Friedrich Hirzebruch (1927–2012) · 2014. 6. 26. · then the seasons in Germany and Australia would coincide, he was punished for insubordination. In the ﬁnal months of the war

Friedrich Hirzebruch(1927–2012)Michael Atiyah and Don Zagier, Coordinating Editors

Friedrich Hirzebruch (universally knownas Fritz) died on May 27, 2012, at the ageof eighty-four. He was the outstandingGerman mathematician of the postwaryears who helped to restore mathematics

in his country after the devastation of the Nazi era.Appointed at a very early age to a full professorshipat the University of Bonn, he remained there forthe rest of his very active life and moved thecenter of gravity of German mathematics fromthe traditional centers of Göttingen and Berlinto Bonn. The famous “Arbeitstagungs” (moreproperly Arbeitstagungen), which he establishedin Bonn in 1957, have been running annually orbiannually ever since and are a focal point ofmathematics worldwide. They carried his personalimprint in their content, attendance, and style,being always broad, topical, and informal anddoing much to educate succeeding generationsand to foster cross-fertilization. Many new ideasand collaborations grew out of these encounters.Another lasting contribution to mathematicalresearch in Germany and in the world is the MaxPlanck Institute for Mathematics, which he founded,operating on the same lines and creating bondsbetween mathematicians from many countries,including those that were otherwise cut off fromthe international scene.

Although Fritz, given his multiple roles, retiredseveral times, he remained active till the very endand was preparing to attend a conference in hishonor in Poland when he was struck down.

In this introduction we will give an overviewof Fritz’s life and of some of his most important

Michael Atiyah is honorary mathematics professor at theUniversity of Edinburgh. His email address is [email protected].

Don Zagier is a scientific member at the Max Planck Insti-tute for Mathematics in Bonn and a professor at the Collègede France in Paris. His email address is [email protected].

DOI: http://dx.doi.org/10.1090/noti1145

achievements. More detailed accounts will thenfollow in the individual articles by the two coordi-nating editors, with the one by Atiyah concentratingon the work in topology and the years before 1970,and the one by Zagier on the work in numbertheory and the years after 1970. The subsequentarticles by the invited contributors describe furtheraspects of his personality, his scientific work, andthe role that he played in the mathematical livesof many individuals, organizations, and countries.

* * *

Friedrich Hirzebruch in 2006 at the Max PlanckInstitute.

Friedrich Ernst Peter Hirzebruch was bornon October 17, 1927, in Hamm, Germany, toDr. Fritz Hirzebruch and Martha Hirzebruch (néeHoltschmit). His father, who was the headmaster ofa secondary school and himself an inspiring teacherof mathematics, gave him his first introduction to

706 Notices of the AMS Volume 61, Number 7

the subject—including, when he was nine yearsold, the proof that

√2 is irrational—and the love

of it that was to last throughout his life.Still a teenager, Fritz was drafted into the

German army during the final year of World War II,but his military career was mercifully short andhe was never sent into combat, being assignedinstead to an antiaircraft battery with the task ofcomputing artillery trajectories. He was even ableto attend some scientific courses, though whenhis commanding officer asked him on one suchoccasion to confirm that winter and summer arecaused by the earth’s varying distance to the sunand Fritz dared to contradict him, pointing out thatthen the seasons in Germany and Australia wouldcoincide, he was punished for insubordination. Inthe final months of the war the Americans put himinto a prisoner of war camp, and even there hemanaged to do mathematics (partly on toilet paper,still preserved today). He was released in July 1945and entered Münster University that winter.

The city and the university lay in ruins andconditions were very difficult, with lectures beingheld only at long intervals, but he had very goodteachers, especially Heinrich Behnke, from whomhe learned the function theory of several complexvariables, and Behnke’s assistant, Karl Stein, aformer pupil of his father’s. His third teacher wasHeinz Hopf, a German who had gone to Switzerlandbefore the Nazi takeover and who invited the youngFritz, first to be his house guest and then for oneand a half years to be a research student at theETH in Zürich. Fritz returned to Münster withthe essentials of a beautiful doctoral thesis aboutthe resolution of certain singularities in complexsurfaces. Already this earliest work showed thecharacteristics of all of his mathematics: eleganceand brevity of thought and exposition, an effortlesssynthesis of sophisticated theoretical ideas withinsights inspired by nontrivial concrete examples,and the fusion of ideas from analysis, topology,and number theory.

In 1952 came the development that was notonly to be a turning point in Fritz’s mathematicalcareer but, as it transpired, to have a majorinfluence on the later development of mathematicsin Germany and in Europe: he was invited to theIAS in Princeton, where he remained for two years.At the IAS, he came into contact with many of themost brilliant mathematicians and most excitingnew ideas of the period and where he made thetwo discoveries with which his name is moststrongly associated: the Signature Theorem andthe Hirzebruch-Riemann-Roch Theorem. Thoseyears and also the early years in Bonn, whenthe core of Fritz’s research was still in topologyand its applications to algebraic geometry, will be

discussed in detail in the contribution by MichaelAtiyah.

Friedrich Hirzebruchca. 1985 in Bonn.

This period also in-cluded three majorevents in Fritz’s personallife: his marriage to IngeSpitzley in August 1952just before taking theboat to Princeton, andthe birth of his first twochildren, Ulrike (1953)and Barbara (1956). Histhird child, Michael, wasborn a little later, in 1958.Inge, known to and lovedby countless mathemati-cians, was a big part ofeverything he built upduring his life. Both ofhis daughters later studied mathematics andeventually worked in related areas (Ulrike in mathe-matical publishing and Barbara as a schoolteacher),while Michael was to become a doctor. Ulrike’scontribution to this article gives us a vivid pictureof Fritz as a father.

When Fritz returned to Germany in 1956 totake up his duties at his new chair in Bonn, hehad a clear ambition and a mission: to establish acenter that would attract mathematicians from allover the world. After the First World War, Germanmathematics had been ostracized by the interna-tional community, led by France. This lasted formany years and embittered relations. Fortunately,the 1945 generation of French mathematicians,led by Henri Cartan, was more enlightened, andprewar mathematical friendships were rapidlyrenewed. The Münster school members underHeinrich Behnke were welcomed back into the fieldby Cartan, while Fritz, part of the Behnke team,

Karl Stein, Reinhold Remmert, FriedrichHirzebruch, and Henri Cartan in the 1950s.

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Friedrich Hirzebruch with Gerd Faltings in theMPIM Library ca. 2000.

played a full role in this rapprochement. So with hisPrinceton contacts, including Kodaira from Japanand the new talent emerging from Paris (Serre,Borel, Grothendieck, …), Fritz was well placed toreintegrate German mathematics into the worldcommunity. Contacts with Britain came initiallyvia the Cambridge geometry school of Hodge andTodd and later the younger generation (Atiyah,Adams, Wall, …). The division of Germany and,more generally, the cold war partition of Europewere particularly challenging, but Fritz spent manyyears of his life forging links between East andWest, including notably the German DemocraticRepublic and the Soviet Union.

He achieved his goals remarkably quickly. AtBonn University he built up the mathematicsdepartment to a high level, doubling the numberof full professors and attracting people suchas Klingenberg, Tits, Brieskorn, and Harder. TheArbeitstagung, which he established in 1957, soonserved as a worldwide meeting point and attractedmany who would never otherwise have returned toGermany. But Fritz’s main goal stemmed from hisexperience in Princeton: to set up a visitors’ centermodeled after the IAS. A first attempt to createthis as a Max Planck Institute failed because of theopinions of various referees (including Courantand Siegel) that, at least at that time, there werebetter ways to use both Fritz’s talents and thetaxpayers’ money to further mathematical researchin Germany. But some ten years later, when theGerman Research Council (DFG) set up a newresearch program for German universities whoseunits (called Sonderforschungsbereich or “SpecialResearch Domains”, abbreviated SFB) would besupported for a limited period of time, Fritzpresented his ideas of a visitor center to thedecision committee and came back with two SFBs:one (SFB 40, with himself as Sprecher or chairman)for theoretical mathematics and one (SFB 72, withRolf Leis and Stefan Hildebrandt) in a more applied

direction. With his Sonderforschungsbereich, hestarted the envisaged visitor program on a limitedbasis. This turned out to be so successful thatwhen the support ended and Fritz applied for atakeover from the Max Planck Society, his projectno longer met with the former reservations, and apermanent Max Planck Institute for Mathematicswas founded in Bonn in 1981 and has beenflourishing ever since. Through the Arbeitstagung,the Sonderforschungsbereich, and finally the MaxPlanck Institute, Fritz created an extensive visitorprogram that he guided with his many outstandingqualities: his personal tastes in mathematics werebroad and generous—he was no narrow specialist;his international contacts were extensive; hisefficiency became legendary; and above all, heencouraged an informal and friendly atmosphere,far removed from the traditional rigidity of Germanacademia.

