1995, remembering lipman bers

RememberingLipman Bers

8 NOTICES OF THE AMS VOLUME 42, NUMBER 1

There are many of us in the mathe-matical community whom LipmanBers profoundly affected. These fourarticles will refresh our memories andintroduce this great figure to another

generation: his unusual background, his turningto mathematics, his important scientific contri-butions, his role in human rights, and the pow-erful support he lent his women students andcolleagues. I myself worked with Lipa long agoin the early 1950s during a period when, as C.C.Lin once told me, Lipa was masquerading as anapplied mathematician. In Abikoffs biographyand mathematical story I followed the thread of

Lipas transformation as he leftsteady fluid dynamics and at-tacked quasiconformal map-pings. Both areas profited fromhis deep understanding of com-plex analysis.

Lipas life was never far fromjust causes, and in Corillon andKras article we see just howdeeply involved he was andhow he drew his colleagues into worry and do somethingabout political wrongs.

Tilla Weinstein was a Bersstudent at the Courant Institute when I was ayoung faculty member. In her article she broughtback to me those times when Lipa was sharingwith me his exuberant love of mathematics, es-pecially trying to teach me a mathematical vo-cabulary. Gilmans article reminded me in turnnot only of how savvy Lipa could be on the de-

partmental political front but also of how sup-portive Lipa and Mary were as I struggled tohandle mathematics and children.

One memory of my own. When Stalin died, weheard about it around midnight on our car radio.We rushed over to the Bers house. The lights wereon, so Mary, Lipa, Herbert, and I were soon drink-ing toasts to a new era. It was a long time com-ing, but, especially since his father had been im-prisoned for many years in the Gulag, I am veryhappy that Lipa lived to see the fall of the So-viet Union.

Cathleen S. Morawetz

Lipman BersWilliam Abikoff

Lipman Bers was born in Riga, Latvia, on May 22,1914. He died in New Rochelle, New York, on Oc-tober 29, 1993, at age seventy-nine. Here I pre-sent the story of his life, both personal andmathematical; often it is delivered in his ownwords. Bers was a mathematician whose workpossessed power, grace, and beauty; it has con-tinuing relevance in both mathematics andphysics. He also set the highest of standards as

His workpossessed

power, grace,and beauty

William Abikoff is professor of mathematics at the Uni-versity of Connecticut in Storrs. His e-mail address [email protected].

bers.qxp 3/16/99 3:27 PM Page 8

JANUARY 1995 NOTICES OF THE AMS 9

leader in our community, as human rights ad-vocate, teacher, and mentor.

No person who was close to him can write anarticle about someone named Lipman Bers (pro-nounced bears). To colleagues he was Lipa or, in-evitably, Papa. Within his family, Lipa or Flipwere common. Lipa was a gentleman in the tra-ditional European stylehe usually wore a tie.Formality, even in the form of a tie, was for-saken in deference either to hot weather or in-tense activity. When he first arrived in the UnitedStates, he had to learn that we do not shake thehand of a colleague each time we pass in thehalland we even address each other, indepen-dent of rank, by first names. Linda Keen relatesthe story of the time when she received her doc-torate. Lipa made clear to her she was free to con-tinue to use an honorific title in conversation withhim (to her it was a sign of respect), but he wouldrespond in kind. Henceforth he would refer toher using every title she had ever earned.1 Lipapossessed a grace and charm that convincedrather than bludgeoned. His linguistic skillshewas fluent in at least four languagesand hissense of humor enhanced his ability to presenthis ideas.

His humor was often self-effacing; it was de-livered with the confidence that captured the at-tention and affection of his audience. He said,when asked to give an autobiographical lecture,

[Dennis] Sullivan asked me what arethe thoughts underlying my research.This is already a compliment, becauseit assumes that there is a thoughtunderlying the research.

The Early YearsLipa was born into a family that emphasizedlearning and insight. His mother was principalof the secular Yiddish language elementaryschool in Riga; his father was the principal of thecorresponding gymnasium. His mother later be-came a psychoanalyst. His earliest years werespent in Petrograd amidst the turmoil of theRussian revolution, yet he recalled those yearsas peaceful, a luxury open only to the very young.His youth was spent in Riga and Berlin, wherehis mother trained at the Berlin PsychoanalyticInstitute. In the chaos that was Europe betweenthe world wars, Bers learned the love of mathe-matics and ideas and began his life-long love af-fair with Mary Kagan Bers.

He studied for a short time at the Universityof Zurich, then returned to Riga and its univer-sity. There he led the life of a politically involved

social democrat in a Europe self-destructing dueto the extremism and violent tactics both of theBolsheviks and a collection of right-wing tyrantsand killers.

In 1934 Latvia contributed its own local dic-tator to the general atmosphere. Lipa wrote foran underground newspaper and spoke publiclyagainst the new regime. Riga soon became toodangerous a place to remainhe left as a war-rant for his arrest was issuedand he fled toPrague, where he continued his studies at CharlesUniversity under the direction of Karl Lwner.Mary joined him in Prague and they married.

Lipa spoke of feeling neglected, perhaps evennot encouraged, by Lwner and said that only inretrospect did he understand Lwners teachingmethod. He gave to each of his students theamount of support neededLipa used the sametechnique with the nearly fifty students he ad-vised and the multitude of others who, like me,fell under his spell. It is obvious that Lipa did notappear too needy to Lwner. In 1938 he receivedthe degree of Doctor of Natural Sciences. His doc-toral dissertation was never published; indeedLwner urged him to submit my thesis withouttrying to add more results or it will be too late.The thesis concerned potential theory.

The year 1938 was cataclysmic for all of Eu-rope but especially for Czechoslovakia. TheFrench and British gave the country to the Nazisin the forlorn hope of achieving peace in ourtimes. Among the most endangered people inNazi Europe were the stateless, the leftists, andthe Jews. Bers satisfied all of these conditions.Lipa and Mary crossed a Europe fraught with dan-ger and lived in Paris until the Nazi occupation.While waiting for an American visa,2 and livingunder the stressful condition of being a state-less Jew in France at the start of World War II,Bers demonstrated his devotion to mathematicsby continuing his work. He wrote two short noteson Greens functions and integral representa-tions, ideas to which he later returned in the con-text of Teichmller theory and Kleinian groups.

1 Im exaggerating here, ever so slightly, to make apoint. I learned that from Lipa.

2 Functionaries in the American State Department,whether simply afraid of a flood of Europeans of fargreater intellect than they possessed or a simple lackof humanitarian concern, created artificial obstacles tothe granting of American visas to people fleeing HitlersNew World Order. This additional level of obstructionadded to a common American xenophobic reaction toa flood of European (especially Jewish) refugees.

The official reaction is in stark contrast to the wel-come offered by the American mathematical commu-nity. Bers pointed out that it was only in countries likethe United States, already possessing a strong mathe-matical environment, that European refugee mathe-maticians could be appreciated. The refugees foundassistance and a collegiality that was graciously of-fered and gratefully received.



Those representations were then in vogue dueto Stefan Bergmans work on kernel functions.

While the Berses lived in Paris, their daugh-ter Ruth, now a psychoanalyst in New York Cityand professor of psychology at John Jay Collegeof the City University of New York, was born.

American Mathematics Goes to WarIn 1940, ten days before Paris fell to the Nazis,the Berses moved into unoccupied France. Theyfinally obtained American visas10,000 wereissued for political refugees after the personalintervention of Eleanor Rooseveltand the fam-ily sailed to New York. Lipas mother and step-father, Beno Tumarinlater an actor, theatricaldirector, and teacher at the Juilliard Schoolwerealready in New York. Lipa, like so many of theyoung mathematicians of our time, sought em-ployment in his specialty only to find that no po-sitions were available. The Berses lived in the cir-cle of unemployed refugees in New York until1942; he received some support from YIVO, aYiddish research organization; the result was apaper in Yiddish about Yiddish mathematicstextbooks.

It was roughly in this period that he startedjoint work with Abe Gelbart on -monogenicfunctions, which later developed into the theoryof pseudoanalytic functions.

In 1942 the Berses moved to Providence,where their son Victor, now professor of classicsat Yale, was born. Bers had accepted the invita-tion to participate in the Brown University pro-gram Advanced Research and Instruction in Ap-plied Mathematics. Here he was soon joined byLwner who, in recognition of the haven offeredby the United States, had anglicized his name andthenceforth was known as Charles Loewner.

