as you con-tinue your lifelong engagement withtheir illimitable splendors.

Serge Lang was born near Paris on May 19, 1927.His family lived in St. Germain en Laye. Serge’smother was a concert pianist and his father was abusinessman. His sister, with whom Serge main-tained an affectionate relationship all his life, cur-rently lives in Los Angeles and is a stage and filmactor. Serge’s twin brother was a college basketballcoach.

The family decided when Serge was a teenagerto move to Los Angeles, California. Serge attendedCaltech as an undergraduate and finished with aB.A. degree in physics in 1946. After spending 1.5years in the U.S. Army, Serge entered graduateschool at Princeton University in philosophy. Heabandoned that study after one year and turned his

attention to mathematics. That attention never de-viated (except occasionally for his politics) for therest of Serge Lang’s life.

At Princeton Serge Lang fell under the spell ofthe great algebraic number theorist Emil Artin.Along with John Tate, a fellow student of Artin, Langdeveloped a passion for algebra and algebraic num-ber theory. In later years, Lang and Tate co-editedthe collected works of Artin. Lang earned his Ph.D.in 1951.

Lang’s first academic position was as an in-tructor at Princeton. Lang also had an instructor-ship at the University of Chicago from 1953 to1955. Lang’s first permanent position was at Co-lumbia University beginning in 1955. In additionto producing some terrific mathematics and di-recting five Ph.D. theses, Lang became passion-ately involved with the politics of the time (inprotest against the Vietnam war). Serge ultimatelyresigned his position at Columbia in 1971 (withoutyet having arranged for another job) in protestagainst Columbia’s treatment of anti-war protest-ers. It is also remarkable that, during his tenure atColumbia, Lang directed two Princeton Ph.D. stu-dents: Marvin Greenberg (1959) and NewcombGreenleaf (1961).

After leaving Columbia University, Serge Langlanded a job at Yale University (beginning in 1972),where he spent the remainder of his career. Langdirected nine additional Ph.D. degrees while atYale. He was awarded the AMS Frank Nelson ColePrize (1959) for his mathematical research and theAMS Leroy P. Steele Prize (1999) for his writing. Hewas elected to the National Academy of Sciencesin 1985.

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Although Lang’s first mathematical loves werealgebra and number theory, his interests rapidly ex-panded to cover an astonishing panorama of mod-ern mathematics. Areas that he influenced includenumber theory, algebraic geometry, diophantinegeometry (in which he was a pioneer), diophantineapproximation, differential geometry, analysis, hy-perbolic geometry, Arakelov theory (in which he wasa pioneer), modular forms, and many other areasas well. The scope of Lang’s books and papers isastonishing not only for its magnitude but for itsbreadth.

Serge Lang resigned from the AMS in 1996 in adispute concerning an article in the AMS Notices byDenise Kirschner. He retired from Yale in the springof 2005.

It gives a sense of Serge Lang to quote from hisformal note of acceptance for the Steele Prize(which in fact had to be heavily edited because itwas formulated in such strong language):

I thank the Council of the AMS and theSelection Committee for the Steele Prize,which I accept. It is of course reward-ing to find one’s works appreciated bypeople such as those on the SelectionCommittee. At the same time, I am veryuncomfortable with the situation, be-cause I resigned from the AMS in early1996, after nearly half a century’s mem-bership. On the one hand, I am now un-comfortable with spoiling what couldhave been an unmitigated happy mo-ment, and on the other hand, I do notwant this moment to obscure impor-tant events which have occurred in thelast two to three years, affecting my re-lationship with the AMS.

… …

Torn in various directions, sadly butfirmly, I do not want my accepting theSteele Prize to further obscure the his-tory of my recent dealings with the AMS.

Serge Lang was a remarkably energetic individ-ual with eclectic and broadly ranging tastes. In ad-dition to his passion for mathematics he lovedmusic and the arts. He himself was an accom-plished pianist and lutenist, and he enjoyed play-ing in public. He took a keen interest in politics,especially as it manifested man’s inability to facethe truth. Lang loved to bring down individuals whoobfuscated, who hid behind their rank, or whoabused power. He engaged in a great many ratherpublic battles with a wide-ranging collection ofpeople, from social scientists at Harvard to re-searchers at the National Institutes of Health to ed-ucation researchers at Stanford. As Lang himself

put it, he “put scholarship in the service of actionto stop the nonsense.”

Serge also was a prolific writer. He wrote morethan 120 research articles and sixty-one books(and this does not count multiple editions and for-eign translations). In fact he has 198 citations onMathSciNet. It is amazing to examine the range ofmathematical topics covered by Lang’s opus: cal-culus, real analysis, complex analysis, differentialgeometry, algebra, algebraic geometry, diophan-tine geometry, hyperbolic geometry, math talksfor undergraduates, the heat kernel, and much,much more. Perhaps Lang’s most famous and mostinfluential book is Algebra, now in its third edition.In it, Lang single-handedly reorganizes and revi-talizes this fundamental and central subject. Thebook has had an enormous impact.

Serge Lang was a man with incredible focus andself-discipline. Mathematics and politics (which hecalled “troublemaking”) were his primary inter-ests, and everything else was secondary. As hegrew older, he felt that he had to conserve his en-ergy and he set other interests aside. He madehard decisions and stuck by them. As an example,when he decided to stop listening to music, he putall his recordings on the shelf, never to be pickedup again.

It is astonishing how Lang’s books affected peo-ple at all levels. One high school teacher who reg-ularly used Lang’s calculus book in his teaching saidthis:

As a high school teacher, I used thistext with great success several timesfor both AP Calculus BC and AP Calcu-lus AB courses. It is my favorite calcu-lus text to teach from, because it is veryuser-friendly and the material is pre-sented in such an eloquent way. Thereare no gratuitous color pictures of peo-ple parachuting out of airplanes here.Opening this book is like entering atemple: all is quiet and serene. Epsilon-delta is banished to an appendix, where(in my opinion) it belongs, but all of theproofs are there, and they’re presentedin a simple (but not unsophisticated)way, with a minimum of unnecessaryjargon or obtuse notation.

A somewhat recondite joke is the query “Whydid Bourbaki stop writing?” The answer is thatthey discovered that Serge Lang is one person.Lang’s output of text connected to his many po-litical disputes was voluminous. He also has someunpublished books of a political nature (others ofhis political tracts were actually published). Langliked to say that the best way to learn a new topicis to write a book about it.


Perhaps Serge Lang’s greatest passion in lifewas learning. For Serge, learning manifested itselfin many guises; but one of the most important ofthese was his teaching. He saw himself as a rolemodel for his students, and he spent a great dealof time with them. He often said that the best wayto learn about a university was to eat in the stu-dent cafeteria. He did so frequently. He often tookhis students out to eat, or invited them to his res-idence to listen to music. Although he did so qui-etly and discreetly, Serge was known to provide fi-nancial assistance to students and mathematicianswho were in need. Serge is remembered fondly forentreating his students, cajoling his students,screaming at his students, and especially for throw-ing chalk at his students.

Serge’s graduate courses frequently followedthe track of the book he was currently writing. Hisundergraduate courses could be more freewheel-ing. An important point to note is the joy thatSerge Lang derived from all things mathematical.It can certainly be said that most of us mathe-maticians experience some sort of “high” when welearn to tackle and tame new ideas. As we get older,we become more jaded; as a result, this “high” isharder and harder to achieve. Not for Serge. He wastruly engaged and fulfilled when he discoverednew ideas on any level, be that an illuminatingproblem for one of his undergraduate texts or aninsight into a new mathematical landscape. As a re-sult, Serge Lang always remained mathematicallyyoung.

Serge Lang spent the fall of 2004 at U. C. Berke-ley as a Miller Visiting Professor. He gave a num-ber of lectures and made his presence known inmany other ways. As an example, budget cutbackshad caused severe curtailment of departmentalteas. People now had to pay daily for their bever-ages and cookies. Serge quietly contributed a sub-stantial amount of money so that the Monday teaswould be lavish: many fine cakes and pastries andlots of nice things to drink. Certainly this had a verypositive effect on departmental life, and Sergeasked for no particular credit for this gesture.

Serge had wide-ranging interests. He visitedBerkeley every summer for the past several decades(in fact he kept an apartment there) and he wouldattend colloquia in departments ranging fromphysics to history to political science to medicineto mathematics. Of course he did not simply attend.His habit was to confront the speaker with detailedand probing questions. Frequently the sessionswould become so heated and protracted that in-tervention was necessary.

One memorable incident—just to illustrate theeclecticism and vehemence of Serge’s interests—has Serge threatening to clobber with a bronzebust a very distinguished Princeton mathemati-cian in the Fine Hall Professors’ Lounge because the

latter would not accede to the self-evident asser-tion that the Beatles were greater musicians thanBeethoven.