After about 1970 the main thrust of Fritz’smathematical work slowly moved from pure topol-ogy and algebraic geometry to the connectionsof these domains with number theory. They willbe discussed in more detail in the contributionby Don Zagier. During these years he also be-came more and more active and influential inthe development of mathematics, both nationallyand internationally. These activities, which willbe described in more detail later, included mostnotably his unflagging efforts to build up relationswith the countries of the Eastern Bloc, his manycontributions to rehabilitating Germany’s imageafter the years of the Third Reich and to creatingnew scientific and human bonds with Israel, his twopresidencies of the German Mathematical Union,and his roles as first president of the EuropeanMathematical Society (described by Bourguignon)and as honorary president of the InternationalCongress of Mathematicians in Berlin in 1998.

During all the years before the Iron Curtainfell, Fritz indefatigably kept up contacts withmathematicians in the Eastern Bloc, no matterhow much effort this required and how unavailingit seemed. Russian mathematicians were alwaysinvited to the Arbeitstagung, though only once—perhaps because of an oversight by the Russianbureaucracy?—were some of them allowed to come:in 1967, Anosov, Manin, Postnikov, Shafarevich,and Venkov took part, and all of them gave atalk. But these efforts were not in vain, because,as we learned later, the yearly invitations tocome to Germany, even when they had to remainunanswered, often helped their recipients bydemonstrating to the authorities their visibility inthe West. Fritz himself was seen quite positivelyby those same authorities and in 1988 was electedas a foreign member of the Academy of Sciencesof the USSR. After 1990, of course, many more


possibilities of exchange opened up, and the MPIMtoday is never without some Russian conversationin its corridors. Another Eastern country thathe became deeply involved with was Poland. Hiscontributions here, in particular in connectionwith the Stefan Banach International MathematicalCenter in Warsaw, are recounted by StanisławJaneczko.

By a coincidence that seemed willed by fate,Fritz was elected twice to the presidency ofthe DMV (German Mathematical Society) at keymoments in the history of postwar Germany andpostwar German mathematicians: in 1961 whenthe Berlin Wall was built, and again in 1989–90when it fell. The separation of Germany into twoblocks fell in the middle of his first term, andhe solved the problem of the inability of theEast German mathematicians to cross into WestBerlin by repeating in its entirety the first DMVmeeting that he chaired after the separation. Butof course such makeshift measures could not last,and soon the DMV was split into a new East Germanbranch (MGDDR = Mathematische Gesellschaft derDDR) that for almost three decades was no longerofficially connected with the West German one.When the political world changed again and thetwo halves of Germany were reunited, Fritz wasable to preside over the reunification also of theMathematiker-Vereinigung and to ensure that thetransition took place in a spirit of reconstructionrather than of recrimination or retaliation.

For several years after the wall fell, he travelednearly every week to Berlin, where he had the task ofhelping the nearly two hundred mathematicians ofthe previous Karl Weierstrass Institute of the EastGerman Academy of Science to find new positions.The individual cases were very dissimilar, andthe solutions he came up with were varied. Thecases where no adequate solution could be foundhaunted him, and he sometimes spoke to hisfriends at the institute of the sorrow he felt, but inthe vast majority of cases provisional or permanentpositions could be set up, whether in temporaryMax Planck Working Groups, in permanent newinstitutes that he helped establish, or in schoolsor universities in Germany or abroad. His contactswith the GDR during its years of isolation andthe respect in which he was held on both sidesof the previous dividing line made him effectivein this role in a way that no one else could havebeen, and his achievements, though little known tooutsiders, were received with enormous gratitudeby the people involved.

Of the many other countries with which Fritzbuilt up or maintained intensive contacts, two mustbe mentioned individually. One is Japan, whichFritz visited many times, starting in 1972, andfrom which a huge number of visitors came, first to

the Sonderforschungsbereich and later to the MaxPlanck Institute, at a period when the possibilitiesof scientific interchange between Japan and Europewere still severely limited. His contributions aredescribed in detail by Kenji Ueno. The other isIsrael, which is dealt with by Mina Teicher. Fritzalmost never mentioned overtly, but very clearlyalso never forgot, what Germany had done in theyears of the Third Reich, and a leitmotif of muchof what he did in his life was to help reestablish animage of the country that would be characterizedby decency and tolerance.

Not surprisingly, Fritz was showered with manydistinctions of every imaginable kind. His first halfdozen honorary doctorates came roughly at thesame times and with the same frequency as hisgrandchildren, and he used to make jokes aboutthis ongoing race, but with only three children,the competition was an unequal one and thedoctorates finally won 15:6. He was a full orassociate member of more than twenty academies,in several of which he was scientifically active, andalso of the order “Pour le Mérite”, which has as itsmembers Germany’s most distinguished scientists,writers, and artists and to which he was particularlyattached. Among the many prizes that he receivedthe most notable were the Japanese Seki Prize,usually given to institutions and which he receivedfor his role in developing the contact betweenJapanese and non-Japanese mathematicians; theLobachevsky Prize and the Lomonosov Medal fromthe USSR; the Polish Stefan Banach Medal; theGerman Krupp Prize and Georg Cantor Medal;and, most important of all, the Israeli Wolf Prize,which he received in 1998 and which, quite apartfrom its immense prestige, had a huge symbolicsignificance. At one point the honors were arrivingso thick and fast that his secretary once famouslyremarked, after checking his morning’s mail, “Wirhaben heute keine Ehrungen bekommen!” (“Wedidn’t get any honors today!”)

That “we” somehow characterizes in two letterswhat was so exceptional about Fritz and the wayhe made those around him feel. We hope that thearticles collected here will convey to those who didnot know him some feeling for this extraordinarypersonality.


The Hirzebruch signature dish being admired byLily Atiyah and Fritz, Edinburgh, 2010.

Michael Atiyah

Fritz Hirzebruch played a major part in mylife, particularly over the early formative period.He became a close personal friend, a long-termcollaborator, and, through the Arbeitstagung, myintroduction to the mathematical world. I learneda good deal from him on how to write papers, howto present talks, and, most importantly, how tohandle people. In short, he was an ideal role model.

I first met Fritz in 1954 when I was a younggraduate student and he visited Cambridge at theinvitation of Hodge, my supervisor. Hodge andTodd had been much impressed by what Fritzhad been doing at Princeton and were keen to bebriefed on the Riemann-Roch Theorem and Chernclasses. What I remember about the occasion ishow friendly and informal Fritz was. Although hewas already an assistant professor at Princetonand I was merely a graduate student, there were nobarriers between us, and we quickly established afriendship which blossomed over the subsequentyears. We met again at the Amsterdam ICM of1954 and then, for a longer period, when I wenton a postdoctoral fellowship to the Institute forAdvanced Study in Princeton.

The Princeton YearsThose Princeton years were, for me, for Fritz, andfor many others the “golden years”. Algebraicgeometry and topology were being transformedby the new ideas of the French School. Sheavesand spectral sequences from Leray combined withcomplex analysis by Henri Cartan produced pow-erful machinery to tackle classical problems. Thiswas taken up by Kodaira and Spencer, while Serre

burst on the scene with spectacular applicationsto both algebraic topology and algebraic geometry.When I arrived in September 1955, brilliant youngmathematicians were absorbing the new ideas andcarving out new routes for the future. I rememberin particular the gang who regularly attendedKodaira’s lectures: Fritz, Serre, Bott, Singer.