Lipa wrote most eloquently of Loewner, inwords that describe their author as well: He

was a man whom everybody liked,perhaps because he was a man atpeace with himself. He conducted alife-long passionate love affair withmathematics, but was neither com-petitive nor vain. His kindness andgenerosity in scientific matters, tostudents and colleagues alike, wereproverbial. He seemed to be inca-pable of malice. Without being re-ligious he strongly felt his Jewishidentity. Without having any illu-sions about Soviet Russia he was aman of the left. He was a good sto-ryteller, with a sense of humor whichwas at once Jewish and [humanistic].

It was at Brown that Bers had the first threeof his forty-eight doctoral students.

The American universities had become boththe training grounds for the war effort and thecenters for war-related scientific research. It wasthere that Bers started his studies of two-di-mensional subsonic fluid flow. This was partic-ularly relevant in the late war years as aircraftwings (airfoils) were being designed with sharpedges, jet engines were introduced, and flightspeeds nearing that of sound became conceiv-able. The potential function for the planar flowshared many of the properties of harmonic func-tions, which are the potentials of an incom-pressible flow. The techniques used by Bers areparticular to the plane. With respect to suitablychosen coordinates the complexified potentialis an analytic function. The choice of such co-ordinates, in the form of isothermal coordinates,is a subject to which we will later return in termsof solving the Beltrami equationit is central toBerss major work.

Pseudoanalytic Functions and Subsonic FlowBerss work on pseudoanalytic functionsa sub-ject developed independently by I. Vekua in theSoviet Union at a time when East-West scientificcommunication was often considered treason-ablewas one of the generalized complex func-tion theories hinted at in earlier eras by Bel-trami, Picard, and Carleman. It was brought tofruition only much later. Quasiconformal map-pings, of which we will have more to say, gen-eralized conformal maps, while pseudoanalyticfunctions were based on generalized Cauchy-Riemann equations. We would like all complex-valued functions in plane domains to have com-plex derivatives, but, of course, that is impossible.Pseudoanalytic functions come as close as pos-sible to having complex derivatives. Experts willrecognize them, after a bit of thought, as non-singular quasiregular functions.

In some domain D in C, choose two Hldercontinuous functions F;G : D ! C with=(FG) > 0. The class AD(F;G) of pseudoana-lytic functions with generators F and G consistsof the functions

w = F + G

with the properties that and are real-valuedand the dot derivative

w (z) := limh!0

[(z + h)(z)]F (z + h)+ [ (z + h) (z)]G(z + h)

.h

exists and is finite for all z 2 D . Notice that the(1; i)-pseudoanalytic class defines the class ofcomplex analytic functions.



Some basic properties of pseudoanalytic func-tions are:

the notion of pseudoanalytic function is con-formally invariant;

with respect to the dot derivative, the func-tions are infinitely differentiable and there isa Cauchy theory; and

the zeroes of functions in the class are iso-lated and each such function admits uniquecontinuation.On closed subsets of D, w can be written as

w = es(z)f (z) where f is analytic and s is Hldercontinuous. Bers and Nirenberg later showedthat this factorization of w remains valid whenw is a solution to an elliptic system having onlymeasurable coefficients.

To the (F;G)-pseudoanalytic function w wemay associate ! := + i which is pseudoana-lytic with respect to F 1 and G i. The func-tions ! have strong geometric properties, e.g.,they are open, orientation-preserving, and lo-cally quasiconformal.

Berss work in the period from the early 1940suntil the mid-1950s concerned an amalgam ofthis generalization of single variable complexanalysis, together with linear or mildly nonlin-ear elliptic partial differential equations in theplane and their applications. The early develop-ment of the theory of pseudoanalytic functionscoincided with Berss work on subsonic flow.

The system describing the steady-state planarflow of a compressible fluid is given by the ve-locity components u; v of the fluid which satisfythe system

uy vx = 0and

(c2 u2)ux uv(uy + vx) + (c2 v2)vy = 0where c is the speed of sound in the fluid. Usingcomponents of the velocity vector as indepen-dent variables, the equation can be linearized as

(c2 u2)vv + 2uvuv + (c2 v2)uu = 0:In 1945 Gelbart was instrumental in bringing

both Bers and Loewner to Syracuse University.Led by William Ted Martin and later StewartCairns, the department prospered for a time; itsfaculty included Paul Erds, Paul Halmos, ArthurMilgram, Dan Mostow, Murray Protter, PaulRosenbloom, Hans Samelson, and Atle Selberg.Bers left Syracuse in 1949.

It was during his stay at Syracuse that he firstworked on the problem of removability of sin-gularities of nonlinear elliptic equations. Ear-lier, Sergei Bernstein had proved that any solu-tion of the minimal surface equation

(1 +2y )xx 2xyxy + (1 +2x)yy = 0

which is defined in the whole complex planemust be linear. There are standard techniquesthat reduce the study of the minimal surfaceequation to the linear equation

(1 + 2)! + 2! + (1 + 2)! = 0:

The latter equation may be studied using themethods of pseudoanalytic functions. Bersproved that any finite isolated singularity of asingle-valued parametrized minimal surface isremovable.3 This profound extension of Rie-manns theorem on removability of singulari-ties of analytic functions shows that nonlinearelliptic equations can have significantly morerigid behavior than their linear counterparts.This removability theorem was both announcedand discussed by Bers at the International Con-gress of Mathematicians in 1950. The Annalspaper containing the proof is a magnificent syn-thesis of complex analytic techniques which re-late the different parametrizations of minimalsurfaces to the representations of the potentialfunction for subsonic flow and thereby achievesthe extension across the singularity.4 Bers, along

Lipman Bers at Columbia University in 1984.

3 I like to think of this theorem as the physically obvi-ous statement that a pinprick will burst a soap bubble.4 Shmuel Agmon, among others, has noted that Lipawas always doing complex analysisno matter thefield in which he was working.



with several others, later extended the theoremto a wide class of nonlinear elliptic equationsusing generalized complex analytic techniques.

The Transition to QuasiconformalMappings and Their ApplicationsBers then spent two years at the Institute for Ad-vanced Study. It was at this time that he begana ten-year odyssey that took him from pseudo-analytic functions and elliptic equations to qua-siconformal mappings, Teichmller theory, andKleinian groups. He said,5

I was looking for an a priori estimatefor an inequality which would saythat if a solution to a certain equationexists, then it is less than a fixednumber. I thought there might bean inequality involving quasiconfor-mal mappings which would help meto prove an existence theorem. Iwas trying to prove the existence ofa flow that no sane physicist doubtedexists. I found the inequality in apaper of Lavrentiev. [He then wroteon the blackboard] If z 7! f (z) [hethen turned to the audience and said]This arrow wasnt invented yetmaps the disk onto itself homeo-morphically and

jdilatation(f )j K

and f (0) = 0, then

jf (z1) f (z2)j 16jz1 z2j1=K:

The interesting thing is [that] thisis called Theorem (Ahlfors-Lavren-tiev). This was in a paper by Lavren-tiev and I looked in the literature butdidnt find the proof anywhere. I wasin Princeton at that time. Ahlforscame to Princeton and announced atalk on quasiconformal mappings.He spoke at the University so I wentthere. And sure enough, he provedthis theorem. So I came up to himafter his talk and asked him, Wheredid you publish it and he said, Ididnt yet. So why did Lavrentievcredit you with it? He [Ahlfors] said,He probably thought I must know itand was too lazy to look it up in theliterature. Well, as a matter of fact,three years later I met Lavrentiev in

Stockholm. I asked him, Why didyou credit Ahlfors with this theo-rem? Lavrentiev said, literally, Ithought he must know it and I wastoo lazy to search the literature. I im-mediately decided that, first of all, ifquasiconformal mappings lead tosuch powerful and beautiful resultsand, secondly, if it is done in thisgentlemanly spiritwhere you dontfight over prioritythis is somethingthat I should spend the rest of my lifestudying.

I read or tried to read the papers byTeichmller. At that time his repu-tation was not yet established; peo-ple doubted whether his main theo-rem was proved correctly. Ahlforspublished a new proof, but a verydifficult one, in the Journal dAnalyse.Some of you may know that Teich-mller was a Nazi.6

Teichmllers theorem(s) assert the existencesand uniqueness of the extremal quasiconformalmap between (for simplicitys sake) two compactRiemann surfaces, of the same genus, moduloan equivalence relation. The equivalence relationis given by quasiconformal maps fi from a basesurface to a member Si of a family of surfaces.S1 S2 if there is a map g : S1 ! S2 so thatg f1 f12 is homotopic to a conformal self-map of S2. The equivalence classes form the Teichmller space Tp of compact Riemann sur-faces of genus p. Teichmller also showed howto identify Tp with the unit ball in the space ofquadratic differentials on the base surface . TheTeichmller space Tp is the setting for the so-lution to Riemanns problem of moduli. It wasthe initial triumph of the Ahlfors-Bers collabo-ration to give a solution of the moduli problem.