At Yale in 2001, Serge was invited to be thekeynote speaker at a Pierson College Master’s Tea.He dressed in a courtier’s outfit and regaled thepacked room with his theory of the similarities be-tween Elizabethan music and classic rock of the1960s. Lang illustrated his points by playing (clas-sical LP records of) Ding Dong the Witch is Dead, a1969 hit by the Fifth Estate, a 1612 piece by MichaelPraetorius, and We Can Work it Out by the Beatles(1965).

Serge loved to challenge people—friend and foealike—just for the sake of challenging them. As aninstance, James Borger recalls

I remember one time when I was a gradstudent, I was standing next to him attea while he was explaining to a first-year student that analysis is just num-ber theory at infinity. I said, “Come on,that’s not true.” He immediately turnedup the volume, challenging me to stopbullshitting and give an example. I said,“OK, p-adic analysis,” and then walkedaway. But I’ve always wished I hadstayed to see what his reaction wouldhave been. We need more trouble mak-ers like him.

In 1998 Serge Lang published a book calledChallenges. This editor (Krantz) found the volumeto be particularly inspiring, for it recounted, fromLang’s personal perspective, some of his most in-volving and exciting political battles. The book istruly outstanding for its honesty and incisiveness.Two particular battles that stand out are

The Case of Ladd and Lipsett. In the late 1970sthe distinguished social scientists Everett Carll

Ph.D. Students of Serge LangMarvin Greenberg Princeton 1959Newcomb Greenleaf Princeton 1961Warren May Columbia 1963Stephen Schanuel Columbia 1963William Adams Columbia 1964Bernard Berlowitz Columbia 1966Allen Altman Columbia 1968Joseph S. Repka Yale 1975David E. Rohrlich Yale 1976Donald T. Kersey Yale 1980Jing Yu Yale 1980Minhyong Kim Yale 1990William A. Cherry Yale 1993Michael J. Nakamaye Yale 1994Lisa A. Fastenberg Yale 1996Eliot P. Brenner Yale 2005


Ladd Jr. and Seymour Martin Lipsett set out toevaluate the American professoriate. They con-cocted a questionnaire to be distributed across thecountry, asking professors detailed questions abouthow they plied their trade, what values they heldas members of the academic profession, and soforth. Their results were published as The 1977 Sur-vey of the American Professoriate in 1979. Langfound the questionnaire, and the premises for thestudy, to be repugnant. He conducted a massive ef-fort to discredit their work. In fact Lang publisheda rather massive tome, The File (Springer-Verlag,1981), containing all his correspondence and in-formation about the battle. In the end, Lang causedLadd/Lipsett to lose much of their funding and agreat deal of their credibility.

The Case of Samuel P. Huntington. In 1968Samuel P. Huntington wrote a book entitled Polit-ical Order in Changing Societies. In it Huntingtonuses what might charitably be characterized aspseudomathematical hucksterism to “prove” thatSouth African society in the 1960s was a “satisfiedsociety”. Serge Lang decided that nothing could befurther from the truth, and in any event Hunting-ton’s methodology was suspect if not corrupt. Heconducted a vigorous campaign to derail Hunt-ington’s credibility, and he twice successfullyblocked Huntington’s election to the National Acad-emy of Sciences.

Serge Lang was quite proud of his efforts to in-still a sense of truth and honor into our public dis-course. For years after his battle with Huntington,he would give his students “Huntington tests” toascertain their ability to think critically. Serge’sbattle cry was to demand whether his listenersknew “a fact from a hole in the ground”. EvidentlySerge did. In a particularly earthy moment, Sergeliked to say that “he was inside the tent pissing in”(with allusion to Lyndon Johnson commentingabout J. Edgar Hoover). For each of his battles,Lang would create what he called a “File”. This wasa detailed and copious collection of all his corre-spondence and all his data connected with anygiven case. Often a file would consist of several hun-dred pages of closely knit text. Lang would, at hisown expense, send copies of his files to mathe-maticians and other interested parties all over theworld. The Serge Lang files have been a staple ofmathematical life for over forty years.

Serge Lang said of himself

I personally prefer to live in a societywhere people do think independentlyand clearly. One of my principal goalsis therefore to make people think. Whenfaced with persons who fudge the is-sues, or cover up, or attempt to rewritehistory, the process of clarifying the is-sues does lead to confrontation, it

creates tension, and it may be inter-preted as carrying out a “personalvendetta” …I regard such an interpre-tation as very unfortunate, and I rejectit totally.

Serge spent hours every day on the telephone,wheedling, cajoling, instructing, and most oftenyelling. His collaborators relate that Serge wouldoften phone several times a day—every day. Hewould learn what was the best time to phone andthen phone regularly at that time. Often one wouldpick up the phone and hear “Serge! Let me continueto instruct you about …” But it should be stressedthat Serge was disciplined to the extreme. He didnot waste time. It was amazing to watch him eatlunch in five minutes and dash back to his officeto resume his writing.

In the last twelve years of his life Serge Lang de-veloped a deep and energetic program to fight thecurrent directions of research on the disease AIDS(Acquired Immune Deficiency Syndrome). A naiveassessment of Serge’s position is that HIV does notcause AIDS. But this would be an injustice to Serge.First of all, he was very careful. He very rarelymade an error of fact. Secondly, he was quite a sub-tle thinker. His cause and his complaint, in fact, wasthat the search for a cure to AIDS had becomepoliticized. At a certain point, the federal govern-ment simply commanded the National Institutes ofHealth to declare that HIV caused AIDS. The causalmechanism had not been identified, and the con-nection not logically established. To be sure, thereis considerable ad hoc evidence of a link betweenHIV and AIDS. Certainly many of the modern treat-ments for AIDS are premised on that link. ButSerge’s assessment was that the existing data analy-sis does not support the conclusion that HIV causesAIDS.

The present article is a celebration of the life ofSerge Lang. We present a number of vignettes, con-tributed by mathematicians, former students, col-leagues, and friends. These are divided into piecesabout Serge the man, pieces about Serge the writer,pieces about Serge the tilter at windmills, andpieces about Serge the mathematician. Our aim isto give a well-rounded picture of what a diverse andmulti-faceted person we have lost. He was in manyways a thorn in our collective sides, but he was afriend to us all.


Memories of Serge Lang

Friedrich Hirzebruch, Max-Planck-Institutfür Mathematik

Serge Lang was a close friend of my wife and me,of our three children, and even of some of ourgrandchildren. We miss his frequent telephonecalls—“It’s me”—the last one was on September 10,2005. We shall miss his visit next summer and allthe following summers.

My wife and I met Serge 53 years ago in Prince-ton when he and I were 25 years old. We becamegood friends. It was old Europe that all three of usliked. Serge rarely spoke about his personal past,but by asking questions we slowly learnt the basicfacts. He came with his father and his sister fromParis to the United States after France had been oc-cupied by Germany. He was a soldier in the U.S.Army from 1946 to 1947 and was stationed in Italyand Germany for part of the time. The fact that hislife was disturbed by the Nazi war was not a bar-rier between us. How little Serge spoke of himselfcan also be seen from the Curriculum Vitae in hisCollected Papers. The CV has thirteen brief lines,from the first one, “1927 born” to the last one“1972–present Yale”.

We kept close contact with Serge, also after ourreturn to Germany. In the summer of 1955 he vis-ited us in the house of my parents in Hamm (West-falen) (see top photo, right).

I was appointed to the University of Bonn in1956 and began the series of Arbeitstagungenwhere the speakers are chosen by “public vote” atthe beginning of the meeting. With very few ex-ceptions Serge attended all Arbeitstagungen until2003. During the thirty Arbeitstagungen I orga-nized from 1957 to 1991, Serge gave thirteen lec-tures. The second photo from the top, right, showsSerge lecturing at one of the Arbeitstagungen. Thenext photo down shows him at some other Ar-beitstagung activity.

During each of the twenty-five years from 1979to 2003, Serge spent one month in Bonn, usuallyJune; in addition, he came for three sabbatical fallterms in 1993, 1997, and 2000. He financed his Junevisits from 1984 to 1989 by the funds of his Hum-boldt Prize.

He had a stable routine: In the summer he wentfrom Yale to Europe. For many years he visitedParis for a month until he stopped. “To everythingthere is a season,” he used to say. In other yearshe went Zurich or Berlin. He never omitted Bonnuntil the season also ended for the Max-Planck-In-stitute. In 2004 and 2005 he only visited us privately

for a few days. After his European visit he went toBerkeley where he enjoyed the cooler climate andwhere we met him in a number of years.