This had been for Fritz the experience thattransformed him from a promising novice to aworld figure capable of competing with the greatesttalents of the time. Within a short period of time hecame up with two great triumphs. Both were basedon the innovative way of associating multiplicativeclasses to formal power series in one variable. Firstthere was his formula

Sign(M) =∫ML(M) =

∫M

n∏i=1

xitanhxi

,

where (formally) p(M) =n∏i=1

(1+ x2i )

for the signature of a 4n-dimensional mani-fold in terms of the Pontrjagin classes pj(M).This was a beautiful application of Thom’snew cobordism theory. But Fritz’s second tri-umph, his generalization (now known as “HRR” orHirzebruch-Riemann-Roch)

χ(X,V) =∫X

ch(V)Td(X)

of the Riemann-Roch Theorem, was even moreimpressive. Here χ(X,V) = σ (from q = 1 toq =m) of (−1)q dimHq(X,V) is the holomorphicEuler characteristic

∑mq=0 dimHq(X,V) of the sheaf

cohomology groups of a holomorphic vector bun-dle V (of dimension d) over a complex projectivealgebraic manifold X of dimension m. The Cherncharacter is defined in terms of the total Chernclass c(V) by

ch(V) =d∑i=1

exi ,

where (formally) c(V) =d∏i=1

(1+ xi) ,

and the Todd class Td(X) is defined similarly interms of the total Chern class c(X) of the tangentbundle of X by

Td(X) =m∏j=1

yj1 − e−yj ,

where (formally) c(X) =m∏j=1

(1+ yj) .

Of course HRR built on fundamental work byKodaira, Spencer, and Serre, but the proof was atour de force that had the hallmark of Fritz’s ownmathematical style.


Characteristic classes had been developing formany years, from the algebraic geometry of theItalian School through significant advances by Toddand the later topological approach of Steenrod andChern. But all this emphasized their geometricalorigin and significance. It was Fritz, in collaborationwith Borel, who took the dual route of cohomologyand, connecting it to the theory of Lie groups, gaveChern classes their formal algebraic setting, whichhas now become standard. With his command ofthis algebra and with his insight into the rightalgebraic framework, Fritz had developed histheory of multiplicative sequences, which providedthe right tools to tame the horrendous-lookingformulae.

When Todd, no slouch at algebraic computa-tions, had computed the first half dozen “Toddpolynomials”, it had been a matter of brute force.In the hands of Fritz as “magician” the calculationsbecame elegant and transparent. After seeing this,Todd remarked that he now had to reverse the ear-lier view he had held of the “Princeton School” that,while they might be good at general theory, theywere not adept at calculations. The old maestroconceded defeat to the young contender.

It was fortunate for the new generation likeme, eager to learn about the great advances inalgebraic geometry, that Fritz was also a brilliantexpositor. His book Neue Topologische Methoden inder Algebraischen Geometrie (Springer, 1956) gavean impeccably clear account of sheaf theory, Chernclasses, and all the new machinery that culminatedin the Hirzebruch-Riemann-Roch Theorem. Thebook and its subsequent English edition TopologicalMethods in Algebraic Geometry (with appendicesby R. L. E. Schwarzenberger, one of my earlystudents) has remained the standard work for overfifty years.

The Early ArbeitstagungsWhen Fritz returned to Germany as a full professorat the University of Bonn, a new day dawned forGerman mathematics. With his enthusiasm, ability,efficiency and drive, Fritz soon transformed Bonninto a major center of the mathematical world.Modelled on Princeton, it aimed to introduce intoEurope the features that had so attracted Fritz andothers across the Atlantic.

Because of the friendship I had forged withFritz in Princeton and because of the proximity ofCambridge to Bonn, I was fortunate to have beeninvited to the very first of the annual meetingsthat became the famous Arbeitstagung. I wenton attending these meetings for almost thirtyyears. It became an obligatory part of the academiccalendar where new results were announced,many famous mathematicians regularly attended,and the whole event was under the careful but

Program discussion at the Arbeitstagung, 1987.

loving care of the “maestro”. Fritz’s talents werefully exploited, but not exposed, in these annualgatherings. With their relaxed atmosphere, theRhine cruises and the skillful selection of speakersby what has been described as “guided democracy”,the Arbeitstagungs were unique. Happy familygatherings they may have been, but much seriousmathematics was always being presented andfostered. Ideas flowed, collaborations emerged, andsuccessive years reflected the latest movements.

Moreover, as the years passed, Fritz was alwayskeen to attract new talent, and he encouraged meto send promising graduate students to attend. Iwas happy to respond, and, over the years, mystudents were introduced to the internationalscene through the Arbeitstagung. Graeme Segal,Nigel Hitchin, Simon Donaldson, Frances Kirwan,and many others came and became, in their turn,regular participants.

But if the entire series of Arbeitstagungs becamehighpoints of the academic calendar, the initial one(in 1957) on a very modest scale was particularlynoteworthy for launching Grothendieck. He hadjust developed his brilliant new approach to theHirzebruch-Riemann-Roch Theorem, based on K-theory. I remember him lecturing for many hours onhis ideas. In fact he seemed almost to monopolizethe timetable, but the novelty and importanceof his work fully justified the time devoted toit. The fact that the program was sufficientlygenerous and flexible to allow this to happen wasan early indication of the way Fritz wanted theArbeitstagungs to work. No set plans, and fullsteam ahead for novel and exciting mathematics.

Grothendieck’s explosive entry on the scenewas a hard act to follow, but the Arbeitstagungsin those early years saw a succession of new andexciting results, including Milnor’s discovery ofexotic spheres and their subsequent realization


Michael Atiyah and Friedrich Hirzebruch in frontof Mathematisches Institut Bonn, 1977.

by Brieskorn (a student of Fritz) via isolatedsingularities of algebraic varieties (a study initiatedby Fritz). In fact, so many new ideas filled theArbeitstagung air that most of my own work (andprobably that of many others) emerged from thisbackground. We learned many new things fromdisparate fields, and cross-fertilization became thenorm. I will elaborate on this in the next section.

My Collaboration with FritzIn the three years 1959 to 1962 Fritz and I wroteeight joint papers, all concerned with topologicalK-theory and its applications. This had emergednaturally from the early Arbeitstagungs and in par-ticular from Grothendieck’s K-theory in algebraicgeometry, as expounded in the very first Arbeitsta-gung. But there were many other ingredients in thebackground, notably the Bott periodicity theorem.

Topological K-theory was mainly developed byFritz and me in 1959 when we both had a sabbaticalterm at the IAS in Princeton. A preliminary accountappears in [1], and we planned to write an expandedversion in book form. In fact, we never had time forthis project, but a book [4] did eventually appearunder my name based on a Harvard course oflectures.

These joint papers are a mixture of general the-ory and concrete problems. For example, [3] showedthat the famous Hodge conjectures were false forinteger cohomology (still leaving the case of ra-tional cohomology as one of the Clay InstituteMillennium Prize problems). Other papers wererelated to some of Fritz’s earlier Princeton period,such as his discovery of a relation between Steen-rod squares and the Todd polynomials [2]. Some ofour joint papers appeared in German (written byFritz), while others appeared in English (written by

either of us), but one appeared in French (writtenby neither of us!). That one gave bounds on thesmallest dimension in which various manifoldscould be embedded. While a primitive version wasan idea of mine, the final very polished versionwas an exquisite illustration of Fritz’s elegancewith algebraic formulae. But my mathematicalinteraction with Fritz extended far beyond thesejoint publications and the three years they cover.Much of my work was influenced in one way oranother by Fritz, and a later publication [5] isone of my favorites. Here we proved that a spinmanifold that admits a nontrivial circle actionhas vanishing A-genus. This emerged as a newapplication of index theory, which first appearedin the Arbeitstagung program of 1962. Fritz tookgreat interest in the development of index theory,which owes so much to his pioneering work.

While our later mathematical paths may appearto have diverged, this is only superficially true.We met frequently in Bonn and elsewhere, andwe followed each other’s work with great interest.One notable example is Fritz’s beautiful results onthe resolution of the cusp singularities of Hilbertmodular surfaces (as explained by Don Zagier). Hiskey result gave the signature defect of such a cuspsingularity as the value of a suitable L-function ofthe number field. He then conjectured that thisresult would continue to hold in higher dimensionsfor arbitrary real number fields. This was one of themain sources of inspiration that eventually led tothe index theorem for manifolds with boundary [6]and its application [7] to prove Fritz’s conjecture.

Fritz also followed with great interest theexciting interaction between geometry and physicsof recent decades. He organized several meetingsof mathematicians and physicists (in Bad Honnefin 1980 and in Schloss Ringberg in 1988, 1989,and 1993). He also extended [8] the work of Wittenand others on the elliptic genus, a subject close tohis heart.