The problem asks for a holomorphic param-etrization of the compact Riemann surfaces ina fixed homeomorphism class. The complexstructure of this space must be chosen so thatthe periods of abelian differentials of the first

5 In 1986, he gave an impromptu lecture on his workon quasiconformal mappings in Dennis Sullivans sem-inar. As was common there, the talk was videotaped.

6 It is still a source of controversy that Teichmller wasa mathematician of the highest order and a humanbeing of the lowest. Some ten years ago, I wrote a bi-ography of Teichmller and collected recollections ofhim. Werner Fenchel once told me that Teichmllerlacked the opportunism of a Bieberbach. His life offersa vivid demonstration that a person could simultane-ously ascend the heights of insights scientific and de-scend the depths of human experience as typified byblind obedience to the Nazi credo.



kind are holomorphic functions of the parame-ters. The idea can be stated far more simply. Rie-mann worked with algebraic equations like

P (z;w ) :=Xijaijziwj = 0:

If we assume that aij is a holomorphic func-tion of a parameter t 2 CN, then we get a holo-morphic family of Riemann surfaces St. If we fillin the points that sit over 1 and desingularize,each St becomes a compact Riemann surface.These compact surfaces are topologically equiv-alent to spheres with p 0 handles. Riemannsmoduli space Rp is the space of holomorphicequivalence classes [S] of Riemann surfaces offixed genus p. Riemann wanted to parametrizeRp so that maps of the form t 7! [St ] are holo-morphic. Ahlfors and Bers proved that Rp hassuch structure. The structure projects from theorbifold (or branched) covering by the Teich-mller space Tp of genus p. Ahlfors was first togive a complete description of the complex an-alytic structure; his work is a tour de force. Butthere is an easier way, given by Bers, which wewill soon describe at length.

In 1951 Bers moved to the Courant Institutebefore it had that nameof New York University,where he remained until 1964. At Courant, Berschaired the graduate program. His teaching andpersonal style were merged in the lunches thathe held each week. Linda Keen recalls that dur-ing her graduate years, Bers taught his classevery Friday after lunch. Before class, he wouldalways have lunch with his children; these in-cluded Bernie Maskit, Ronny Wells, Bob and Les-ley Sibner, and a number of others. He called itthe childrens lunch. Of course others oftencame, including students like Jerry Kazdan andJoan Dyer and NYUs young faculty, like Ehren-preis, Garabedian, Morawetz, Moser, Nirenberg,and Lax. I also remember Serge Lang coming.These lunches were both social and mathemat-ical, certainly initiating us into the mathemati-cal culture. Bernie Maskit adds, One of the rulesof the gathering was that the bill was alwayspresented to the youngest person presentthisusually meant Lindawho had to add the tip anddivide by the appropriate number. We onceturned the tables and insisted that Lipa do it; hegot flustered and didnt get the arithmetic right.We then went on to have some discussion aboutwhether mathematicians could do arithmetic(Lipa, of course, said that they couldnt; I, whohad kept the books in my familys businessfor a few years, did not add that I could doarithmetic).

After the move to the New York City area, theBerses lived in the enclave of European-bornmathematicians in New Rochelle, New York, that

had formed around Richard Courant. Membersof that community included, at various times,Fritz John, Kurt Friedrichs, Cathleen Morawetz,Harold Grad, Jrgen Moser, Wilhelm Magnus,and Joan Birman. Lipa and Mary created an en-vironment of warmth and excitement in theirhouse on Hunter Avenue. Mathematicians pass-ing through New York, dissidents just expelledfrom the Soviet Union, friends, and studentsfound a home with the Berses. This fosteredboth interest in mathematicsthe excitementshared between generationsand in humanrightswhich was hard to ignore after sharingcoffee and ideas with incredibly brave people likeChalidze and Litvinov. It was also the place webrought all aspects of our lives; the Berses sharedour joys and our sorrows.

One of Lipas great skills was to coin phrases.In a filmed lecture, Marc Kac told of describingto Lipa his work on the eigenvalues of the Lapla-cian for plane domains. Lipa said that he was ask-ing whether one can hear the shape of a drum.The answer is no. The proof was given by Gor-don, Webb, and WolpertScott Wolpert is Berssmathematical grandson.

By 1954 quasiconformal mappings had begunappearing in Berss pa-pers, and his emphasismoved into new areas.He signalled his depar-ture from a field bywriting up what heknew in the form of abook or survey article.7

Between 1954 and 1958there appeared booksor major surveys onsub- and trans-sonicflows, pseudoanalyticfunctions, and influen-tial sets of lecture noteson algebraic topology and Riemann surfaces.The notes on topology served as an introductionto topological methods for a generation of hardanalysts. In particular, it was one of the earliesttreatments of fixed point methods and topo-logical obstructions which was accessible to theworking analyst. The Riemann surfaces volumedescribes the state of the field just before theAhlfors-Bers Theorem revolutionized the field.

7 He felt that one should write a book in order to leavea field. After going through the process of writing a bookand preparing it for publication, he asserted that itwas easy to leave because by then one had come to hatethe field. I should note that the planned book on Te-ichmller space, which was to be a joint undertakingwith Fred Gardiner and Irwin Kra, was never written,although several drafts were produced. I presume thatBers was never quite ready to leave the field.

...a lifelong,passionatelove affair

withmathematics



A later set of lectures on several complex vari-ables was also quite influential.

Moduli Theory and Kleinian GroupsIt was during this period of extensive expositorywriting that Berss research blossomed as well.In his 1958 address to the International Congressof Mathematicians, Bers announced a new proofof the so-called measurable Riemann mappingtheorem. He then essentially listed the theoremsthat followed directly from this methodin-cluding the solution to Riemanns problem ofmoduli. He was outlining the work of severalyears, much of which was either joint with or par-alleled by work of Lars Ahlfors. Their profes-sional collaboration was spiritually very close;they were in constant contact although theywrote only one paper together, the paper on themeasurable Riemann mapping theorem.8

Nonetheless, their joint efforts, usually inde-pendent and often simultaneous, and their per-sonal generosity inspired a cameraderie in thevaguely defined group which formed aroundthem. The group has often been referred to asthe Ahlfors-Bers family or Bers Mafia; c.f.the recent article by Kra [O-1]. It is gratifying tosee that the spirit of cooperation they fosteredlives on among many members of several gen-erations of mathematicians.

First let us set some notation. With @ and ( )denoting partial derivative with respect to ,define the partial differential operators

@z := (1=2)(@x i@y ) and @z = (1=2)(@x + i@y ):The Ahlfors-Bers theorem asserts the follow-

ing: Let : C! C be measurable withj(z)j < k < 1 almost everywhere. Then the Bel-trami equation

(1) wz = wz

has a unique solution w fixing 0, 1 and 1which is a homeomorphism of C onto itself.Given any point z0, the map 7! w(z0) is aholomorphic mapping from the unit ball in L1(C)to C. The most important consequence of theAhlfors-Bers approach is that, if depends ona parameter t either to some degree of smooth-ness or real or complex analytically, then w de-pends on t as well as it possibly can.

A solution w to (1) is conformal with respectto the Riemannian metric

ds2 := (z)jdz + (z)dzj2 with > 0and gives an isothermal local coordinate for thatmetric.

In his International Congress paper, Bersasked whether there is an embedding (holo-morphic, of course) of the Teichmller space asa bounded domain in CN . In a paper entitledSpaces of Riemann Surfaces as Bounded Do-mains, Bers gave an argument which shouldhave proved the theorem. The argument wasnot correct. In fact, Bers referred to the correc-tion as his most important paperit is one andone-half pages long, including the references. Inthe latter work, he described the object nowcalled the Bers embedding of the Teichmllerspace.9 The embedding brings the idea of mod-uli (parameters) for Riemann surfaces back totheir roots in the work of Gauss, Riemann,Schwarz, and Poincar on parametrizations of so-lutions of ordinary differential equations andtheir monodromy groups. Bers embedded theTeichmller space of a (say, compact, hyper-bolic) Riemann surface S in a space of Schwarz-ian derivatives [f ] of univalent functions f. [f ]is a quadratic differential for a Fuchsian groupG0covering S and f conjugates G0 into a quasi-Fuchsian group Gf, that is, a group of complexmbius transformations quasiconformally con-jugate to G0. This gives an elegant geometric for-mulation to deformations of S in terms of hy-perbolic 3-geometry while holomorphicallyembedding the Teichmller space as a cell in thecomplex vector space of holomorphic quadraticdifferentials.