During his visits to Bonn he gave many lectures,in seminars on his own research and for studentsof beginning and advanced level. In an official re-port to me (13 February 1997) he wrote as follows:“While at the Max-Planck, I also visit other mathe-maticians, both in Germany and elsewhere such asHolland. I have substantial con-tacts with students. I used tolecture every year in your analy-sis course. Last year I lectured tothe high school class ofKarcher’s son. Thus my days atthe Max-Planck, regularly for onemonth in June every year, andonce for four months in fall of1993, have been important pe-riods in providing proper envi-ronment for establishing math-ematical contacts at all levels, aswell as learning and doing math-ematics.”

Serge also lectured once tomy algebra course (120 stu-dents) where I used his algebrabook. When I came to the lecturehall, I saw that the official stu-dent representatives were sell-ing cheap photocopied versionsof Serge’s book. I told them thatthis was illegal. The studentssaid “The author is far away.” Ireplied “You are wrong. He willbe here in two minutes, becausehe is taking over my lecturetoday.” Serge came and did notmake a great fuss about it. Heeven signed some of the copiesbefore he began his lecture.Serge also lectured to the pub-lic. He wanted not only to teachmathematics, but also how to becritical and responsible. “I wantto make people think.” Amongthe public lectures I mentionthe “Beauty of Mathematics”. Heexplained the hyperbolic 3-dimensional manifolds withcuspidal ends which are like the arms of an octo-pus. The bottom photo above shows Serge sittingat my desk in our home drawing octopuses. Behindhis back there is one shelf with 40–50 of his books.When he presented me a new edition of one of hisbooks, he threw the old edition into the wastepa-per basket from where I retrieved it later.

Mathematics was the most important part ofSerge’s life. He worked with great self-discipline for

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many hours seven days a week. I admired the wayhe could turn courses into books and how he con-tinuously did fundamental research. In the earlyyears there was time for the piano (including com-posing), for playing guitar and lute, for going to con-certs, to the theatre and opera, and to enjoy liter-ature. We could often do all this together with himwhen visiting him in his apartment in New York,where he played his piano compositions for us, andwhen we went out in New York to the theatre. Sim-ilar activities took place in Bonn during his visits.He enjoyed the musical life around Bonn. But thencame the time of the file making. He was able togive up things he loved and to concentrate on thetwo parts of his later life (mathematics first andthen political work). All other things had to go. “Toeverything there is a season.” The file concerningThe 1977 Survey of the American Professoriate byLadd and Lipsett developed from 1977 to 1979.Serge put a lot of time and energy into it. We al-ways received Serge’s mailings in installments of20-30 pages. It was exciting reading, full of sus-pense. Other files followed. The mailings came reg-ularly, the last one on the day of his death. It wascertainly not easy to discuss the files with him insuch a way that he did not begin to yell. “Ich binein unbequemer Mensch,” he said. We admired hissincere way to rely only on facts, “to distinguish afact from an opinion.” He fought for honesty andprecision in research and in journalism. He hated“big shots who throw their weight around”. He ob-jected to covering up because of collegiality.

His heritage is his Collected Papers, his scientificand his political books. But we miss Serge as afriend.

Norbert Schappacher, University ofStrasbourg

Two years ago, I gave a seminar in Zürich on thetopic of intellectuals among twentieth-centurymathematicians. My list included the EnglishmanG. H. Hardy, the Germans E. J. Gumbel and E. Kamke,the Frenchman L. Schwartz—and the French-bornAmerican Serge Lang.

The term intellectual (intellectuel ) used here isa French invention of the Dreyfus affair, from thefinal years of the nineteenth century. Emile Zola,Anatole France, Marcel Proust, and others were thefirst self-declared intellectuals. The expression hasa built-in partiality: it is only used for people whoseopinions you sympathize with, and whose opinionsand ways of expressing them are loathed by thosewho are on the other side.

Serge Lang was an intellectual in this Europeansense of the word, and he was one of the rare math-ematicians of the second half of the twentieth

century who can lay claim to this epithet. If col-leagues sometimes felt he was overdoing things,this may actually confirm what he represented.

But Serge Lang lived and acted in the U.S. whereno heritage of intellectuals exists, in spite of liter-ary figures like Arthur Miller. So Lang had to cutone out for himself, as the Yale professor whomade The New York Times by blocking SamuelHuntington’s admission to the National Academyof Sciences. In doing so he was surely helped bythe ambient climate of the late 1960s and 1970s,the Free Speech Movement, etc. But his personal de-vice, “the file”, was his own creation.

Let me add something more personal: The mostwonderful thing about Serge was that he was alwaysaround, and meeting him would always matter. Ifirst saw him as a young student in Bonn duringthe Arbeitstagungen of 1970 and 1971; I went tohis talks because I knew the name from hisAlgebra book. At the time I did not understand theleast bit of the mathematics he was talking about;but I distinctly remember the presentation: his talkseemed to be about presenting things from theright point of view, which others working in the fieldhad failed to see or to adopt.

I kept meeting him over the years in many places,and each time I was greeted by his charm andthrilled by his intensity. The last two summers I in-vited him for talks to Darmstadt and Strasbourg.Even though he would never complain aboutmediocre accommodation or food offered to him,Serge enjoyed being taken out to good restaurants.At least by the time we got to dessert he invariablyhad to raise his voice (for instance, because I stillwould not understand his way of putting to restthe philosophy of the Vienna circle, that he had fig-ured out after a few months as a student of phi-losophy), and posh people around us would startraising their eyebrows. I loved this kind of scene(with him, not in general), and I will now miss it alot.

Barry Mazur, Harvard University

In 1958, at Princeton, I had accidentally slipped intothe room in which Serge was giving his seminar inAbelian Varieties; I was transfixed by the metallicurgency, the vitality, of the voice of this chalk-wielding person; I understood absolutely nothingof the subject, but was instantaneously convinced,with that utterness of conviction that is the gift ofignorance, that abelian varieties—whatever theywere—were of breathtaking importance, and fur-thermore, of breathtaking importance to me. ThatSerge (a “mathematical grown-up”) would, shortlyafterwards, collar me and request a series of pri-vate lectures in differential topology was as-tounding to me. I treasure the halting lectures I gave


him, as a rite of passage, of immense importance.And Serge did this sort of thing, through thedecades, with many of the young: he would prof-fer to them gracious, yet demanding, invitations toengage as a genuine colleague—not teacher to stu-dent, but mathematician to mathematician; he didall this naturally, and with extraordinary generos-ity and success. Serge was a gadfly with formida-ble tenacity. That we are personally responsible forthe web of compromises that we have all come toaccept, and to think are inevitable, is something hewould never let us forget. That we, as editors or ref-erees of journals, make our judgments based onsome presumed social, or sociable, contract (e.g.,no political articles in a math journal) does not letus off the hook when asked to examine withoutprejudice the underpinnings of our (usually onlyimplicit) social contracts. Serge seemed to be, overthe decades, of one age, and that age was young(with its virtues and drawbacks). He had, when heplayed the piano, something of a brilliant Frencharticulation to his style, and there was a hint of thisin everything he did, from his walking gait (stac-cato) to the way in which he pronounced certainkey words in mathematics, like idea which, fromSerge, would sound like EYE-dee, which has a kindof platonic zing to it.

Over decades of mathematics Lang was led,more specifically, by an over-arching vision, whichhe pursued through the agency of various fields ofmathematics. The vision, baldly put, is that geom-etry is an extraordinarily striking dictator of qual-itative diophantine behavior. The still open Con-jecture of Lang in higher dimensions continues toserve as a guiding principle to the way in which thegrand subjects of geometry and number theorymeet, just as Serge himself served as an inspirorof generations of mathematicians, and a spokesmanfor intellectual honesty.

Paul Cohen, Stanford University

I was a graduate student when Serge Lang arrivedin Chicago as an assistant professor. My interestswere tending towards number theory, but werenot very focused. The arrival of Serge made a hugedifference to me and to many other graduate stu-dents. He immediately gave courses in algebraicgeometry and in algebraic number theory, also I be-lieve, accompanied by a constant output of notes.Suddenly I had a different idea of what mathe-matical research was. One could just attack prob-lems without a huge background of knowledge.His lectures were entertaining, of course, but alsoa little intimidating to the poor souls who mightask silly, or too elementary, questions. It was ad-visable to be nimble enough to dodge flying chalkcoming from his direction. Another powerful

memory is watching him in classes by André Weilon abelian varieties. Whereas the rest of us weretotally cowed by Weil’s personality, Lang seemedto be able to follow anything and even to make cor-rections and amendments to Weil’s presentation.This seemed to me to be nothing short of miracu-lous. After Serge left Chicago to pursue his very il-lustrious career, I had only intermittent contact withhim. When we did meet, the memory of those earlydays in Chicago would come flooding back, and ina way, he was a powerful force in my life. From adistance, of course, I watched his erratic battles andeven was very rarely a victim of some of his out-bursts. But his textbooks, his successes, gave megreat pleasure. I regard him as a great mathemati-cian, much more than an expositor, as I believesome regard him. I don’t know if we will ever seehis like again, with anything approaching the enor-mous energy and insight which he brought to every-thing he touched.