Final CommentsI knew Fritz and was a close friend for nearly sixtyyears. We were mathematical brothers and shareda common love of geometry in the broadest sense.We had very similar tastes, even if I could nevermatch Fritz’s algebraic virtuosity. I was a greatadmirer of his lecturing style, and, with my limitedGerman, I found he was the only German lecturerI could understand. He was also a magician whocarefully crafted his lecture so as to produce asurprise at the end. Alluding to this skill of his, Ionce said that “rabbits do not appear out of hatsunless they are put there!”

A close mathematical partnership leads to aclose personal friendship and also evolves from it.This extends to families on both sides. Lily and I got


to know Fritz and Inge in Princeton when we bothhad small children, and we have remained closefriends ever since, meeting occasionally in Bonn,Oxford, Edinburgh, Barcelona, and elsewhere. InBonn, at all the Arbeitstagungs, Inge was always awelcoming hostess, and the friendly atmosphere ofthe Hirzebruch family was an important ingredientin the success of both the Arbeitstagungs and theMPI.

References[1] M. F. Atiyah and F. Hirzebruch, Vector Bundles and

Homogeneous Spaces, Proceedings of Symposium inPure Mathematics, Vol. 3. Amer. Math. Soc., 1961.

[2] , Cohomologie-Operationen und charakteristis-che Klassen, Math. Zeit. 77 (1961), 149–87.

[3] , Analytic cycles on complex manifolds, Topol-ogy 1 (1962), 25–45.

[4] M. F. Atiyah, K-Theory, Benjamin, New York, 1967.[5] M. F. Atiyah and F. Hirzebruch, Spin manifolds and

group actions, in Essays on Topology and Related Topics,Mémoires dédiés à George de Rham (A. Haefliger andR. Narasimhan, eds.), Springer-Verlag, New York–Berlin,1970, pp. 18–26.

[6] M. F. Atiyah, V. K. Patodi, and I. M. Singer,Spectral asymmetry and Riemannian geometry. I,Math. Proc. Camb. Phil. Soc. 77 (1975).

[7] M. F. Atiyah, H. Donnelly, and I. M. Singer, Signaturedefects of cusps and values of L-functions, Annals ofMath. 118 (1983), 131–177.

[8] F. Hirzebruch, Elliptic genera of level N for com-plex manifolds, in Differential Geometric Methodsin Theoretical Physics (Como, 1987), Kluwer, 1988,pp. 37–63.

Don Zagier

Entering Fritz’s OrbitMy first meeting with Fritz Hirzebruch was anever-to-be-forgotten moment in my life (not leastbecause I also met Egbert Brieskorn and SilkeSuter, my future wife, on the same day). It wasMay 1970 and I was not yet nineteen. I had beena precocious but incompetent topology graduatestudent in Oxford for two years, the first underthe supervision of Michael Atiyah, who tried toteach me the basics I should have learned asan undergraduate, and the second with no realsupervision, because Atiyah had left for Princetonin 1969. I had been studying Professor Hirzebruch’sbooks and papers on applications of the signaturetheorem to constructing exotic spheres and the likeand had found some amusing formulas relatingthese to cotangent sums and other elementarynumber theory, which I had sent to him, inquiringon the same occasion about the possibility ofcoming to Bonn to complete my D.Phil. studiesunder his supervision (an idea supported by Atiyah).He had responded with an invitation for a shortvisit to meet both him and Professor Brieskorn,

Thirty-two years later at the eightieth birthday ofSir Michael Atiyah, Edinburgh, April 20–22, 2009.

who would be in Bonn for a few days, and nowreceived me with all the friendliness and interestin my work that he would have shown if I hadbeen an established mathematician and which as abeginner I had certainly not expected.

That first meeting lasted several hours (in theevening Hirzebruch had to go home, but Brieskorninvited me to a Chinese restaurant to continue thediscussion) and resulted in new research projectsfor me and invitations to come to Bonn a monthlater for my first Arbeitstagung (also memorable!)and again in the fall as Hirzebruch’s doctoralstudent. (I remained immatriculated in Oxford,and Hirzebruch received a salary of £5 a year forhis work.) As my advisor, he met me frequently,listened to my reports with great attention, andmade such minimalistic comments that I alwaysfelt the new ideas that emerged were my own,although I did sometimes wonder why everythingwas working out so much better than it ever hadbefore.

My actual thesis was on a somewhat differentsubject from the cotangent sums that had providedthe initial contact with Hirzebruch, but during thetwo years that I spent in Bonn as his student andStudentische Hilfskraft , we also had many morediscussions about those things, and he gave acourse on the subject which turned into our jointbook [2] on relations between index theoremsand elementary number theory. One of the topicstreated in that book, the calculation of invariantsof torus bundles over the circle, was to lead himlater to his beautiful discoveries, discussed below,on the geometry of Hilbert modular surfaces. Someof this work and of Hirzebruch’s own work in thisarea is beautifully told in his article [1], whoseintroduction ends with the words

In the second half of this lecture weshall point out some rather elementary


Silke Suter, Friedrich Hirzebruch, and Don Zagier,1982 in Bonn at the MPI.

connections to number theory obtained bystudying the equivariant signature theoremfor four-dimensional manifolds. Perhapsthese connections still belong to recre-ational mathematics because no deeperexplanation, for example of the occurrenceof Dedekind sums both in the theory ofmodular forms and in the study of four-dimensional manifolds, is known. As atheme (familiar to most topologists) underthe general title “Prospects of mathematics”we propose “More and more number theoryin topology.”

As we will see, these last words were to be propheticin his own case.

I had intended to come to Bonn only for thetime needed to complete my thesis, but ended upstaying there for my whole life. This development,which I could never have imagined (not onlybecause I knew no German when I came andhad no relationship with the country, but alsobecause I am half Jewish and much of my father’sfamily had perished in the concentration camps),was due exclusively to Hirzebruch’s tremendouspersonality and to the atmosphere that he created.In the first period after my thesis, I began workingmore and more intensively with him, first oncotangent sum-related topics and then on Hilbertmodular surfaces. Part of this collaboration tookplace on long train trips to Zürich, where he wasgiving a course on the latter subject and whereI regularly accompanied him because it was theonly chance to get him to myself for long periodsat a time. In the evenings we often ate together atthe elegant Zunfthäuser (guild halls turned intorestaurants) of Zürich, gradually becoming betterfriends and increasing our alcohol consumptionfrom a modest single glass each at the beginning

to a full bottle. On one occasion this was increasedto one and a half bottles, and Professor Hirzebruchformally proposed the use of “Du” and first names.Henceforth he was always “Fritz” to me, and sohe shall remain for the rest of this article. Duringthese years I also got to know his family well,and this too made Bonn become a true home. Hisdaughters, Barbara and Ulrike, also attended mycourse on elementary number theory. Both hadreal mathematical talent, but in the end neither oneopted for a research career, though Ulrike wrotea master’s thesis on elliptic surfaces with threeexceptional fibers that is still quoted regularlytoday.

Fritz’s Work in Number TheoryFritz had already done earlier work that is importantin the theory of algebraic and arithmetic groups,most notably his fundamental papers with ArmandBorel about homogeneous spaces (in particular, thedetermination of their characteristic classes) andhis proportionality principle, which has provedenormously important in the theory of automorphicforms. But starting around 1970 his interest inthe relations between topology and number theorybecame much more intense and led to what onemight call a second spring in his mathematicalresearch career. The high point of this was hiswork on Hilbert modular surfaces, which I nowbriefly describe.

In the classical theory of modular forms a crucialrole is played by the modular curve H/SL(2,Z)(H = complex upper half-plane) and its cousins.The higher-dimensional generalization of thiscurve is the Hilbert modular variety Hn/SL(2,OK)associated to a totally real number field K. HereOK is the ring of integers of K and SL(2,OK) is theHilbert modular group, embedded into SL(2,R)nby the n different real embeddings of K andhence acting naturally (and discretely) on Hn. Thisvariety can be compactified by adding “cusps”to give a projective algebraic variety XK , butthese cusps are highly singular points, with theboundary of a small neighborhood of each cuspbeing a Tn-bundle over Tn−1 rather than a (2n−1)-dimensional sphere. In particular, for n = 2 theseneighborhood boundaries are precisely the torusbundles over a circle that Fritz had already beenstudying in connection with the equivariant indextheorem, and it was this that led him to the studyof Hilbert modular surfaces.