The connection between complex mbiustransformations and hyperbolic 3-dimensionalgeometry had already been discovered by Poin-car. The action of any mbius transformationon the extended plane can be represented asthe composition of two or three reflections in cir-cles or lines C. Poincar extended the action tothe upper halfspace by using reflection in thespheres or planes S meeting the complex planeorthogonally. These reflections on the upperhalfspace give the full group of isometries of hy-perbolic space.

Using the Ahlfors-Bers Theorem, Bers gavewhat remains the most intuitive proof of Teich-mllers existence theorem. Teichmllers proofof his uniqueness theorem remains the stan-dard proof.

8 They called it the Riemann mapping theorem for vari-able metrics. The theorem should properly be called ei-ther that or the Ahlfors-Bers theorem or the measur-able Riemann mapping theorem with dependence onparameters. Without the dependence on parametersthe theorem is due to Morrey. It is precisely the depen-dence on parameters that displays the complex analyticstructure of the Teichmller space.

9 In the correction, Bers states that the same result wasobtained, simultaneously and independently, by Ahlfors.So Ahlfors must have named the embedding.



By 1964-65 both Ahlfors and Bers hadchanged research direction. Also in 1964, Bersmoved uptown from the Courant Institute toColumbia University. He remained there untilhis retirement in 1984 afterhaving served as chairmanfrom 1972 to 1975.

Berss embedding of theTeichmller space gives it aboundary, which was firstdescribed by Bers and Maskitin separate, but closely co-ordinated, papers.10 Theboundary points correspondto degenerate Riemann sur-faces and also to a class ofKleinian groups whose exis-tence had not previouslybeen suspected. At the timethey were called totally de-generate groups; now, fol-lowing further explorationinitiated by Troels Jrgensenand Bill Thurston, they arecalled singly degenerategroups.

While Bers was exploring the boundary of Te-ichmller space, Ahlfors had gone off in a relateddirection and had proved his finiteness theo-rem for Kleinian groups.11 In filling a minorgap in Ahlforss argument, Bers gave numericbounds for the hyperbolic areas of the Riemannsurfaces under discussion.12 Within the span often years, Ahlfors and Bers had led a major por-tion of the complex analytic community from thesolution to Riemanns moduli problem, hence aclose interaction with the algebraic geometrycommunity, to the study of groups of motionsof hyperbolic 3-space.

In his earliest papers, Bers had examined in-tegral representation theorems. In the 1960s hesimply wrote down an operator which general-ized the reflection operation (z) 7! (z) to re-flection across quasicircles. Convergence is onlyguaranteed if is a holomorphic q-differentialwith q 2. Since the operator is antianalytic, itgives global holomorphic sections of the bundleof q-differentials. He also introduced a fiberspace whose natural quotient space attaches, in

a holomorphic fashion, to a point S in the Te-ichmller space, the surface S. Topologically, thequotient space is Tp S, but using the Bers em-bedding we actually obtain a holomorphic bun-

dle structure.Yet another major

stream of Berss workconcerned Eichler coho-mology. First studied byEichler in number-theo-retic investigations, thecohomology classeswere later used by AndrWeil to parametrize theinfinitesimal deforma-tions of discrete sub-groups of Lie groups.Soon thereafter, Bersused the Eichler theory,in the context of Fuch-sian groups, to prove thefiniteness theorem thatAhlfors refers to as themodel for his finitenesstheorem for Kleinian

groups. This represents yet another view of howquadratic differentials parametrize deforma-tions. Berss Area Theorem is still another useof this cohomology theory. Later work of Ahlfors,Kra, and Bers uses the theory to get a corre-sponding deRham theory, to find spanning setsfor automorphic forms represented as Poincarseries, and later to find bases for these sets andto determine when the Poincar series of a ra-tional function vanishes identically.13

In the late 1970s and beyond, hyperbolic 2-and 3-dimensional geometry became a subjectof intense interest to topologists and dynamicalsystems people. The revolutionary insights of BillThurston lead to a classification of surface dif-feomorphisms in a fashion that complementedthe one previously obtained by Nielsen, butThurstons has a structure and unity all its own.Bers gave a proof of Thurstons classificationusing quadratic differentials and Teichmllertheory. In a sense, Berss proof is easier becausethe flexibility obtained by working with classesof diffeomorphisms is replaced by the rigidityof working with a unique element, namely theTeichmller map, in each diffeotopy class. Berssproof is simply the classification of diffeomor-phisms of surfaces via the existence or not of thesolution to a new extremal problem on Teich-mller space. The same philosophy of using therigid map in a class of homeomorphisms un-derlies the Bers-Royden proof of the Sullivan-

10 Of course, in the spirit of the Bers-Ahlfors school, itwas Ahlfors whose 1962 International Congress addressasked what can be said about the boundary.11 Ahlfors tells us that the proof is an extension of aproof, due to Bers, of the finiteness theorem for Fuch-sian groups.12 At the same time, Leon Greenberg used a com-pletely different and likewise powerful method tofill the same gap.

13 Hejhal had previously solved this last problem in amore restricted case.

Lipman and wife Mary Bers in Hawaii,1979, attending the U.S./Japanconference on Riemann surfaces.



Thurston improvement of the -lemma of Ma,Sad, and Sullivan, as well as Berss proof of Sul-livans unified approach to both Ahlforss finite-ness theorem and Sullivans eventual periodic-ity theorem (wandering domains cannot existfor iterated rational maps of the Riemannsphere).14 Berss paper on the Sullivan unifica-tion was proved after he had moved to the (NewYork) City University Graduate Center follow-ing his retirement from Columbia in 1984. He re-mained at CUNY as visiting professor until 1989.

Professional and Personal RecognitionBers received many honors throughout his ca-reer. Here it is particularly appropriate to men-tion that he served as vice-president of the AMSfrom 1963 to 1965 and as president from 1975to 1977. In 1975 he received the Steele Prizefrom the Society. He was elected to the NationalAcademy of Sciences in 1964 and served as chairof the Mathematics Section from 1967-70.15 Inone of the other articles in this issue of the No-tices, you will find an exposition of Berss workas the inaugural chairperson of the Academyshuman rights committee starting in 1979. In1961 he became a Fellow of the American Acad-emy of Arts and Sciences. He was elected Fellowof the American Association for the Advance-

ment of Sciencesin 1965 andchaired its SectionA in 1973 and in1983-84. He was amember of theCouncil of the As-sociation from1969-73 and from1984-85. He wasparticularly proudto have beenelected to theAmerican Philo-sophical Society,because it is theoldest learned so-ciety in the UnitedStates, foundedby Ben Franklin,whom he particu-larly admired. Hewas the first G. H.

Hardy lecturer for the London Mathematical So-ciety in 1967.

When New York City began giving awards forScience and Technology, Bers was one of thegroup of five people first chosen. The citationsaid that he was chosen for his influential andcreative contribution to modern mathematics, hisinspiring guidance to generations of students,and his tireless campaign in support of thehuman rights of persecuted scientists though-out the world.

Lipa, ever the optimist, did not possess thepower of prophecy. During the years 1966through 1968, he chaired COSRIMS, the Com-mittee on Support of Research in the Mathe-matical Sciences, which was a joint endeavor ofthe National Academy of Sciences and the Na-tional Research Council. He took responsibilityfor the conclusions about the future of the pro-fession which appear in the final report of thatcommittee. It concluded that the future lookedincredibly bright for employment in the mathe-matical sciences. Two years later the job marketcrashed, leaving a much chagrined Lipman Bers.

During the period from 1969 to 1971, hechaired the Division of Mathematical Sciencesof the National Research Council. That was aninteresting time for Lipa, an antiwar activist, tobe an official of a quasigovernmental agency.Lipa gave heartfelt, impassioned speeches at Co-lumbia against American military involvementin Southeast Asia. The FBI started asking ques-tions about himthey said he was being con-sidered for a high government position. Hechecked and none was open. He was proud.Later, Lipa felt deflated when his name did not

Lipman and Ruth Bers.