When I learned of his death, a profound sadnesscame upon me. It was a feeling of incompleteness,that somehow I could not express my closeness tohim personally, nor help him avoid some of hismore acrimonious disputes. But Serge would havescoffed at such thoughts, and said that he managedperfectly well.

Stephen Smale, University of California,Berkeley

I had met Serge Lang by 1960. In fact in that yearhe initiated a very nice offer to me to leave Berke-ley to join the faculty at Columbia University, whichI accepted. During the three years I spent at Co-lumbia I became close friends with Serge and it washis support and friendship that helped make theColumbia years such a memorable period.

During my first years back at Berkeley, I becameinvolved in the Free Speech Movement and espe-cially the Vietnam anti-war protests. In this periodI invited Serge for a visit and we shared an officefor a year (1966–67, I believe). I introduced him tosome local activists and Serge himself became a po-litical activist, writing a book on the Bob Scheer cam-paign for political office.

During the following decades we kept in touchespecially during his summer visits to Berkeley. Itried without success to get a permanent appoint-ment in the mathematics department for Serge.We got along well and I seemed to have some im-munity from his occasional outbursts of anger. Isaw much less of Serge in the last decade partly be-cause of my life in Hong Kong and Chicago.

While leaving to others assessments of Serge’smathematical research, I want to make some briefremarks on other contributions.


Serge’s Books: My opinion is that these bookswere a great contribution to mathematics. He gavecopies of almost all to me, and I frequently usedsome for textbooks in courses I was teaching. Theywere characterized by economy and elegance andwere written from a broad point of view, mathe-matics as a whole. I especially enjoyed his gradu-ate analysis book. This was a book written for stu-dents in mathematics, (and mathematicians, notjust analysts) and reached the heart of the subjectquickly. In contrast, other texts I have encounteredspend a semester or even a year on foundationalmaterial as a step in the training of an analyst. Onthe other hand, his book on differential geometryhad the virtue of giving infinite dimensional foun-dations to the subject, which I found important inmy own research.

Serge’s Teaching and Inspiration to Students:Serge made a special and constructive effort toreach out to students of all ages. For example, hewrote a mathematics book for high school stu-dents and gave annual lectures to math club stu-dents at Berkeley. His contributions included mak-ing elementary expositions of topics in currentresearch. On two different occasions Serge gave lec-tures—in German!—to Don Zagier’s wife’s son Bern-hard’s school class. The talks were enjoyed by all,and greatly increased Bernhard’s prestige with bothhis teachers and his classmates.

Serge’s Files: These files and their accounts con-tained extensive documentation of hypocrisy of“The Establishment”, in science, in and outside ofmathematics. Although acknowledging their pos-itive role in science I sometimes disagreed with himin these matters. In particular, though his critiqueof the AIDS bureaucracy sometimes made sense,it was hard to go along with his attack on the HIVtheory of AIDS.

In ending, let me emphasize how big an influ-ence Serge has been in my life and how much I willmiss him.

John Coates, Cambridge University

I think Serge’s most remarkable quality as a col-league was his unstinting support for young math-ematicians. I personally benefited from this myselfwhen I was a young postdoc at Harvard in about1970, and Serge came as a visitor for a semester,shortly after he had resigned from Columbia. Wetend to forget when we are more established in themathematical world how precious it is when oneis trying to make one’s way in research to have thesupport and encouragement of an older mathe-matician. Serge was not at all distant to young peo-ple, but went out of his way to find out what onewas thinking about, and took time to discuss hisown ideas and feelings about the subject with one.

We only wrote one small joint paper together (ondiophantine approximation on abelian varieties),but his encouragement and friendship came at acrucial time in my own mathematical evolution, andI have always been immensely grateful for it. Thesecond striking quality of Serge’s was that he re-ally did live for mathematics, and somehow his be-lief in the goodness of the endeavour to do math-ematics was profoundly moving.

Dorian Goldfeld, Columbia University

Of the many people who had serious interactionswith Serge, I am one of those who came away withfierce admiration and loyalty. In the mid-1960s, Iwas an undergraduate in the Columbia engineeringschool on academic probation with a C–average. Inmy senior year I had an idea for a theorem whichcombined ergodic theory and number theory in anew way, and I approached Serge and showed himwhat I was doing. Although I was only a C–level stu-dent in his undergraduate analysis class he tookan immediate interest in my work and asked Lorchif he thought there was anything in it. When Lorchcame back with a positive response, Lang imme-diately invited me to join the graduate program atColumbia the next year, September 1967. In the fallof 1967 I found an unfixable error in my ergodic-number theoretic theorem. Lang was not at all per-turbed. He said these things happen all the timeand encouraged me to move on to something else.The following year Serge refused to discuss math-ematics with me or with anyone else. He said hewas taking a year off from mathematics and doingpolitics instead, but he kept encouraging me toprove theorems and told me to talk to Gallagher,who became my official advisor.

Despite his kindness to students, everyone closeto Serge has seen him explode, and this happenedto me on several occasions. For example, in 1992I organized a special year on number theory at Co-lumbia University. I invited some of the well-knownestablished leaders of the field such as Bombieri,Lang, Mazur, Manin, Schmidt, Szpiro, as well asmany younger people. Lucien Szpiro was chief ed-itor at Asterisque and invited me to submit a pro-ceedings of the conference to Asterisque. Lang be-come utterly infuriated and blew up at me whenAsterisque refused to accept his paper with Jor-genson, which I previously had invited him to sub-mit. I ultimately told Asterisque that I would resignas editor of the proceedings and withdraw my ownsubmitted paper unless they accepted theJorgenson-Lang paper. Asterisque refused to budge,and I immediately followed through on my threat.The proceedings were, nonetheless, eventually pub-lished. My paper and the Jorgenson-Lang paper


were published in 1994 by Springer-Verlag as abook.

I was very shaken for several days when I heardthat Lang is with us no more. He has had a profoundinfluence on my life and I will miss him enor-mously.

Jay Jorgenson, City College of New York

During the past fifteen or so years, Serge Lang andI were colleagues, co-authors, and close friends.Since Serge was known to never discuss his personallife, it is my inclination to not comment on ourfriendship beyond the mathematical collaboration.For instance, the reporter for the New York Timeswho interviewed me for Lang’s obituary did not un-derstand why I would not answer questions re-garding Lang’s family. Whereas I do feel compelledto respect his privacy after his passing, I have de-cided to accept the invitation of Steven Krantz andinclude comments regarding my own interactionswith Serge.

As with so many others, I first learned the nameSerge Lang as an undergraduate mathematics majorwhen I purchased textbooks for my mathematicscourse. I met Serge for the first time in 1987, dur-ing my second year in graduate school at Stan-ford. I remember the level of excitement among thegraduate students in anticipation of Serge’s talk atStanford. It was thrilling to see his energy duringhis lecture. Lang seems to be someone, I remem-ber thinking, who has discovered what will give himthe most out of life, namely his mathematics andhis politics (what he himself called “trouble-making”), and he is doing it.

In 1990 I joined the faculty at Yale as a Gibbsinstructor, and during that spring semester I gavea graduate course which Lang attended. As I hadexpected, from the audience Serge directed my lec-tures for himself, insisting on immediate changesin notation and topics. In one particular lecture, Ipresented an evaluation of spectral determinantson elliptic curves which avoided the usual ap-proach, namely Kronecker’s limit formula, and in-stead relied on a trick I developed in my thesis. Langwas silent during the entire presentation. When Ifinished, he insisted that I wait in the classroomso that he could go to his office. When he returned,he had two papers with him, one by Artin from 1923and another by himself from 1956. He pointed outthat the technique I presented in the setting ofheat kernels on elliptic curves was conceptuallyidentical to Artin’s ideas in the setting ofL-functions of number fields and Lang’s ideas intopology, in the context of the characteristic poly-nomial in linear algebra. That conversation grewinto our first joint paper which was published inCrelle’s journal in 1994.