He set himself three main goals:

(i) to describe the geometry of XK and calculateits numerical invariants,

(ii) to give for n = 2 the resolution of thesingularities at the cusps, and

(iii) to apply this to the classification of XK inthe sense of the theory of algebraic surfaces.


He achieved these goals in a series of paperspublished between 1970 and 1980, partially incollaboration with A. van de Ven and me in thecase of part (iii). Each part was mathematicsof the highest order. The calculations of thenumerical invariants involved deep results fromboth differential geometry and number theory,including Günter Harder’s extension of the classicalGauss-Bonnet theorem to noncompact manifoldslike Hilbert varieties and classical results of Hecke,Siegel, and Curt Meyer about Dedekind zetafunctions and class numbers of number fields andtheir relationship to cotangent sums. The resolutionof the singularities in terms of periodic continuedfractions was an amazingly beautiful result initself and also spawned many generalizations,including the theory of toroidal compactifications(work of Mumford, Faltings, and many others)that now plays a central role in the theory ofmirror symmetry. The results in part (iii), whichculminated in the complete determination of theposition of the Hilbert modular surfaces withinthe Kodaira classification, provided a beautifulcollection of algebraic surfaces having particularlyinteresting properties because of the interplaybetween their transcendental aspects (descriptionas quotients of H2) and their algebraic aspects(description as projective varieties). This interplayleads to many insights that are not availablefor varieties possessing “merely” an algebraicdescription. All aspects of the theory are describedin the masterful exposition [3].

Fritz’s investigation of the geometry of theHilbert modular surfaces led him to an intensivestudy of the modular curves TN (N ∈ N) that arenaturally embedded in these surfaces. This ledto a joint paper with me [4] showing that thegenerating function

∑N[TN] qN of the classes

of these curves in the second homology groupof the surface is itself a modular form in onevariable, a result that in turn has given rise to manylater applications and generalizations (work ofKudla-Millson and many others). There is anotheramusing anecdote connected with this. Serre,who had studied Fritz’s work on the topologicalinvariants of Hilbert modular surfaces, wrote hima letter pointing out a coincidence between thenumbers occurring here and the formulas for thedimensions of certain spaces of modular forms.His letter and Fritz’s giving the explanation interms of our modularity result crossed in the mail,a nice example of a question being answered beforeit is received. I should perhaps also mention thatthis collaboration was one of the most excitingmathematical events of my own life and, I think,meant a lot to Fritz too. On the day when wesent off the final manuscript, we celebrated witha dinner together with our families in a fancy

Don Zagier and Friedrich Hirzebruch at theLeonhard Euler Congress, June 10–12, 2007,St. Petersburg.

restaurant at which Barbara famously reacted tothe bill by computing how many portions of Frenchfries she could buy with the same money.

In later years Fritz worked on many othertopics at the interface between number theory andtopology that for lack of space I will not describein detail. A prime example arose in the late 1980swhen Ochanine and Witten introduced ellipticgenera, which attach modular forms to manifolds.Not surprisingly, Fritz was very interested in thisdevelopment and wrote some beautiful papersand a book [6] (joint with his students Th. Bergerand R. Jung) on this topic. Other topics includedthe study of Fuchsian differential equations (aloneand in collaboration with Paula Cohen) alludedto above and his really beautiful work applyingthe Miyaoka-Yau inequality and other deep resultsabout characteristic classes of surfaces to classicalquestions going back to Sylvester (1893) aboutconfigurations of points and lines in the plane [7].

Final RemarksFritz Hirzebruch was the most important person inmy life outside my own family, and it is impossiblefor me to say everything he meant to me. It is hewho taught me how to be a mathematician, butmore important than this were his human qualities:his empathy, his gentleness with everybody, andhis ability to correct without criticizing. His moralrectitude and the straightness of the path hefollowed made one wish to also act in a way hewould approve of. In many almost invisible ways,he made the people around him slightly betterpeople and the world around him a slightly betterworld.


Friedrich Hirzebruch giving a talk in Berlin, 1980.

References[1] F. Hirzebruch, The signature theorem: Reminiscences

and recreation, in Prospects in Mathematics, Annals ofMath. Studies, 70, Princeton University Press, Princeton,1971, pp. 3–31.

[2] F. Hirzebruch and D. Zagier, The Atiyah-Singer The-orem and Elementary Number Theory, Math. LectureSeries, 1, Publish or Perish, Boston, 1974, 262+xii pages.

[3] F. Hirzebruch, Hilbert modular surfaces, EnseignementMath. 19 (1973), 183–281. (Appeared in the same year asMonographie No. 21 de l’Enseignement Mathématique,103 pages.)

[4] F. Hirzebruch and D. Zagier, Intersection numbers ofcurves on Hilbert modular surfaces and modular formsof Nebentypus, Invent. Math. 36 (1976), 57–113.

[5] F. Hirzebruch, Elliptic genera of level N for com-plex manifolds, in Differential Geometric Methodsin Theoretical Physics (Como, 1987), Kluwer, 1988,pp. 37–63.

[6] , Notes written by Th. Berger and R. Jung, Man-ifolds and Modular Forms Friedr. Vieweg & Sohn,Baunschweig, 1992..

[7] , Arrangements of lines and algebraic surfaces,Arithmetic and Geometry, Vol. II, Prog. Math., 36,Birkhäuser, Boston, 1983, pp. 113–140.

[8] N. Schappacher, Max-Planck-Institut für Mathematik—Historical Notes on the New Research Institute at Bonn,Mathematical Intelligencer 7 (1985), 41–52.

Yuri I. ManinFriedrich Hirzebruch was eighteen years old inDecember 1945 when he started his study atMünster University. Reminiscing about this timein 2009 he wrote:

Wenn ich damals einen kurzen Lebenslaufabgeben musste, dann enthielt er immer denSatz “Von Mitte Januar 1945 bis zum 1. Juli1945 durchlief ich Arbeitsdienst, Militärund Kriegsgefangenschaft.” (In those days,whenever I had to supply a short CV, it

Yuri I. Manin is professor of mathematics at the MaxPlanck Institute for Mathematics. His email address [email protected].

always contained the sentence: “Betweenmid-January 1945 and July 1, 1945, I servedfatigue duty, military duty, and was detainedas prisoner of war.”)

This statement puts a double distance between thepresent day and painful youth of war years, defiesany attempt to express this pain more eloquently,and does so by silence.

After settling in Bonn in 1956, Hirzebruch putgreat effort into the restoration of the Europeanmathematical community, destroyed, like so manyother institutions and lives, by the war. Thebrilliant idea of annual Arbeitstagungen and laterthe founding of the Max Planck Institute forMathematics (MPIM) bore rich fruit. Hirzebruchstruggled for the new Europe, like Henri Cartan inFrance, using all the influence that he possessedas an internationally renowned researcher.

My first close contact with Fritz and IngeHirzebruch came in 1967. I spent six weeksat the Institut des Hautes Études Scientifiquesin Bures–sur–Yvette, where Grothendieck taughtme the “fresh-from-the-oven” project of motiviccohomology. After that I got permission and aGerman entry visa, which enabled me to visit Bonnand to participate in the Arbeitstagung on my wayback to Moscow.

The blissful stress of study with Grothendieckand of Paris magic did something to my body, butin Bonn, Inge and Fritz treated me as their son andhelped my healing. Their kindness and generosityforever remain in my memory.

In 1968 an abrupt end came to these buddingdirect contacts between mathematicians of WesternEurope and their colleagues in the Soviet Unionand Eastern Europe. The next generation, comingafter Hirzebruch’s and then mine, had differentconcerns. As one of those young men recalledrecently, “We thought it highly likely we would beblown off the planet, and that, somehow, it was upto us—children after all—to prevent it.” 1

We were not blown off the planet. The existingorder of things again started to seem stable—orstagnating. I had not the slightest premonition thatthis epoch would also pass during my life and thatalmost a quarter of a century afterwards I wouldmeet Fritz again and become a colleague of his inthe MPIM. After 1990 and the fall of the Berlin Wall,Friedrich Hirzebruch, through an immense effort,helped many mathematicians from East Germanyfind jobs and continue their scientific lives in anew environment.