14 The editor, on reading Berss approach to these the-orems, recalled Eli Steins approach to finding a nee-dle in a haystack. Eli advised us to pick a smallhaystack.15 He once commented in jest that the Academy hadtwo reasons to exist: to make jealous those who did notbelong and to bore those who did.



appear on Nixons enemies list, an honor role,as it were, of American dissent. It was not thatdissent, in the American political landscape,was his pose but rather that America had givenhim so much that his expectations were loftierthan that of the leadership demonstrated byNixon and his coterie.

A Personal NoteI worked with Lipa for over twenty years, but Iwas not his student. I was welcomed to his houseby both Lipa and Mary and shared in the warmthof that meeting ground with generations of math-ematiciansperhaps two senior to me, my own,and a generation junior to me only in age. I knowthe mathematics that we shared, the joy, and theenthusiasm that so many people shared withhim.

Lipa Bers was a man who had a tremendousinfluence on me. From the time I was a gradu-ate student, trying to decide what would be agood thesis topic, to the time that, as a (hope-fully) less immature member of the community,I was choosing a name for a series of books, Ilooked to Lipa for counsel, for advice, for a senseof who I am as a mathematician and, more im-portant, of who I would like to be both as a manand as a mathematician.

I first met Lipa at a conference at Stony Brookin 1969. The first lecture was given by LarsAhlfors. The graduate student in me saw that thetheorems that I thought were the end result ofmathematics were simply the tools when placedin the right hands. I learned from Lars Ahlforsthat mathematics grows on itself. But I learnedsomething else from Lipa. He gave a simple sem-inar talk, and there I learned about his embed-ding of Teichmller space and its boundary. Helectured in typical fashion about quasi-Fuchsiangroups; he drew a wiggling simple arc on theblackboard. That was the limit set of a quasi-Fuchsian groupPicasso would have been proud.Almost nothing else was written on the board.He spoke to the audience and created a visionof a set and made us feel that we not only un-derstood it, but that we were seeing it. In real-ity it is a horribly complicated fractal.

I felt that I was a child and he was an oldermanhe always had the image of an older, wisermanwho was taking me through a garden andshowing me the flowers. The flowers were thetheorems, the beauty. I did not know the field,did not even know the definitions; I could see,only through Berss eyes, the beauty of his ideas.When I left the room, I knew I needed to betterunderstand that garden. That was twenty-fiveyears ago, and I am still finding delight in thefreshness of this most classical of mathemati-cal disciplines.

In a way we were children, and Lipa and MaryBers taught us, by their example, to be thinkingand caring people.

A mutual friend, Bill Harvey, first introducedus. I knew of Bers, both because of his lecturenotes and because his sons father-in-law waspresident of (as it is now called) the PolytechnicUniversity, where I was a student. I used a Yid-dish word, and he corrected me by pointing outthe one I should have used. We did not meet againfor the better part of a year. At Bills urgingac-tually he twisted my arm, and my wife pushedme out the doorI went to show Lipa some ex-amples of Kleinian groups; I was a nervous wreck.After all, why should a really important and re-spected mathematician be interested in the sillyexamples of a graduate student from an engi-neering school? Along with other students, fac-ulty, visitors, and whoever else wanted to seehim, I stood on line outside his office. When I fi-nally entered, he was on the telephone but bademe start to talk. At a certain point in my dis-course, he smiled and said into the phone, Letme call you back. At that moment I knew I hadbecome a mathematician.16

Lipas highest compliment to the discovererof a new idea was the comment,You sly dog.He attributed to others the observations that,Mathematics is a collection of bad jokes andcheap tricks and that Mathematicians workfor the grudging admiration of a few closefriendsbut it was acceptable to steal jokes.17

Lipa was a master storyteller and jokester, but,for all his joking, he knew the good fortune thatwas his. In the year of the centenary celebrationof the founding of the American MathematicalSociety, he gave a moving talk in which hethanked the collective body of American math-ematicians, both for himself and for his gener-ation of European immigrants, for saving hislife and for providing him the opportunity to con-tinue to work at the mathematics he loved.

Lipa loved the beauty and clarity of mathe-matics yet, as an old man, he remarked,

In mathematics you have completefreedom with a complete lack ofarbitrariness. Certainly the con-cept of what is beautiful has changed.And in general when an older man

16 It was only in preparing this article that I becameaware of Berss comments on his becoming a mathe-matician. He said,When I went to Lwner to ask himfor a thesis topic, I expected him to grab me by the neckand say What makes you think that you can write athesis on mathematics? OUT!!!17 Lesser artists borrow, great artists steal can befound in the LATEX manual . I t i s attr ibuted toStravinsky.



said something was not beautiful, hewas wrong.

As a committed social democrat, he was some-what self-conscious of having devoted his life tothe mathematics from which he derived so muchpleasure. He said,

Here I am, a grownup man, worryingabout whether the limit set of a Klein-ian group has positive measure andwilling to invest a great deal of effortto find the answer.18

In his power as a mathematician, his dignity,his enthusiasm, and his caring for others, he seta standard for the people who knew him. In thecircle of people who surrounded Lipa and Mary,devotion to mathematics and people, respectfor one anothers work, public service, and po-litical activism were neither demanded nor ex-pectedthey were just done. The personal mod-els set by Lipa and Lars Ahlfors placed nodemands on us. We were simply offered a per-sonal ideal, a personal standard, a goal which wecould not expect to attain but found pleasure intrying.

Lipa possessed a joy of life and an optimismthat is difficult to find at this time and that issorely missed. Those of us who experienced itdirectly have felt an obligation to pass it on.That, in addition to the beauty of his own work,is Lipas enduring gift to us.

Bibliography:19

There is original autobiographic material. The follow-ing are videotaped lectures:

(V-1) The European mathematicians migration toAmerica, AMS-MAA Joint Lecture series, Amer. Math.Soc., Providence, RI, 1988.

(V-2) My life with QC mappings, unpublished videoof an impromptu lecture in Dennis Sullivanss semi-nar.

The following are printed interviews:(P-1) D. J. Albers, G. L. Alexanderson, and C. Reid

(eds.), More mathematical people, Harcourt Brace Jo-vanovich, Orlando, 1990, pp. 321.

(P-2) Allen L. Hammond, MathematicsOur invisi-ble culture in Mathematics TodayTwelve informalessays, Springer-Verlag, 1978, pp. 1534.

Berss autobiographical writings:(A-1) An as-yet unpublished autobiography of his

early years.

(A-2) Charles Loewner, Collected papers (L. Bers,ed.), Birkhuser, Boston, 1988 (see the introduction).

Articles about Bers:(O-1) I. Kra, The Ahlfors-Bers family (to appear).(O-2) L. Keen, Lipman Bers (19141993), AWM

Newsletter 24 (1994), 57.His selected works, which are being edited by Kra

and Maskit, will contain lists of his papers and all hisstudents.

Related commentary may be found in Ahlforsscomments on his work with and paralleling Berss in(A-1). L. V. Ahlfors, Collected works, Birkhuser, Boston,1982.

Major surveys and books by Bers.(W-1) An outline of the the theory of pseudoanalytic

functions, Bull. Amer. Math. Soc. 62 (1956), 291331.(W-2) Uniformization, moduli and Kleinian groups,

Bull. London Math. Soc. 4 (1972), 257300.(W-3) What is a Kleinian group, Crash Course on

Kleinian Groups, Lecture Notes in Mathematics, vol.400, Springer-Verlag, 1974, 114.

(W-4) Finite dimensional Teichmller spaces andgeneralizations, Bull. Amer. Math. Soc. (N.S.) 5 (1981),131172.

(W-5) Mathematical aspects of subsonic and transonicgas dynamics, Wiley, New York, 1958.

(W-6) (with F. John and M. Schechter), Partial dif-ferential equations, Lectures in Applied Mathematics,vol. III, Interscience, New York, 1964.

(W-710) Courant Institute lecture notes on pseudo-analytic functions (1952), topology (1957), Riemannsurfaces (1958), several complex variables (1964).

On the SocialActivism of LipmanBersCarol Corillon and Irwin Kra

Throughout his life, Lipman Bers created excit-ing mathematics and courageously strove to fur-ther his social and political ideals. Bers was amathematician who charted new research paths;he was a humanitarian who set new standardsin social activism and responsibility. He stimu-lated and challenged generations of mathe-maticians; he cajoled, coaxed, inspired, and oc-

18 This is Ahlforss measure problem, and, for generalfinitely generated groups, it remains open to this day.19 The bibliographic references given here are only aguide to Lipman Berss autobiographical and bio-graphical material, together with sufficient guides tohis mathematical work and exposition, to permit fur-ther study.