During the summer of 1990, Lang called meevery day, several times, as he traveled through Eu-rope and to Berkeley. We always spoke about math-ematics, and he challenged me on the same pointI attempted to make to him earlier: Why did I be-lieve that one can use heat kernels and heat ker-nel techniques, perhaps formally in a way to be de-veloped, in a wide range of mathematical questions?We discussed, argued, and debated, as only Sergecould, until he returned to Yale that fall. He invitedme to lunch one of the first days he was back. Dur-ing lunch he asked if I would be willing to work withhim on the mathematics we discussed. One cannotimagine my thoughts at that moment. To have asenior mathematician express interest in one’sideas is remarkable enough, but to have Lang sayhe wanted to work with me simply cannot be de-scribed. I found out later a more touching aspectto that conversation with Serge. Apparently, in1988 Lang had told some faculty at Yale that hismathematical abilities were gone, and he couldn’tcontinue; after we began working together, he thenwould comment to others that our work was keep-ing him alive.

During the first few years of our joint investi-gations, we spent countless hours developing along-term program of research. Serge was a veryprivate person, more so than I have seen with any-one else. Although I knew his private telephonenumbers, I never called him at home, though hewould not hesitate to call me at any time. It wasvery rare that he mentioned his family or his non-mathematical, or non-trouble-making, interests.We became friends, and we made a point of talk-ing, perhaps quite briefly, each morning when hearrived in his office, each evening when he left togo home, and many times during the day.

I left Yale in December 1996. Serge was hospi-talized in December and later in February, and heinsisted that I visit him each day to continue ourmathematical conversations. His February stay inthe hospital was reported in the Yale student news-paper, and the dean of Yale College, a person Sergevery much disliked, was quoted as saying that Langwas an excellent mathematician. I showed Langthe article while he was still in the intensive careunit, and, when he read the dean’s comments, hescreamed, “That ∗∗∗∗∗∗∗∗ isn’t qualified tojudge.” Serge was stunned when his outburst re-sulted in my expulsion from his hospital roomuntil the next day. As in so many other instances,Serge was right, but perhaps his message could havebeen delivered differently.

Later in the spring of 1997, as I was seeking em-ployment for the upcoming year, Serge asked meto visit him at Yale so we could discuss our pro-jects. At the time, I considered leaving academicmathematics and seeking a career change awayfrom a university environment. We discussed the


matter in detail, and Serge pointed out to me theeffect it would have on him if I were to quit math-ematics. At the end of the day, I honored his requestto continue our program of study and promised tonot seek a nonacademic position.

We spent less time together during the next twoyears when I was at Oklahoma State during 1997–98and in Greece during 1998–99. In the fall of 1999,I accepted a position at the City College of NewYork, in part because it allowed me to visit Sergefrequently, which I did on most weekends. At thattime, we began focusing our ideas, even going sofar as writing a document for ourselves, estab-lishing a “wish list” for our mathematical program.We created an outline of the articles, monographs,and books which must be completed in order to ful-fill the steps of the program which we envisioned.One of the earliest words my son knew was “Serge”,which he said each time the phone rang; frequently,my son was right, and indeed the call was fromSerge.

It was evident to me, even in 1999, that at age72 Serge was growing tired. Even though the learn-ing process did keep him alive, time was catchingup with him. He was always very sharp, with fas-cinating ideas and insight. However, he left the of-fice earlier in the evening, and he required restduring the day. Without acknowledging the act,we altered the pace of our work, finishing what wecould given the energy he had. When he was at Yale,I visited him most weekends, and when he was inBerkeley, I traveled there to continue our work.

Most everyone saw Serge as the forever young,highly energized individual. For the most part, hedid not allow many people to see that time was af-fecting him. The fact that hardly anyone noticedthat Serge was aging was, I believe, another mani-festation of his level of privacy.

In August 2005, we completed another book. Onthe day we submitted the manuscript to Springer-Verlag, Serge was excited. To me, he momentarilyregained his youth as I knew it when we deliveredthe manuscript to the post office. That day I sawagain the dynamic person I remember lecturing at

Stanford during my graduate school years. How-ever, when we returned to the office, he needed torest, and that evening during dinner he directed theconversation away from the specific ideas for ournext project. Instead, he spoke of his wish that Ialways pursue our program of research, hopefullyarriving at the goals we set for ourselves. Duringthe next weeks, we spent time revising our wish listand reviewing our original mathematical plan. Wecontinued to speak several times each day, up toand including September 12, 2005, the day Sergedied.

Having spent so much time with Serge, there aremany stories I could tell. Serge had many sideswhich affected everyone, including me, many dif-ferent ways. He had friends and enemies, and per-haps I inherited some of both. As he pointed outto me quite some time ago, it is possible that myassociation with him was both positive and nega-tive for me. Serge once told me that he made cer-tain personal decisions early on in his life, and hestuck by those decisions. For me, I stand by my de-cision to work with Serge. He was a close and loyalfriend, and I believe that I was for him. Serge wasa part of my daily life for nearly fifteen years, andfor me his absence is great.

Paul Vojta, University of California,Berkeley

Although I had seen Serge at Harvard once or twicein the Common Room (invariably arguing withsomeone), I first met him when I went down to Yaleto ask him to write a letter of recommendation forme. At one point in the discussion, he asked meabout my definition of integral point. I started todescribe Serre’s idea of an infinite set of points withbounded denominators, but he angrily interruptedme: “I don’t know what an integral point is, and nei-ther do you!!!” However, later on that day he addeda flattering paragraph to his new edition of Dio-phantine Geometry describing work in my thesison integral points relative to a divisor with suffi-ciently many irreducible components.

Later, upon hearing about my conjectures, hecalled me up and invited me to come to Yale. I wasa little hesitant about this, given his propensity foranger, but my advisor Barry Mazur convinced methat I should accept.

During my time at Yale, I gave two or three grad-uate courses. Serge always sat in the front row, pay-ing close attention to the point of interrupting memidsentence: “The notation should be functorialwith respect to the ideas!” or “This notation sucks!”But, after class he complimented me highly on thelecture.

Lang at Berkeley.

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While on sabbatical at Harvard, he sat in on acourse Mazur was giving and often criticized thenotation. Eventually they decided to give him aT-shirt which said, “This notation sucks” on it. Soone day Barry intentionally tried to get him to sayit. He introduced a complex variable Ξ, took its com-plex conjugate, and divided by the original Ξ. Thiswas written as a vertical fraction, so it looked likeeight horizontal lines on the blackboard. He thendid a few other similar things, but Serge keptquiet—apparently he didn’t criticize notation un-less he knew what the underlying mathematicswas about. Eventually Barry had to give up andjust present him with the T-shirt.

Once, close to the end of my stay at Yale, I wasin his office discussing some mathematics withhim. He was yelling at me and I was yelling back.At the end of the discussion, he said that he’d missme (when I left Yale). Now that he has left, I willmiss him, too.

Gilles Lachaud, Institut de Mathématiquesde Luminy

I met Serge Lang in 1972, during the AMS sympo-sium on Harmonic Analysis on HomogeneousSpaces, held in Williamstown. He had just left Co-lumbia for Yale, and I was with Paris 7.

At that time, French and American universitieswere a hotbed for the antiestablishment ideas, andSerge was involved in the Free Speech Movement.Also, he was writing his book on SL(2,R): thus, inWilliamstown, we discussed both alternative poli-tics and spherical functions.

In my mind, Serge was a hunter, in mathemat-ics as in his political and social struggles. He waschasing a precise game and nothing was able tomake him deviate from this goal.

As a polemicist, he was very proud of the Fileprocess, consisting of bringing lies on some topicto full light by sending letters to opponents, to waitfor contradictions in the answers, and to sendXerox copies to all the people involved in the con-test. He was sure that victory would emerge fromthis confrontation.

The special feature was that he wanted to workalong scientific lines and to prove the statementshe was defending, an uncommon and irritating po-sition outside the mathematical community. Hismodel was in mathematics, as opposed to social sci-ences, about which he used to say: “In mathemat-ics you cannot say, ‘I disagree with this statement.’You can say, ‘this is false’ or ‘this is of poor inter-est,’ but there is no disagreement to express what-ever.”

Among American mathematicians, Serge was oneof those who were closest to France: the souvenirs

of his life during his youth with his parents wereenduring, in particular vacations on the Mediter-ranean seashore. Later in his life, he became amember of the Bourbaki group, stayed in Franceand lectured at length, in Paris and elsewhere.

In some respects, Serge was rather austere. Buthe had a cheerful and authentic enthusiasm formathematics: this is reflected in his books, and thisenthusiasm was contagious.

At the beginning of 2005, we exchanged ourwishes by phone. As usual, he was infuriated, thistime by his own illness: he was conscious of his badhealth, and desperate not to be able to fulfill be-fore his death the program on zeta functions hehad in mind, after twenty years of work on analy-sis on groups.

We miss a great intellectual figure, and I miss afriend.