Mathematics is a travail de longue haleine.Leonard Euler (born in Basel and working inSt. Petersburg), inspired perhaps by the seven

1“It was about Cold War”, letter by Geoffrey Wells, LRB,5 April 2012.


bridges of Königsberg (mostly destroyed by bomb-ings in 1944 and 1945), discovered the notion ofEuler characteristic of a graph. This notion hadevolved during two centuries and by the timeFriedrich Hirzebruch was maturing as a mathe-matician, found reincarnation as an alternatingsum of dimensions of cohomology groups of(invertible) sheaves on an algebraic manifold. Thecelebrated Riemann–Roch–Hirzebruch formula of1954 (described by Atiyah) expressed this numberthrough geometric invariants of the base, cruciallyusing the Todd genus, discovered by J. A. Toddfrom Liverpool. At the first Arbeitstagung in 1957,Alexander Grothendieck, son of a Russian anarchistand an eternal exile in France and everywhere else,presented its great generalization.

Perhaps the Riemann–Roch–Hirzebruch–Grothendieck Theorem, which fused andcrowned efforts of a dozen great spiritsfrom all corners of Europe, deserves to be put onthe flag of the united Europe more than any othersymbol.

Gerard van der GeerThe first time I saw Hirzebruch was when he visitedmy thesis advisor, Van de Ven, at Leiden University,where I was a Ph.D. student. I got to know himslightly better when Van de Ven took me to Bonn,where we visited Hirzebruch for a few days in 1974to discuss Hilbert modular surfaces. At the time Iwas quite surprised to see how seemingly relaxedhe was, though he must have been extremely busyat the time. He took ample time to talk to us,and the same happened about a year later when Ivisited him alone.

He invited me as a postdoc in 1977 to the Son-derforschungsbereich Theoretische Mathematik,the predecessor of the present Max-Planck-Institut.Shortly after my arrival there we celebrated hisfiftieth birthday, the beginning of a long series ofsimilar celebrations.

What struck me when we discussed mathematicswas his instinct for the beauty of mathematics,and in fact all that he did bore the hallmark ofelegance. The charming way he could lead theprogram discussion for the Arbeitstagung wasanother instance of this.

During my time in Bonn he would often inviteme to his office and ask my opinion or even advice.In the beginning this surprised me, though I foundout that weighing opinions of various people waspart of his way of forming an opinion or coming to adecision. This applied especially to his preparationfor the Arbeitstagung, where in the month before

Gerard van der Geer is professor of mathematicsat the Korteweg-de Vries Institute for Mathematics. Hisemail address is [email protected].

The Fine Hall faculty in 1956. First row: Wigner,Tucker, Bargmann. Middle: Hirzebruch, Fox,Moore, Steenrod, Feller. Back: Spencer, Church,Artin, Wilks, Milnor, Tukey.

he was collecting suggestions for speakers andtitles. It was surprising to see how he managed,seemingly without effort, to have the outcome ofthe public program discussions be guided by theideas he had assembled.

In 1981 he invited me to join him for a SummerAcademy of the Studienstiftung in Alpbach in theTirolean Alps in Austria. This was a two-weekseminar where we would work with twenty-fivevery bright German students on a topic, studyingin the morning, hiking in the afternoon. This wasthe first of seven such summer schools, the lastone held in 1997. That was a fantastic experience,and during these seven summer schools I got toknow Hirzebruch very well. From an awe-inspiringand paternal mathematician he became a very goodfriend. Professor Hirzebruch became Fritz. Howdifficult it was in the beginning to use “Du” insteadof “Sie”! He enjoyed these days enormously andoften in the later years would recall the happy daysin Alpbach.

The charming way in which he would leadthe summer school and discourse with studentsonly fed my admiration for him. We would havelectures by students and ourselves in the morningand go hiking the whole afternoon. After dinnerthere would be interdisciplinary talks, becausethe Sommerakademie comprised groups fromvarious disciplines, ranging from astronomy, say,to linguistics. After those talks we would gather inthe Roter Salon of the Böglerhof Hotel for a beerand discussions with the students. Around 11 p.m.we would change location with our group to thedisco, where we would dance—yes, Fritz too!—andcontinue to discuss as far as the noise admitted,and where we awarded drinks for prize-winningsolutions to the exercises and problems. In theearly hours of the morning we would return to


At the third Arbeitstagung in 1959: Thom,Grothendieck, von Viereck, Milnor, and

Dombrowski.

our rooms and decide whether quick preparationfor the lecture the next morning was better donethen or after sunrise. To do things efficiently wasanother lesson he taught by example.

In the later years, besides recalling Alpbach,he would often refer to the “golden fifties”, theyears he spent in Princeton, where he proved hislandmark Riemann-Roch Theorem. For somebodywho lived as a young man in the horrible Nazi time,those years must have been paradise. Inspiredby this and his desire to rebuild mathematicsin his own country, he formed the idea to havesuch an institute in Germany. That he succeededin creating in Bonn one of the world’s bestmathematical research institutes is just one proofof his many talents. That it possesses such apleasant atmosphere is another.

I often noted how he exerted a positive influenceon other people just by being there. Or, evenwithout him being there: I often noticed that facedwith a difficult situation or decision, I asked myselfhow Fritz would have acted in such a case, andhow much it helped. He was a wonderful person.

John MilnorIn 1955–56 Fritz and I were fellow assistantprofessors at Princeton. I don’t believe that I reallygot to know him that year. However, I was certainlyvery much impressed by his mathematics. HisNeue topologische Methoden in der algebraischenGeometrie had just appeared and was extremelyexciting.

This was a time when many radically new ideaswere beginning to completely transform the fieldof topology. Both Norman Steenrod’s theory ofcohomology operations and Jean-Pierre Serre’sthesis, which brought the previously intractable

John Milnor is professor of mathematics at Stony BrookUniversity. His email address is [email protected].

study of homotopy groups under control, providedpowerful new algebraic tools for studying homo-topy theory. René Thom’s ingenious geometricarguments exploited the work of both Serre andSteenrod to provide a completely new way of study-ing smooth manifolds. Hirzebruch’s book addeda whole new dimension, grounded in algebraicgeometry and the study of complex manifolds.His theory of multiplicative sequences providedan important complement to Thom’s work. Forthe first time, this brought Bernoulli numbersinto topology, where they are related not onlyto groups of differentiable spheres but also tostable homotopy groups of spheres and the Adamsconjecture.

I certainly got to know Hirzebruch well in thefollowing years. He jumped from an assistantprofessorship in Princeton to a full professorshipin Bonn and almost immediately established theannual Arbeitstagung, a true stroke of genius:It provided an annual get-together for mathe-maticians from all over Europe and from the USwho wanted to follow the latest developments intopology and geometry. The relaxed atmosphereand low-keyed organization were a marvel ofbenevolently supervised democracy. The annualexcursion on the Rhine provided a special opportu-nity for interaction. Visits to Bonn in the followingyears were always a pleasure, and Fritz and Inge’shospitality was much appreciated.

Mina TeicherI want to start with the day that Hirzebruch receivedthe Wolf Prize. It was on May 12, 1988, in theKnesset (the parliament) of Israel in Jerusalem—avery structured ceremony in the presence of thepresident of the country and five hundred guests.

Hirzebruch was sixty years old at the time hewas awarded the Wolf Prize. He was the youngestperson and only the second German to havereceived it.

Two prizes in mathematics were awarded, andHirzebruch was chosen to respond on behalf ofhimself and the other laureate. He came to thepodium to deliver his speech. With his strong anddirect voice, he expressed his gratitude to the WolfFoundation for awarding them the prize.

He then added a few sentences on behalf ofhimself only. When he completed his speech, theaudience was dead silent for a few seconds, andthen with tears in their eyes they started to clap ina fashion that is usually not seen in the academicworld. They clapped and clapped more and more.He had spoken from his heart and had exposed hissoul:

Mina Teicher is professor of mathematics at Bar-IlanUniversity. Her email address [email protected].


…As a professor at the University of Bonn,I am one of the successors of the famousmathematicians Felix Hausdorff and OttoToeplitz. Hausdorff committed suicide in1942, together with his wife, when deporta-tion to a concentration camp was imminent;Toeplitz emigrated to Israel in 1939 anddied there the following year. The memoryof these mathematicians is with me alwayson this trip.