Carol Corillon is the director of the Committee onHuman Rights of the National Academy of Sciences, Na-tional Academy of Engineering, and Institute of Medi-cine in Washington, D.C. Her e-mail address [email protected].

Irwin Kra is professor of mathematics and dean of theDivision of Physical Sciences and Mathematics at theState University of New York at Stony Brook. His e-mailaddress is [email protected].



casionally browbeat colleagues and associatesinto joining together in the cause of humanrights. And in that cause, as in all the other as-pects of their life together, his wife Mary was athis side. With Erna and Lars Ahlfors, Lipa andMary created an international extended family,a mishpoche1 bound by common mathematicaland humanitarian principles. Because of theforce of their personalities, Mary and Lipa Bersoccupy a very special place in the hearts of themembers of that family.

One may wonder whether Berss social ac-tivism was an outgrowth of his enthusiasm formathematical research and teaching or whetherhis mathematical achievements were a spring-board to a career as a social activist.2 He wasprobably born an activista social consciencewas in his genesand he developed a taste formathematics. Lipman attributed his talents andpopularity as a teacher to his youthful experi-ences as a soapbox orator; he credited his re-search accomplishments to his interactions withbrilliant colleagues. While enjoying an active andwarm family life, Lipman devoted himself fulltime to politics and to mathematics, and he ex-celled in each of these seemingly disparatehuman endeavors.

Bers devoted the greater part of his activescientific career to the study of moduli of Rie-mann surfaces. In an irony not lost on Lipa, oneof the most outstanding contributors to thisfield was the well-known function theorist, Os-wald Teichmller, who was also an ardent Nazi.In one of Lipas earliest papers on the subject,he quoted Plutarch to describe his strong feel-ings: It does not of necessity follow that, ifthe work delights you with its grace, the onewho wrought it is worthy of your esteem.

Two major themes, at times interrelated, dom-inated Berss social agenda: the need for struc-tured reform of the AMS3 and the constant needto promote and protect human rights world-wide. The example set by Bers in his profes-sional behavior was, in effect, an integral part of

this agenda. Lipman was an activist long beforehe became part of the American (United States)mathematics establishment. Even his choice ofa place (Prague)4 to pursue graduate study hada political component: the need to escape fromthose who put him on a wanted list in his na-tive Riga, Latvia. His education and political ac-tivities in Europe presaged the main themes ofhis career in the United States: as a superb re-searcher/scholar/teacher, social reformer, andhuman rights advocate. His service to the AMSon the Council (19571959), on its ExecutiveCommittee (19601961 and 19741977), as oneof its vice-presidents (19631965), and finally asits president (19751976) came at critical timesfor the organization.

Bers was active in AMS affairs during the1960s and was president of the AMS during the1970s, a crucial time in American and AMS his-tory. The unrest and social activism of that pe-riod strongly affected the lives and careers ofAMS members. Nevertheless, throughout thatdecade the Society continued to be run alongmore or less traditional, if not authoritarian,lines. Council meetings, for example, were notopen to the general membership, and there waslittle regard for the wishes of the great majorityof mathematicians.5

Mathematical research is, almost by definition,an elitist activity. The AMS has two faces: it isboth an elitist organization serving the highestdemands of what most of us, perhapsparochially, consider the queen of the sciences,

1 Family in Yiddish, a language dear to Lipa. Fluentand eloquent in English, Russian, and German, amongothers, Lipman had a special love for this language.When he and Mary arrived in the United States in 1940,Lipas familiarity with Yiddish and its rich literatureserved him well and facilitated his entry into American-Yiddish cultural life. Among Lipas large number ofmathematics papers is one in Yiddish on Yiddish arith-metic books.2 One reader of a preliminary version of these articlesremarked, After reading Abikoffs article, it is clear thatBers was a political activist whose taste for mathe-matics developed after his escape from Latvia. This re-mark is indeed correct if mathematical life starts withgraduate education.

3 Meant in the broadest sense. Bers was not concernedonly with the reorganization of the Council and im-proving its committee structure, although he delightedin such activities, he was interested in broadening andhumanizing the scope of the AMS mandate and indeedthe outlook of the entire mathematical community.4 Prague because it was in a democratic country, be-cause they let him in, because he had an aunt there(hence he could manage without workinga conditionof entry for most students to most countries), and per-haps because Charles Loewner was there.5 H. Hasse was invited by the Mathematical Associationof America (MAA) to give a lecture during the August1963 Boulder meeting in a session on the life and workof E. Artin. Bers, M. Kac, and later E. Moise, among oth-ers, denounced the invitation to a former Nazi partymember to speak at a session honoring a mathemati-cian who left Germany because of Nazi policies and triedto get the AMS to react appropriately to this invitation.The foremost aim of most, though not all, officers of theAMS and MAA was to sweep the whole affair under therug. Proposed letters to the Notices by Bers and Moisewere eventually withdrawn, and how to describe the dis-cussion of the issue in the Council minutes was the sub-ject of much negotiation. In the end, Hasse spoke, andthe general mathematical community knew nothingabout the controversy.



and a professional society serving the interestsof its members. There was in the past, and therestill is today, an ongoing debate within the AMSbetween those who would narrow and those whowould broaden the scope or definition of the So-cietys mandate. Lipman Bers was among thosewho succeeded in broadening the Societys hori-

zons to include social and political concerns.Before the Bers presidency, and because of in-ertia for some period after it, most of those gov-erning the AMS defined the scope of the Societysactivities very narrowly. At the same time, agroup of radical mathematicians was pushing theAMS towards more openness, more involvementin a social agenda, and reform. Bers understoodthese tensions, and because of his mathemati-cal stature and political past, he was able tocommunicate with all sides. He was charming;he was political.

Lipmans style imposed civility. He was prob-ably the person most responsible for saving theAMS from a possibly nasty civil war. He respectedallies and opponents, and he took both sides se-riously. As president of the Society, he once re-sponded to strong accusations of inaction andimplied threats by a young radical with, Relax,xxx, we won.

Lipman was proud of his role in creating theCommittee on Committees (in part because ofironic delight in its bureaucratic name) to advisethe AMS president on committee assignments,

and he was the key player in the founding of theCommittee on the Human Rights of Mathemati-cians (CHRM). In fact, the Council of the AMS au-thorized the establishment of CHRM in 1976only after President Bers agreed to prepare acharge for CHRM for the Councils consideration.Politician Bers decided that the initial charge for

this new line of activities forthe AMS should be narrow(restricted to the activities offoreign mathematicians, forexample) since optimist Berswas certain that its scopewould surely expand with itssuccesses (as indeed it did).

In the mid-1970s Lipmanand several other membersof the National Academy ofSciences (NAS) initiated amove to create a humanrights committee at the Acad-emy. It was to be a commit-tee that would use the pres-tige of the institution and itsmembers to pressure gov-ernments to resolve the casesof scientific colleagues whohad disappeared or who hadbeen imprisoned or threat-ened. By 1976 the commit-tee was functioning and hadalready publicly denouncedthe treatment of eight col-leagues in three countries:Argentina, Uruguay, and the

Soviet Union. Always active on its behalf, Bersbecame chair of the committee in 1979 andserved as a member until 1984. Today, after al-most twenty years, more than 1,000 cases havebeen championed by the committee and over halfhave been successfully resolved. The HumanRights Committee is now an integral part of theNAS; it is regarded as the conscience of the Acad-emy and its sister institutions, the National Acad-emy of Engineering and the Institute of Medicine.

Lipman was not merely brilliant and endowedwith superb political instincts. Few could ex-press so simply, so strongly, and yet so very elo-quently, peoples obligations to other humanbeings. Lipman was an unabashed, unrepenting,and unfailingly articulate fighter for the humanrights of all people. He communicated his con-victions directly and forcefully, with little re-gard to personal cost. Those convictions werecharmingly expressed in what he himself calledthe international language of science: heavily ac-cented English.

He never took freedom for granted, not forothers, not for himself. In his successful plea to

Bers with Andrei Sakharov, 1988.



the Academys Council to make a public state-ment on behalf of Andrei Sakharov, Lipman said:When Sakharov began speaking out about vic-tims of injustice, he risked everything, and henever knew whether his intervention might help.Lipa then asked the Council members, Shouldwe, living in a free country, do less?