Roger Howe, Yale University

Books: Serge wrote an extraordinary array ofbooks, from widely used texts, including calculustexts, and even a high school geometry text, tostandard references, to monographs which are theonly treatment of their subject in book form.

The topics tend to cluster around algebra, andespecially Diophantine equations, which were hisgreat love in mathematics, but they span a re-markable range.

For a long time, his practice was to give each yeara graduate course on a new topic, and at the end,to turn the notes into a book. He had an amazingcapacity to boil a subject down to its essence,which he often formulated with a few axioms orproperties. He was a consummate axiomatizer. Ashe reached his late 70s, he gave little sign of slow-ing. His last several books presented joint work withJay Jorgenson on applications of the heat equationto analysis on symmetric spaces, with a view to-wards automorphic forms.

Files: Besides books, a lot of Serge’s literary ef-fort went into making files which chronicled hisfights. Serge loved a good fight and he didn’t havetrouble finding them. He was especially concernedwith honesty, especially honesty in public rhetoric.Serge would coordinate multi-party correspon-dence, organizing sets of letters into packets, andcirculating them with supplementary documents toa “cc list” of parties to the correspondence andother interested readers. At the end of the fight,he collected the whole into a file, and gave it a title.

Serge thought of his files as documentation ofthe way life works today, and especially of his FirstLaw of Sociodynamics: the power structure doeswhat it wants when it wants, and looks aroundlater for justification. Serge was willing to follow


his principles and beliefs (almost) wherever theyled.

Students and Teaching: A third focus of Serge’senergy was teaching and students. Generations ofYale undergraduates benefited from his teachingin the broad sense. He spent hours outside of classtalking with undergraduates, about mathematics,politics, music, anything. He frequently ate mealswith undergraduates in the Yale dining halls. He wasespecially concerned with promoting clear think-ing. He often found that students who came tohim had been confused by poor education, butthat by appropriate challenges he could help themto become independent thinkers. He referred to thisprocess as “Recycling their brains”.

Serge’s attention to undergraduates was a partof his concern for the advancement of youngerpeople generally. It was a habit with Serge to en-courage younger mathematicians and be interestedin their work. That was certainly so for me. I recallwith gratitude his wholehearted enthusiasm formy work, and his help in promoting it, and I knowof many others who similarly benefited.

Gisbert Wüstholz, EidgenössischesTechnische Hochschule, Zurich

Presumably my most expensive investment in math-ematics as a student was a book with the titleAlgebra by an author whose name was Serge Lang.A former schoolmate had recommended it to meas a very modern new tract in algebra. He had en-tered university one year before me and just startedwith a course in algebra where the book had beenrecommended. Certainly the top of my list of fa-vorite mathematics books would be Algebra bySerge Lang. The reason I like it so much is that ithad a clear vision for modern and conceptual math-ematics and this was put together with much math-ematical taste.

It was exactly ten years later at the Arbeitstagungin 1978 in Bonn when I first met Serge personally.At the time everybody talked about “Bombieri-Lang”, a paper which influenced enormously the re-search in transcendence. For us young students innumber theory it was a big challenge to try to un-derstand the difficult methods from geometricmeasure theory, the theory of plurisubharmonicfunctions and L2-estimates. It took us away fromthe classical methods in transcendence theory andtaught us that you need mathematics as a wholeto formulate and to prove interesting new resultsin transcendence theory.

At that Arbeitstagung I talked quite a bit toSerge and he eventually helped me to find a post-doc position in Wuppertal. There I got into contactwith people in algebraic groups and this helped me

to enter into another area which had been openedby Serge: he had started in 1962 with a series ofpapers in transcendence theory out of which an-other book resulted in 1966 which turned out—atleast in my eyes—to be the most influential book(Introduction to Diophantine Approximations) thathe ever wrote.

Even the abc conjecture, one of the favoritesubjects of Serge, has now been incorporated intotranscendental context, and this indicates how im-portant Serge’s impact into diophantine geometryand transcendence has become. He had createdthe frame of a very active and broad area to whichhe substantially contributed, he had the right math-ematical visions and supported enthusiasticallyany progress.

For many years Serge visited me at Zürich. Hegave numerous talks in my seminar and to under-graduates, and he enjoyed visiting us at our house.Only one thing I never forgive him: when he to-gether with Schinzel once had been guests at ourhouse in Bonn he came after the main course intothe kitchen where I had started to prepare a souf-flé Grand Marnier. Without stopping he talked tome and distracted me so much that at the end thesoufflé did not rise in the oven. I did not want tooffer it to the guests but Schinzel forced us to eatit since he would not agree to throw away food.After this I essentially stopped cooking soufflés.

Jürg Kramer, Humboldt University

I had my first encounter with Serge Lang as a stu-dent in an indirect way: after having just finishedmy first two years as an undergraduate with the“Vordiplom” in mathematics at the University ofBasel (Switzerland), Martin Eichler proposed a sem-inar on Lang’s new book on modular forms. SinceI was just a beginner in the subject, the book mademe work quite hard, but anyhow, as a result I be-came strongly interested in the subject. Six yearslater, just before completing my Ph.D., Eichlerasked me to accompany him to the “Arbeitsta-gung” taking place in June 1984 in Bonn. After ar-riving at the entrance hall in front of the big lec-ture hall at Wegelerstrasse 10, one of the firstpersons to meet was Serge. At the time, it wasquite impressive for me to have been personally in-troduced to this world-known mathematician. WhatI didn’t know at the time was that this was the startof a relationship lasting for more than twenty years.

In fact, when I returned to Bonn in 1985 to visitthe Max-Planck-Institut für Mathematik (MPIM), Iwas caught by surprise when I met Serge again dur-ing his regular trip to Europe in June and that heimmediately remembered me; as a consequence,we started to talk about mathematics, first on a quite“innocent” level. Our mathematical communication


intensified in the years 1987/88, when I was giv-ing a course on “arithmetic surfaces” at MPIM,while at the same time Serge was preparing his bookIntroduction to Arakelov Theory.

After having completed my “Habilitation” at ETHZürich (where Serge was also visiting on a regularbasis since the late 1980s), I moved to HumboldtUniversity (HU) in Berlin in 1994. From 1995 until2003, Serge regularly visited HU in late May/earlyJune for one week. During this period of almost tenyears, we got to know each other more closely andour relationship deepened. In Berlin, aside from histraditional talk in our number theory seminar(demonstrating the ubiquity of the heat kernel inthe last years), he was intensively arguing, dis-cussing, and interacting with our graduate stu-dents. In addition, he always generously offeredtalks to gifted high school students. In particular,the high school students consider his unexpecteddeath as an infinite loss, and it is very sad that thistradition has come to such a sudden end.

Although Serge tried to stay away from close per-sonal relationships, it seemed to me that in his lastyears when coming to Berlin the ties between him,my wife Ruth, and myself got somewhat closer. Wewill surely miss him.

David E. Rohrlich, Boston University

Shortly after Serge’s death a few people sug-gested to me that I write something about his math-ematical contributions. The suggestion apparentlystemmed from a concern that obituaries wouldfocus on Serge’s eccentricities and temper tantrumsrather than on the highlights of his career. Later,when I was invited to contribute something to thepresent article, I decided to intertwine my personalreminiscences with some glances at Serge’s math-ematics, partly because I was mindful of the con-cern that had been expressed to me, but partlyalso because Serge’s passion for mathematics wasin my view an essential part of his persona. How-ever, the length of my submission ended up ex-ceeding the stipulated limit, and the editors excisedlarge portions of it, including all of the mathemat-ics. For that I had only myself to blame, but the prob-lem was that the mathematics had been the gluewhich joined one paragraph to the next, and withthe glue removed, the entire piece fell apart like aBrunnian link. So I withdrew my submission. But indoing so I realized that this little episode was some-how a fitting memorial to Serge, for one of his note-worthy eccentricities was his proclivity for quarrelswith editors. He could be very hard on editors, andhe certainly would not have withdrawn a submis-sion without a fight; in fact, his fights with editorswere a significant component of some of his files.As with so many of Serge’s “eccentricities”, his

stance here was based on principle—in the presentcase, a defense of an author’s right to self-expres-sion free from gratuitous editorial intervention—and what converted integrity into eccentricity wassimply his stubborn insistence on continuing to dobattle far beyond the point where the battle seemedworth fighting. But you had to hand it to Serge: hehad the courage of his convictions. We can proba-bly all learn something from his example.

Marvin Jay Greenberg, University ofCalifornia, Santa Cruz

I was Lang’s first Ph.D. student. Officially, EmilArtin is listed as my thesis advisor, but he leftPrinceton for Hamburg three years before I wrotemy thesis. What Artin did was ask Lang to commutefrom Columbia to Princeton in academic year1956–57 to continue teaching algebraic geometry,which Artin began toward the end of his graduatealgebra course—an extraordinary request and anextraordinary acceptance on Lang’s part. So Langtaught a course at Princeton on Abelian varieties,in the style of A. Weil, and after a few weeks, I washis only student.