In these three sentences he managed to createcontinuity between the mathematical communityin Bonn before and after the Nazis, to establishlinks between the Jews in Germany and the Israelisociety, and to penetrate the hearts of the listeners.

Hirzebruch’s first visit to Israel was in 1981 bythe invitation of Piatetski-Shapiro, my Ph.D. advisorat the time. He was widely welcomed for beingthe great mathematician he was, as well as forhis leadership role in reestablishing mathematicsin Europe after WWII. But it was only in the late1980s that he started to be actively involved in themathematical research infrastructure in Israel.

Hirzebruch had a fundamental role in the EmmyNoether Institute of Bar-Ilan University. Followinghis advice, we prepared an application for a jointGerman-Israeli Minerva center in mathematics.We named the center after one of the greatestscientists of the twentieth century and one ofthe first female mathematicians, the German-Jewish mathematician (who fled to the USA in theearly 1930s) Emmy Noether. The application wassubmitted to the Minerva Foundation (a subsidiaryof the Max Planck Society), and in 1991 it wasapproved.

The inauguration ceremony of the Emmy NoetherInstitute took place in the house of the Israeliambassador in Bonn in July 1992. A binationalBeirat was appointed by the Bar-Ilan Universityand the BMBF. Hirzebruch was appointed by thedeputy minister of the BMBF as the chairman ofthe Beirat. He served as chairman for twelve years,a role he took on with great commitment. Hecontributed his valuable time (when appointed hewas still the director of MPIM in Bonn), his endlessenergy, his deep wisdom, and his vast experienceto the success of the center. In 2000 he receivedan honorary degree from BIU for his contributions.

Two major conferences in algebraic geometrywere held in the center in his honor. “Hirz 65” washeld in May 1993 and attracted an internationalaudience, including Fields medalists, directorsof research institutes worldwide, collaborators,former students—all came to pay respect. “Hirz 80”was held in May 2008 and was one of the lastbig conferences he attended. Again, five Fieldsmedalists attended, four Wolf Prize winners, andmore. Fritz came, accompanied by his wife, Inge,

At the award of the Stefan Banach Medal inWarsaw, October 26, 1999.

and his son, Michael. He was very pleased to meetold friends, attended ALL the talks, and enjoyedthe celebrations and the tours to the Golan Heightsand Jerusalem. Unfortunately, on the day beforethe last, during a tour in the Western Wall caves, hefell and broke his leg, but then was most concernedthat the conference was continuing as planned andkept apologizing for disturbing the agenda!

I was reflecting on the times (circa 1975) when,for me, the name Hirzebruch was the title ofa yellow book Topological Methods in AlgebraicGeometry, which, as part of my M.Sc. studies, Ihad to read and then give a shorter proof of theHirzebruch-Riemann-Roch Theorem in the specialcase n = 3. In spring 1988 I participated in aspecial semester in Bonn on algebraic surfaces,coorganized by Hirzebruch. Ten years later, insummer 1998, we organized together a specialsemester in the MPI, “Topology of AlgebraicVarieties”. He gave brilliant lectures, presentingcomplex geometrical structures in a simple andnatural way, demonstrating beautiful examples.I learned more about the skills that helped himreestablish the mathematical community in Europeafter the war. He never forgot that mathematicsis made of—and by—mathematicians. Listeningand attending to everybody’s needs, “combining”people, making his own friends into friends of oneanother. A man who followed his values with noexception. A noble man.


Kenji UenoWhen I was a graduate student of the University ofTokyo, Professor Kodaira often told us his memo-ries of his time in the US. The most unforgettableone is the following.

A young German mathematician came tothe Institute for Advanced Study. He calcu-lated the Todd genus of several algebraicmanifolds. I wondered what he was reallytrying to prove. But suddenly he provedthe Riemann-Roch theorem for all algebraicmanifolds. In that summer I wrote a let-ter to J-P. Serre that Hirzebruch provedthe Riemann-Roch theorem, while I couldonly prove that the Hodge manifolds werealgebraic.

Kodaira had proved the Riemann-Roch Theoremfor algebraic threefolds by using the theory ofharmonic integrals. He was trying to prove thetheorem step-by-step as he could not foresee thatone could prove it in a single step. Hirzebruchused cobordism theory to prove it. This was acompletely new approach and paved the way forthe Atiyah-Singer index theorem.

In December 1971 Kodaira told us that Hirze-bruch would visit Japan the following Februaryand deliver a series of talks at the University ofTokyo. So many times we had heard the nameof Hirzebruch from Kodaira and also consultedhis famous book Topological Methods in AlgebraicGeometry, but we had never expected that wewould have a chance to meet him in Japan. Hislectures were the IMU lectures, which means thatthe IMU supported his visit to Japan. At that timethe Japanese economy was growing but still notstrong enough, so that the Japanese governmentsupported universities very little. We had theresearch grants of the Ministry of Education, butstrangely enough it was forbidden to use themfor either inviting foreign scholars or for visitingforeign institutes. For that we had to apply foranother grant, which was quite difficult to get. Thisrestriction was continued for a long period andonly removed around fifteen years ago. Therefore,at that time it was almost impossible to inviteforeign scholars with Japanese funds.

In January 1972 the title of his talks wasannounced. To my surprise his talks were onthe resolution of cusp singularities and Hilbertmodular surfaces. In February Hirzebruch cameto Japan. The lecture room was full of people. Histalk was so clear and beautiful that I thought Iunderstood every detail. Of course, this was his

Kenji Ueno is director of the Seki-Kowa Institute ofMathematics, Yokkaichi University. His email address [email protected].

magic, and later I realized that I had missed manyimportant points. In his lectures he posed severalexercises and problems related to the subject.Since the classification theory of algebraic andanalytic surfaces was popular among us, some ofhis problems were not difficult. After the lectureKodaira introduced us to Hirzebruch. Before hisnext talk I visited him and showed him answersto some of his problems. He was pleased andencouraged me to study further. At that time I wasinvited to Mannheim University, and Hirzebruchwas kind enough to give me suggestions. Heasked me to attend the Arbeitstagung in Bonn andpromised to send an invitation letter.

After Tokyo he visited Kyoto and gave severallectures. Many young active Japanese mathemati-cians attended his lectures and solved severalproblems posed by him. He asked them to apply tothe SFB 40 in Bonn University. Soon some of themgot invitations to Bonn. At that time in Japan therewere several programs to visit foreign universitiesas graduate students but very few possibilitiesto visit foreign countries as researchers, so thathis advice was very helpful for young Japanesemathematicians.

At the beginning of that March I went toMannheim University. In June I received an invita-tion letter to the Arbeitstagung from Hirzebruch.He never forgot his promise. The Arbeitstagungwas very interesting. I met there many mathemati-cians whose names I knew only from their papers.In October 1972 I was invited to the SFB 40 andstayed there half a year. Then I came back toTokyo to get my Ph.D. and went back to Bonn thefollowing spring.

At Bonn University I got a room in the samebuilding where Hirzebruch had his office. Almostevery day I saw him working hard not only onadministrative works but also discussing mathe-matics with students and many mathematicians.He was busy enough, but he always attendedimportant seminars and colloquium talks. Also,at teatime he came down to the tearoom anddiscussed mathematics with us. He was very kindto answer our questions and always encouragedus to do mathematics. If the questions were notin his fields, he introduced us to the appropriatemathematicians.

In the 1970s the only possible way to inviteforeign scholars to Japan was to use the JSPS(Japan Society for the Promotion of Science) pro-gram. Hirzebruch’s second visit to Japan wasunder this program. He stayed mainly in Kyotoand had discussions with many young Japanesemathematicians. After that he visited Japan severaltimes. He always advised young mathematiciansto apply to the SFB 40 and later the Max PlanckInstitute for Mathematics. Following his advice,


many young Japanese mathematicians applied toBonn and many of them had chances to visit Bonn.They could not only concentrate on their researchbut also collaborate with foreign mathematicians,often in different fields. During the mid-1990smore than one hundred Japanese mathematiciansvisited Bonn.