He was steadfast in his defense of freedomof expression, whether or not he agreed with theopinion expressed. Of course, it was well knownthat Lipa was not one inclined to allow those withwhom he disagreed to remain blissfully unen-lightened for very long. Those who expressed noopinion fared even worse. During the 1978 In-ternational Congress, Lipa was heard, sotto voce,in response to the refusal by a colleague to signa human rights petition on behalf of Shcharan-sky and Massera, that The hottest spot in hell[the humanist secular hell, of course] is reservedfor those who are eternally neutral.

Bers held fast to the strength of his personalconvictions, devoid of moralism or self-right-eousness. His concern for human rights was notan abstract policy decision. He was a mensch6

whose heart was open to all; he always sympa-thized with the oppressed and the underdog. Yet,in his social actions or political campaigns, as inmathematics, his head always ruled. He did,however, approach one subject on a slightly less-than-rational basis. He harbored a passionatedisregard for the high esteem that society ac-corded to Henry Kissinger.

In his commencement address, on the occa-sion of receiving an honorary degree at the StateUniversity of New York at Stony Brook in 1984,Lipman turned to one of his favorite nonmath-ematical topics and said:

By becoming a human rights ac-tivist, as I urge you to do, you do takeupon yourself certain difficult oblig-ations.I believe that only a trulyeven-handed approach can lead toan honest, morally convincing, and ef-fective human rights policy. A humanrights activist who hates and fearscommunism must also care aboutthe human rights of Latin Americanleftists. A human rights activist whosympathizes with the revolutionarymovement in Latin America must alsobe concerned about human rightsabuses in Cuba and Nicaragua. A de-vout Muslim must also care aboutthe human rights of the Bahai in Iranand of the small Jewish communityin Syria, while a Jew devoted to Israel

must also worry about the humanrights of Palestinian Arabs. And weAmerican citizens must be particu-larly sensitive to human rights viola-tions for which our government isdirectly or indirectly responsible, aswell as to the human rights viola-tions that occur in our own country,as they do.

At a human rights symposium held during theAcademys Annual Meeting in 1987, Lipman ex-plained why he so strongly believed that theCommittee on Human Rights should focus on po-litical and civil rights, which he called nega-tive rights, rather than put its energy behind apositive economic, social, and cultural agenda.7

He began:

As an old social democratI wouldsay an old Marxist, if the word had notbeen vulgarizedI certainly recog-nize the importance of positive rights.Yet, I think there is a good reasonwhy the international human rightsmovement, of which our committeeis a small part, concentrates on neg-ative rights. It makes sense to tell agovernment, Stop torturing people.An order by the prime minister orthe president to whoever is in chargecould make it happen.

It makes sense to tell a foreign am-bassador that, The American scien-tific community is outraged that youkeep Dr. X in jail. Let him out and lethim do his work. It requires no plan-ning, no political philosophy, and itcan unite people with very differentopinions.

Lipman, who was never accused of mincingwords, added:

The idea that people of the ThirdWorld are somehow less appalled bytorture or by government-sponsoredmurder than citizens of developednations [is, to me] rank racism.

He then added:

6 In Yiddish, human being, in the most positive sense.

7 Negative rights involve restraints by a governmentagainst an individual citizen doing something; for ex-ample, restricting his or her right to associate freely withcolleagues from abroad. Positive rights require a gov-ernment to fulfill a need, such as providing medical care,food, education, employment, or housing.



It is quite a different matter to tell aforeign government of a developingcountry, You really should give thisor that positive right to your people.If we make such a demand in goodfaith, it must be accompanied bysome plan for implementing this rightand by some indication of the costand of who will pay it and how it willbe paid.I think that the basic em-phasis on negative rights by the in-ternational human rights movementis a reasonable thing.

Lipman, ever the politician, continued:

Now, if we want to do things beyondthis and participate in organizing asocial democratic party in America, Iwill gladly discuss this later.

At that point, catching that provocative twinklein Lipmans eye, the symposium leader, GilbertWhite, broke in and, in an unsuccessful effort tokeep a straight face, said he really did not wantto give the panel even a chance to respond to Lip-mans challenge.

Over the years Lipman sent scores of appealsto foreign heads of state on behalf of scientificcolleagues around the world: Leonid Plyushch inthe Ukraine, Jose Luis Massera in Uruguay, YuriOrlov and Anatoliy Shcharansky in Russia,Ibrahima Ly in Mali, Ales Machacek in Czecho-slovakia, Jose Westerkamp in Argentina, Samuel

Greene in Liberia, Sion Assi-don in Morocco, Carlos Ar-mando Vargas in El Salvador,and Ismail Mohamed in SouthAfrica are among the names.Bers organized drives thatwere fully supported by hiscolleaguesin effect, heforged a consensus thathuman rights activities on be-half of, and public protests ofmistreatment of, scientistsare respectable activities forscientists. He also regularly,and without hesitation, en-listed the help of others. In a1984 telegram, Lipman askedWilly Brandt to help on behalfof Andrei Sakharov to pre-vent a tragedy in Gorky. Hewrote, in part: We feel thatSakharovs life is in gravedanger. The death ofSakharov would be a griev-ous blow to the cause ofpeace and to the hope of im-

proved relations between the West and the USSR.I feel that each of us must do not all we can butmore than we can. I appeal to you because of yourunique moral authority.

In summary, did Lipman Bers make a differ-ence? Yes, he did. He did what he could, and thenhe did more; he expected each of us to do no less.

Lipman Bers asMentorTilla Weinstein

Lipman Bers died on October 29, 1993, after along and debilitating illness which slowly tookfrom us a most extraordinary human being. Oth-ers will speak to his mathematical accomplish-ments, his many contributions to the profes-sion, and his passionate efforts to secure basichuman rights for scientists throughout the world.I have the privilege of commenting upon Lipaslegendary talent as a teacher and thesis advisor.People often ask how he managed to produce somany devoted mathematical offspring (forty-

Bers with student Tilla Weinstein, 1984.

Tilla Weinstein is professor of mathematics at RutgersUniversity, New Brunswick, New Jersey. Her e-mail ad-dress is [email protected].



eight in all). The impressions which follow wereformed during the mid- to late 1950s when I washis student at the Courant Institute. But Lipasinimitable style held steady through the years,as did his unwavering faith in the talent of youngpeople.

Lipa did not wait for students to introducethemselves. He generally approached them earlyin their graduate careers, sometimes in the hall-way or at tea. He would engage them in lively con-versation, showing keen interest in whateverthey had to say. Then, having quickly ascer-tained their likely interests and aspirations, hewould tell them which courses to take, whichbooks to read, which seminars to attend, and soon. In short, he often took on the role of advi-sor before being asked for advice.

Lipas courses were irresistible. He laced hislectures with humorous asides and tasty tidbitsof mathematical gossip. He presented intricateproofs with impeccable clarity, pausing dra-matically at the few most crucial steps, giving usa chance to think for ourselves and to worrythat he might not know what to do next. Then,just before the silence got uncomfortable, hewould describe the single most elegant way tocomplete the argument. A student who askedquestions in class was rewarded with more thanhelpful answers. Lipa welcomed questions as anopportunity to digress or to assign some extraproblem or related theorem for the student toreport on outside of class. Lipa liked to listen tostudents talking about mathematics. Then hegot to ask the questions, gently prodding us tonotice what we needed to learn.

Lipa never seemed concerned about what stu-dents could not do. Instead, he concentrated en-tirely upon what each student could do. Heshowed instant delight at any sign of progress.But he greeted each mathematical accomplish-ment with the immediate assignment of a moreformidable task, due next Tuesday. Then he sentus on our way with the clear impression that heexpected us to succeed.

Lipa generally taught one course or seminarclose enough to his research interests so that hecould use it to bare the day-to-day details of hismathematical life. He let us see his delight withthe lemma he had proved the night before, andhe talked openly about any difficulty he washaving in establishing a subsequent result or inframing just the right conjecture. Most notably,he would explain the context for his currentwork, describing the results which lay thegroundwork for current progress and pointingto theorems just over the horizon which mightbe established using the results he was trying toobtain.

When a student was ready to do research,Lipa would suggest not just the open questionshe could see to the end of, but the tougher prob-lems which might not be solvable in any routinemanner. He was honest about likely difficultiesand shared every idea he had on ways to getstarted. While he took enormous pleasure in thesuccess of students who grappled with particu-larly challenging questions (some quite distantfrom his own research interests), he was equallyavailable and respectful to students working onless demanding topics.