Lang went away to Paris the following year, andwhen he returned after that, we continued meet-ing informally at Princeton. He told me a conjec-ture of his about Abelian varieties he wanted meto prove for my thesis. I had no idea how to do so,and I was extremely busy teaching four elementarycourses at Rutgers that year. Then he confrontedme, as he is famous for doing, and shouted at methat if I did not show some progress with his con-jecture in the next two weeks, I would no longerbe his student.

The following weekend, in my attic room in NewBrunswick, after rereading Lang’s thesis, I sud-denly had a flash of insight on how to solve thatproblem. I needed a few more weeks to write outall the technical details, but when I told him I hadthe solution, he was delighted. He took me out toa fine Spanish restaurant in NYC and treated meto paella, which I’d never eaten before. He invitedme to his apartment overlooking the Hudson Riverand played Bach’s dramatic Partita #6 (which I’dnever heard before) and a Brahms Rhapsody for meon his grand piano. I felt as if he had lifted me intoan exalted new world of excellence.

I remained on good terms with Lang for quite afew years after I left Princeton for Berkeley. Manypeople were turned off by his aggressive person-ality, but I always enjoyed that immensely, partic-ularly his brutal honesty and taunting sense ofhumor. He once told me bluntly that I would neverbecome a great mathematician because I was afraidof making mistakes. He certainly made plenty of


them in the first editions of the many books hedashed off so quickly. John Rhodes once told me:“Serge Lang writes great books badly.”

Shoshichi Kobayashi, University ofCalifornia, Berkeley

With no more than amateurish interest in numbertheory, my mathematical contact with Serge Langis mostly through hyperbolic complex analysis.With his several conjectures and his introductorybook on hyperbolic complex manifolds, he was thebest promoter of the subject.

In 1985 I participated in the AMS SummerResearch Institute on number theory at HumboldtState University, Arcata, in northern California.Thanks to Serge, I was invited to give an introduc-tory talk on hyperbolic complex analysis, includingNoguchi’s partial answers to the function-theoreticanalogue of the higher dimensional Mordell con-jecture as formulated by Lang.

Since Serge was the only one driving back toBerkeley on the last day of the workshop, I ac-cepted his offer of ride with trepidation. As I hadexpected, his driving was like his typing—fast. Hedrove German-style, flashing the headlights when-ever slow-moving cars blocked us, which happenedto be all cars ahead of us. We were back in Berke-ley by 10:00 p.m. As a result, the only thing I re-member from the workshop is this experience.

In 1987, Serge published Introduction to Com-plex Hyperbolic Spaces with Springer-Verlag. Whilewriting this book, he called me, not day and night,but 9 in the morning. Quickly he found out mymorning routine and the most convenient time. Sousually I answered when the telephone rang around9. When my wife answered, he invariably said “IsKobayashi out of the shower yet?” When he stoppedcalling after several months, my wife said “I guessSerge finished his book.”

Without Serge, the summer in Berkeley will nolonger be the same. I miss him.

Hung-Hsi Wu, University of California,Berkeley

Serge Lang’s life is easy to characterize: it was 90percent mathematics and 10 percent scientific pol-itics. I happen to be one of the few mathematicianswho forged a friendship with him through politics,thereby getting a glimpse of a side of him that wasperhaps denied other mathematicians.

For a mathematician to have any kind of friend-ship with Serge, it is a given that it could not havebeen completely divorced from mathematics. I wasno exception.

While I had known about Serge since my un-dergraduate days at Columbia when he was stillwith that institution, my first meeting with him wasin Berkeley around 1987, when he called to talkabout Nevanlinna theory. At the time, he was stillcampaigning against Huntington’s election to theNational Academy of Sciences. In any case, Sergeand I began political discussions with increasing fre-quency after that. He was at Yale, of course, and Iwas at Berkeley, but Serge was never shy aboutusing the phone. While I sometimes had reserva-tions about the tone of his writing, there was neverany doubt in my mind about its substance. It maysurprise some that a firebrand like Serge coulduse any encouragement, as it did me, but I dis-covered that fighting the kind of lonely battle thathe did, he probably found it easier to listen tosomeone sympathetic to his views than to engagein a shouting match twenty-four hours a day. Fromtime to time, he would sound me out on his strate-gies. One consequence of all these phone conver-sations was that I was privy to all his fights sinceabout 1990, including the denouement of the Hunt-ington case, the Gallo case, the Baltimore case, andof course the still-ongoing HIV controversy.

Because I had no strong mathematical connec-tion with Serge, our relationship could afford to bemore relaxed. Each time he called my home whenmy wife and I were out, he would leave a messagereprimanding us for “goofing”. Serge was famousfor getting along with young people. He and my sonwere great pals, and each time my son would asmuch as ask a question about mathematics, a fewdays later a book or two on that subject would ar-rive from Springer-Verlag. The author of thosebooks was of course Serge Lang.

In the last year of his life, his big fights were stillover HIV. One involved his submission of a surveyto the Proceedings of the National Academy of Sci-ences on the state of HIV research and governmentactions on the so-called anti-HIV drugs. He and Iknew from the beginning that it would get nowhere,but the cavalier way in which his survey was re-jected was stunning. I went to his office when wordhad just come and, perhaps owing to similar frus-trations in my own work in mathematics education,I lost it and said in less than polite language thatI had had it. In an instant, his role and mine werereversed and he soothed me with the philosophi-cal observation that life was hard and that we justmove on.

The last I heard from Serge was a phone call onSeptember 3, 2005, after he had left Berkeley forNew Haven. Four days after he passed away, his lastfile on HIV and the Proceedings of the NationalAcademy of Sciences arrived. It was addressed tomy son. He lived his life on his own terms to thevery end.


Lisa R. Goldberg, Morgan Stanley

High ExpectationsSerge Lang believed that young people have a spe-cial ability to see the truth. He was a champion ofyouth, and his students loved him. In January 2005,Serge submitted a brief op-ed piece and a dense fourpage advertisement to the The Daily Californian,the U.C. Berkeley campus newspaper. The topicwas a Serge standard: growingdissent against the orthodox po-sition that AIDS is a disease andit is caused by the HIV virus. Thead contained supporting docu-mentation for the op-ed piece,which had previously been re-jected. The submission was ac-companied by a personal checkto pay for the ad.

Serge had been disseminat-ing information on HIV and AIDSfor roughly twelve years. TheDaily Cal was a natural outletfor Serge’s challenges. Aftersome back and forth, Serge re-ceived a rejection letter explain-ing why his material could not bepublished in The Daily Cal. Hereare some excerpts that arepieced together from The DailyCal file, which contains a copy ofthe rejection letter and Serge’sresponse.

Daily Cal: We are confident that you under-stand all newspapers must have the flexibility asa business to reserve the right to refuse any ad-vertisement, letter to the editor, op-ed or press re-lease at the discretion of their publishers.

Serge: In fact, I know that newspapers have thepower to refuse advertisements, letters to the ed-itor, op-ed or press releases, and I have known itfor a long time. It’s not exactly secret information.

Daily Cal: These are the things we do to protectour readership.

Serge: I call this position Nannyism.Daily Cal: Clearly it would have been a serious

ethical error if we had elected to publish all or aportion of any op-ed letter to the editor that ref-erenced an advertisement specifically designed toclarify or provide back-up data for the op-ed.

Serge: What you find “clear” I do not. In fact, Itake an opposite view. You might have stated moreaccurately “clearly to us” to make your assertionmore precise, instead of pretending to a universalethical standard, applicable to others, and imply-ing that I asked you to do something unethical.

The reply is pointed and funny throughout. Ev-ident in every line are Serge’s high expectations of

the Daily Cal staff members. As far as he was con-cerned, they may as well have been running TheNew York Times, or the world for that matter.

Serge was a champion of youth, but he hadplenty of energy left over for grownups. The DailyCal file closes with a letter to the dean of the U.C.Berkeley School of Journalism. This time, the topicwas journalistic responsibility. Serge stated someof his concerns, drawing examples from the mediacoverage of HIV and AIDS. As always, his high ex-

pectations were in evidence. Heasked, non-rhetorically, “Howdoes one make up for defectivereporting over two decades?”And he signed the missive “In-formatively yours, Serge Lang.”