Lipa took mathematical parenthood seriously,showing pride and pleasure in our accomplish-ments, no matter how remote from mathemat-ics. He wanted to know everything about us: ourbackgrounds, our health, our hobbies, the bookswe read, our love lives, our politics, our hopesand ambitions. He encouraged a sense of cohe-sion among us, creating a mutually supportiveatmosphere in which professional friendshipsflourished. Still, it was probably Lipas wife Marywho most made us feel like a mathematical fam-ily. When Lipa invited us home, it was Mary whomade us feel at home. Her interest in each oneof us was spontaneous and unconditional. Herwarmth and loving support were perfect anti-dotes to the doubts and fears which so easilybeset graduate students.

It would be somewhat disingenuous to avoidthe question I am most often asked about Lip-man Bers. Why did he have so many womengraduate students? What did he do (starting wayback before it was fashionable) to produce sucha disproportionate share of the women earningdoctorates in mathematics? I think some of mymale colleagues over the years suspected thatLipa exuded some secret Latvian musklike fra-grance which drew us mindlessly to his door. Onbehalf of Lipas sixteen women Ph.D. students,I am happy to dispel such myths. There was ofcourse a special reason why Lipa succeeded sowell in producing women students. He succeededbecause he wanted to! Because he made ourprogress part of his own agenda, he went out ofhis way to seek us out and to give us the sameextraordinary care and attention which be lav-ished upon his thirty-two male Ph.D. students.The results speak for themselves.

Lipman Bers was a superb mathematicianwho shared his exuberant love of the subjectwith several generations of mathematical new-comers. For all he did to expand and enrichour lives, for his faith and his patience, we willforever be grateful.



Lipman Bers: APersonalRemembranceJane Gilman

I was Lipman Berss student from 1968 to 1971.When Bers students write about what he was likeas a teacher and a mentor, the result is often crit-icized for being too personal. But this is preciselythe point about Bers, that he affected people ona very personal level. It was not an uncommonexperience for me to meet mathematicians, who,when I mentioned that I was a student of Bers,proceeded to tell how Bers had made a differ-ence in their life (e.g., Bers helped them get onemore year of a graduate fellowship just when

they had given up all hope of con-tinuing graduate school, or Bersrecognized and became enthu-siastic about their work when

no one else seemed to careor notice).

These were common oc-currences for people who werenot Bers students. His impact

upon his students was fargreater. Since he treated

each student as anindividual, the de-tails of our storiesvary, but theessence is alwaysthe same. Here aresome parts of mystory.

I entered Co-lumbia in the fallof 1966 an alge-braist at heart.That year Berstaught the firstyear complexv a r i a b l e scourse. Sincehe made hardanalysis seemas simple andelegant as al-gebra, I be-came his stu-dent. I wasnot the onlystudent so in-fluenced by

Bers. Therewere sixteen students in my

year at Columbia. As a result of his course, eightof them became his students. Six of his eight stu-dents eventually received Ph.D.s from Columbia,while only three of the other eight did.

A big fuss is always made about the numberof female students Bers had. There were very fewwomen in mathematics when I was finishing col-lege and entering graduate school. Some insti-tutions such as Princeton simply did not allowwomen to apply to graduate school; and eventhough my undergraduate professors treatedme well, my undergraduate advisor at the Uni-versity of Chicago told me that women were notwelcome in the graduate program there. For methe big question was, Could a woman be a math-ematician? The question was not just, Could awoman actually get a Ph.D.?; it was, How wouldthe world treat a woman with a Ph.D.?; Couldsuch a woman have the other things in life thata woman might want (family, love, children), ordid a woman have to give up her femininity tobecome a mathematician?

Thus I entered graduate school with a greatdeal of ambivalence, and this was reflected byan uneven performance. When I took the quali-fying exam, I crossed out several correct pagesof mathematics on the complex analysis sec-tion, instructing the readers to ignore thosepages. Bers read the pages I had crossed outand in a typical move, used those pages to jus-tify passing me. Later he asked me why I hadcrossed out correct work. I replied, I dontknow.

It was with this understanding of my strengthsand weaknesses and my ambivalence that Berstook me on as a student. He not only was mymentor during graduate school but also contin-ued to play a significant role shaping my pro-fessional life afterwards.

In the early years I wondered whether he everfelt his time with me had been wasted becauseI was not as mathematically active as others whowere not raising small children. But he neversaid anything and was simply very supportive ofwhat I did do professionally.

If one person suggested to him that I give aseminar talk on a recent result of mine, the sug-gestion became translated by the time Bers spoketo me into his telling me, The masses are clam-oring for you to speak.

When my children were small and I had aheavy teaching load, it was difficult for me to getto Columbia on a regular basis for Berss semi-nar. I recall that when I did make it in one Fri-day, Bers overheard someone saying to me that

Jane Gilman is professor of mathematics at Rutgers Uni-versity, Newark, New Jersey. Her e-mail address [email protected].



I had come in to socialize and catch up on thegossip. He quickly stepped into the conversationand rebuked the person by saying I had come into feed my mathematical soul, not to gossip.I was grateful to know that he un-derstood how important theseminar was to me even if Icame irregularly.

Bers did have some gen-tle (devious) ways of prod-ding students into action.Once he called up to askthat I help rewrite a paperof a foreign mathematicianwhose English was poor.After I ghost edited thepaper and it was on its wayto publication, I mentionedthat using different methodsI had previously proved amuch stronger theorem whichimplied the main result of therevised paper. If he had not sentme that paper to edit, I would neverhave shared my theorem with any-one. I subsequently published mytheorem. It was the first piece of mypostthesis work to receive a lot of at-tention, and it played an importantrole in my getting tenure.

Bers was an astute politician. At onepoint I decided that it was time to shakeup my department. A first step in this wasmaking the pilgrimage to New York to in-form Bers that I was going to do this and thatit meant that in two years I would either havebecome chair of my department or be in total dis-grace and in desperate need of another job else-where, at which point I would need his help. Hisonly reaction was to ask, Do you have any al-lies in this venture? In what was a typical in-teraction with him, I replied that I had no allies,that was, except for the dean of the college andDanny Gorenstein (who at the time was alreadyuniversity wide an extremely influential Rutgersfaculty member). He then changed the subject,and we talked about his most recent reprintfrom the Annals of Mathematics. It was not untilmuch later that I understood that he did not dis-cuss further what he would do when my under-taking failed because he had already figured outthat I would succeed. (It actually took ten yearsto transform my department, and most peoplethought until the very end that it could not bedone.)

Bers refused to suggest problems for me towork on and instead encouraged me to have theconfidence to follow my nose. While at the timeI might have welcomed more help from him, inthe long run it made my successes all the more

valuable to me, a fact for which I am deeplygrateful. It also made interactions with him morefun. One day it clicked in my mind that one of

the Nielsen papers I had read beforeThurstons classification of the map-

ping-class group was about one ofthe (four) classes. (Bers had ex-panded Thurstons threeclasses to four by distin-

guishing two types of infi-nite reducible mapping-classes, parabolic andpseudohyperbolic.) It was

very exciting to call Bersup and say, Did you know

that Nielsen studied para-bolic mapping classes?

All Berss students weretaught very early that while

mathematics was king and wasto be loved and respected, our

worth as a person was not con-nected with our mathematical ac-

complishments. This was what theoft-repeated Teichmller litany(Teichmller was a great mathemati-cian but a Nazi) was all about.

Underneath the force of Berss per-sonality and vivacity was the forceof his mathematics. His mathe-matics had a clarity and beautythat went beyond the actual re-sults. He had a special gift for

conceptualizing things and plac-ing them in the larger context. For

him, there were seldom technical lem-mas. The -lemma is an example. If I close

my eyes now and listen, I can visualize him infront of a lecture hall with a gleam in his eye, andI can hear his beautiful and richly accented voicesaying the words the - lemma in a manner thatgave a dramatic persona to what another math-ematician might have passed off as a mere tech-nical lemma. The lemma took on a life of itsown. While I may not be able to give a preciseformulation of the lemma, the idea of the lemmaand its role in Teichmller theory remain etchedin my mind.

There are a few people in my life whose voiceI always hear whatever I am doing. Even duringthe times when my contact with him was not thatextensive, his voice was constantly with me. I stillhear his voice, but I miss him.


1995, remembering lipman bers

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