Allyn Jackson, NoticesDeputy Editor

“It’s me”: This is what some ofus in the AMS headquarters of-fice would hear every now andagain when our phones rang. Nogreeting, no name: Of course itwas Serge Lang. Probably thevery first time he called me heannounced his name, but neveragain. When I got that first call,I mentioned it to a mathemati-cian acquaintance, who said,“Watch out, you are in the roomwith the snake.” Indeed, it waswith some trepidation that we

AMS staffers would take these calls. Usually Langwould rail on about whatever issue he was currentlycampaigning about, and we would give noncom-mital replies to avoid getting drawn into a debatewith this tireless debater. But after a while he wouldsoften up and crack a joke or tell a story; once herelated to me the plot of a play he had seen. Some-times he would end the call with a whimsical clos-ing along the lines of “Toodle-oo”. Carol McConway,a former AMS employee with whom I worked onthe Notices many years back, received calls fromLang on a regular basis. She told me that in one con-versation he remarked that she was very smartand asked where she had gone to college. Carol, likemany of the highly intelligent and capable womenon the AMS staff, had never gone to college. “Aha,”Lang replied. “That’s why you are so smart. Youwere not ruined by the educational system.” Langbefriended another former AMS employee, TerryDrennan, who worked in the editorial department.When Terry got into a serious scrape with her boss,Lang stood up for her.

I was one of the many recipients of Lang’s in-numerable and lengthy “files”. He sent me massesof documentation about his campaign about

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HIV/AIDS. As I recall, he never stated that he be-lieved that HIV does not cause AIDS. Rather, he ad-vocated the need to view this hypothesis with skep-ticism and rigor. He poked holes in research papersand other writings that uncritically assumed the hy-pothesis to be true. At some point I looked care-fully at one of the AIDS research papers that Langhad denounced; this particular one had been writ-ten by the prominent AIDS researcher David Ho, to-gether with some colleagues, and had been muchcited in the subsequent literature. I became con-vinced that Lang was right in saying that the paperlacked rigor and used mathematics inappropri-ately. This is how it was with Lang’s campaigns: Healways had a valid point. Yes, he was obsessive, hecould be antagonistic, he had a bulldog-like at-tachment to his causes and sometimes lacked asense of proportion. And the solutions he pro-posed to the problems he identified were often to-tally impractical and naive. Still, whenever I tookthe time to examine his analysis of those problems,I found myself concluding that he was basicallyright.

I do not believe Lang pursued his causes out ofa desire for fame or notoriety. Rather, he was hor-rified by the falsity he found all around him, andhe would not let others turn their eyes from it. Wehave lost a person with a highly attuned sense ofwhat is truthful and what is sham, and it is a pro-found loss indeed.

John Ewing, Executive Director, AmericanMathematical Society

I had contacts with Serge over many years, begin-ning when I was editor of the Mathematical Intel-ligencer (and he insisted on publishing a long, longarticle). My kids got to know him by phone—hecalled me at home on and off for more than a year.

Later as executive director I came in contactwith him, largely because of his dispute with theNotices of the AMS (about HIV). Serge always calledthe office and simply said, “It’s me.” Everyone hereknew who that was. When I explained that the EDdoesn’t make editorial decisions, he insisted thatthe “higher ups” at the AMS could do whateverthey wanted. He often sent material related to var-ious things, and always mentioned that he wassending it to the “higher ups”. Phone conversa-tions were always long, protracted affairs.

But even with long phone calls and huge quan-tities of written material (we have a giant file here),Serge was really a charming guy whom I instinc-tively liked and admired. The world is better off hav-ing had him fight for his causes, with passion andindignation.

Joseph Gerver, Rutgers University

I met Serge Lang in 1967, my sophomore year atColumbia, when I took his multivariable calculusclass. This was before the days of unified calculus.All of us were math majors and many of us werespoiled by our high school experience of learningmath with very little effort. So Lang would fre-quently throw chalk at us, or yell.

I often ate dinner at the Gold Rail with Richard(now Susan) Bassein and Eli Cohen, and if Langwas also eating there he would always join us andusually pick up the tab. Sometimes we would talkabout math. Lang did not think logicians were truemathematicians, because no real mathematicianwould worry about whether a proof made use ofthe axiom of choice. Why shouldn’t you use theaxiom of choice? It’s obviously true! Think aboutit! How could you not be able to construct a set bychoosing one element from each set in a collectionof sets? Just do it!

We also talked about politics, music, life. Heshared with us his growing unease about the Viet-nam war and what he viewed as Columbia’s com-plicity in it, although he made a point of never dis-cussing politics in class. He offered advice aboutunrequited love: If at first you don’t succeed, try,try again, but after the third time, if you still don’tsucceed, give it up!

Rarely, he took someone seriously who wasmerely pulling his leg. Dorian Goldfeld, at thattime a postdoc at Berkeley, once reported glee-fully that Lang had posed him a problem which hehad in fact been working on for months and hadalready solved. Goldfeld told him with a straightface that he would think about it, and the next daypresented Lang with the solution. Astounded, Langasked him how he had managed to find the answerso quickly, and Goldfeld explained that he hadused ginseng, which greatly enhanced his mentalpowers. Lang excitedly promised to try it himself,but reported disappointedly a few days later thatginseng had had no discernable effect on his brain.

In the summer of 1968, I was on vacation withmy parents in Berkeley. Lang was there as usual,so my folks invited him for dinner. Naturally theconversation turned to the demonstrations at Co-lumbia and the other student protests around thecountry, many of them against the war. Althoughmy mother was opposed to the war in Vietnam, shedeplored the excesses of some of the demonstra-tors, which she attributed to their permissive up-bringings. “But what about France?” Lang objected.“And Poland? Even in Poland students are protest-ing against the government, and you can be surethey weren’t raised permissively! Young peopledon’t want to be used by their governments. All overthe world, students are fighting for their freedom!”


Peter Duesberg, University of California,Berkeley

About twelve years ago I received a package of“files” from Serge Lang. The files objected to theWashington-style cover up of the scandal sur-rounding the American discovery of the hypo-thetical AIDS virus. Impressed by the thoroughnessand mathematical logic of his case, I sent my ownfile in response: “Dr. Lang,” I wrote, “the politicalscandal about who discovered the hypotheticalAIDS virus is from a scientific point of view no morethan a distraction—a catchy story about who stolewhose fake diamonds. The scientific challenge,however, is whether AIDS is a viral or a chemicalalias lifestyle epidemic, caused by the long-termconsumption of recreational drugs and anti-viraldrugs such as the inevitably toxic DNA chain-terminator AZT.”

Not much later Lang and I became allies in theAIDS debate. Lang gave seminars on AIDS, wrotefor the Yale Scientific, included two AIDS chaptersin his book Challenges (1998) and generated asteady flow of AIDS files, the last of which arrivedhere only after his death. But only now, on the sadoccasion of his death, is Lang’s AIDS engagementpresented as an Achilles heel of a mind that seemedotherwise irrefutable in its high standards of ac-curacy and precision not only by the politicallycorrect New York Times and Yale Daily News, buteven by several of his mathematical peers.

In view of this I take a last stand on behalf ofour colleague, who cannot do this anymore, tryingto inform his survivors with “primary evidence”,rather than “condition them” with governmenthandouts, as Lang would have said. Even if one viruscould cause the twenty-six infectious and non-infectious (!) diseases that are now defined as AIDS,the following would be true:

1) AIDS would be contagious. But, there is no casereport in the peer-reviewed literature of even onedoctor who ever contracted AIDS from one of the929,985 (2004) American AIDS patients in twenty-three years. Moreover, not even one of the thou-sands of AIDS virus researchers ever contractedAIDS from their “deadly virus”, as the New YorkTimes calls it.

2) AIDS should appear within days to weeksafter infection, because the AIDS virus, like otherviruses, replicates with multiplication rates of 100to 1,000 within twenty-four hours. But AIDS is saidto appear only five to ten years after infection byits hypothetical viral cause.

3) The epidemic would spread randomly like allviral epidemics. But AIDS cases in the U.S. and Eu-rope are highly nonrandom, 80% are males, ofwhich 1/3 are intravenous drug users and 2/3 aremale homosexual users of toxic, recreational drugs

like nitrite inhalants and amphetamines and areprescribed DNA chain-terminators.

4) The epidemic would have formed a classicalbell-shaped time curve, increasing exponentiallyand then declining exponentially owing to naturalimmunity within weeks to months, like a seasonalflu. Instead AIDS increased slowly over a decade andhas since leveled off, without ever inducing im-munity against itself.

Thus AIDS fits a lifestyle—but not a viral epi-demic. I hope, therefore, that those who saw Lang’sAIDS engagement as an Achilles heel might re-consider.

I already miss Lang as an ally in the politicallyincorrect debate on the cause of AIDS. And I hopethat you all let me join you in missing the Menschthat was hidden behind the machine. Au revoirSerge Lang!

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Serge Lang, 1927â€“2005

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