In 1997 the Mathematical Society of Japan(MSJ) awarded the Seki-Kowa Prize to Hirzebruchfor his outstanding contribution to the Japanesemathematical community in giving many youngJapanese mathematicians the opportunity to studyand collaborate with mathematicians from all overthe world. At the same time the MSJ had applied forthe Order for Hirzebruch through the Ministry ofEducation. The Japanese government awarded himthe Order of the Sacred Treasure, Gold and SilverStar, which was the highest order for foreignersexcept politicians and diplomats. In November 1997the ceremony was held at the Ministry of Education,and the vice minister awarded him the order. Inthe ceremony Hirzebruch answered that he wouldaccept the order on behalf of all the Japanese andGerman mathematicians who had once stayed inBonn and collaborated together, the secretaries andstaff of the SFB 40 and the Max Planck Institute forMathematics who helped their activities, and theDeutsche Forschungsgemeinschaft and Max PlanckGesellschaft for supporting them financially. Hisspeech deeply impressed officials of the Ministryof Education who were present at the ceremony.

Hirzebruch loved mathematical talks, and histalks were always clear. Once when he visitedKyoto, I asked him to give a lecture for high schoolstudents. At that time every two weeks I organizeda mathematics lecture for high school students atthe Kyoto University. He gave a beautiful lecture onthe regular icosahedron. The high school studentsenjoyed his talk and were impressed by how deeplyhe loved mathematics.

It is really sad that I cannot talk with himanymore. He always talked with a gentle smile andnever failed to encourage us to do mathematics.I am quite sure that his warm memory and hisencouragement to do mathematics will survive inall mathematicians who once met him.

Graeme SegalThe month I spent in Bonn as a second-yeargraduate student in the autumn of 1964, when Ifirst encountered Fritz Hirzebruch, remains one ofmy most vivid memories. When I think of all hemust have been involved in I am humbled to thinkof his kindness in spending so much time, not just

Graeme Segal is professor of mathematics at All SoulsCollege. His email address is [email protected].

Shiing-Shen Chern, Samuel Eilenberg, andFriedrich Hirzebruch, 1956 in Mexico.

in talking to me about my work but in makingsure that my wife, Desley, and I were at home andhappy in what was for us a strange new world.

For a young Australian, Germany then was anoverwhelmingly formal place. After two years I hadjust about become accustomed to the increasedformality of England, but in Germany it attainedanother level. In retrospect I see that the countrywas poised on the brink of a great change in socialstyle, and I think this was essential to Fritz’s magic.On one side he was the perfect German professorof the old school: although only thirty-eight, hehad already served a term as dean of the Facultyof Sciences and was a figure of manifest authority.(My status rocketed with the very genteel elderlylady in whose house we were lodging when oneday the Herr Professor arrived in person to pickme up.) He gave wonderful lectures, but what Imost remember about them was his use of theGerman language—his long, elegant, articulatedsentences in which every clause clicked faultlesslyinto place. Mathematicians had long since ceasedto lecture like that in English; I wonder whether itstill happens in Germany?

But there was another side, as Fritz had becomepart of the Princeton mathematical world with itsvery different manners. He had attracted JacquesTits to Bonn as his closest colleague, and theycalled each other “Fritz” and “Jacques” in public,which was constantly remarked upon to me—sometimes with a definite hint of disapproval—bythe Assistenten in the department. (Peter Pearsand Julian Bream came and gave a recital in Bonnat the time, and the informality of their dressand demeanor on stage also caused a flutter.)I had no idea then of Fritz’s great achievement


in rebuilding German mathematics from the late1950s on, but it seems to me that a big partof his success must have come from his abilityto shine in two, at least, very different styles atonce, with always just the tiniest suggestion ofironical detachment from each. He evidently hada remarkable ability to see what was needed andwhat was possible for the mathematical world anda perfectly pragmatic way of pursuing it, withalmost nothing showing of amour propre. Foreignmathematical visitors like me saw little of his“Germanic” style beyond the legendary clockworkperfection of the arrangements for the annualArbeitstagung, but, looking back, I marvel at how,in gathering together such an outstanding panoplyof diverse mathematicians from all over the worldin his institute, he managed to seem—and indeedto be—so uniformly benevolent, sometimes inthe face of much that was surely alien and evenoffensive to his own nature. I sometimes felt he hada special affinity with the Japanese visitors, whosereserved manners had something in common withhis own.

I shall not try to talk about Fritz’s mathematicalwork, as I don’t feel the best-placed person todo that. I always admired his taste for beautifulconcrete geometric examples and all he couldextract from them, though I never myself workedquite in his area. But I cannot resist mentioningone of his earliest achievements. When I became agraduate student in 1962 the first suggestion madeto me by my first supervisor, Sir William Hodge,was to try to read Hirzebruch’s Neue topologischeMethoden in der algebraischen Geometrie, whichhad just appeared. It was far above my head then,but it begins with the piece of algebra whereby aformal power series gives rise to a multiplicativecharacteristic class for vector bundles. I wasbewildered but tremendously intrigued by this,and I remember struggling with the proof that theonly series f (x) such that the coefficient of xm inf (x)m+1 is 1 for all m is the famous one

f (x) = x/(1 − e−x)

which defines the Todd genus. I can only say thatalmost everything I have ever thought about inmathematics, in K-theory, index theory, elliptic co-homology, deformation quantization, or whateverhas involved what I learned then.

Stanisław JaneczkoThe political changes in Poland in 1990 (EasternEurope) caused many necessary reorganizationalefforts. One of the institutions in trouble was theStefan Banach International Mathematical Center

Stanisław Janeczko is professor of mathematics at War-saw University of Technology. His email address [email protected].

(BC) in Warsaw. It needed a new basis and structurefor a secure and prosperous existence. FriedrichHirzebruch, being at that time president of theEuropean Mathematical Society, offered his helpand great involvement to reconstruct the BC andto form new conditions for European cooperation.During the meeting of the Executive Committeeof the EMS in Budapest, the letter of Intent onCooperation between the Institute of Mathematicsof the Polish Academy of Sciences and the EMS wasdiscussed and signed in order to secure the fruitfulcontinuation of the activity of the Stefan BanachInternational Mathematical Center. It was PresidentFriedrich Hirzebruch’s personal effort, made withcare and concern, for the fruitful future activityof the BC. We found him enormously friendly anddeeply involved in any possible undertakings. Hispragmatism, careful attitude, and firm supportallowed all the good working elements of theformer activity of the BC to be maintained.

The agreement was signed on 30 March 1993,and the first meeting of the new Banach CenterCouncil, with three representatives of the Execu-tive Committee of the EMS, four representativesfrom Poland, and three representatives from thefounding countries, was organized on 25 October1993. Friedrich Hirzebruch agreed to serve as itschairman. The council and mainly the chairmanstarted to work very hard to adapt the BC to the newbut still unstable reality. As a master and friendof all of us, Hirzebruch visited the Banach Centerevery year and taught us how to be supportive andreally helpful to other colleagues; how to be honest,objective, constructive and not discouraging toother applicants, how to improve the atmospherefor successful research, how not to be “divisive”and troublemaking, and how to be gentle andresponsible in formulating opinions about others.He taught us that mathematics is unity, that thereare no better or worse branches of mathematics,but that it is engagement in research and strivingfor perfection that are of key importance. He wasalways supportive of the director of the institutein the latter’s difficult fights and efforts. He was anexcellent advisor during my period of directorship,always patient and understanding, friendly, withimpeccable manners. He made an enormous effortto help the institute in its fight to maintain thebasic properties. Under his chairmanship the firsteight years, despite the material difficulties whichwe all suffered in Poland, the Banach Center wasvery successful and prosperous.

In 1997 Friedrich Hirzebruch became a memberof the Polish Academy of Sciences. The next yearan Algebraic Geometry Conference in Honor ofF. Hirzebruch’s Seventieth Birthday was organizedin the Banach Center in Warsaw. It was an unusualevent with extreme importance also for Polish


mathematicians. Then in 1999 he got a prestigiousaward of the Polish Academy of Sciences—theStefan Banach Medal.

Since my first visit to the Max Planck Institutefor Mathematics in Bonn in 1984, through ourdaily meetings and walks along the paths in thepleasant neighborhood of the institute to my lastvisit there in November 2011, I experienced Fritz’sgenerous, warm, and extremely eager ideas andadvice on mathematics as well as on everyday life.He was always so pleased with mathematicians’new achievements and at the same time deeplyworried about his colleagues’ material status andthe financial conditions of mathematics in general.He showed us that we have to be extremely carefulnot to lower the value of mathematics and not toisolate it from the global efforts of mankind.

It was an extreme pleasure and satisfaction forme when he agreed to come to Warsaw in May2012 to celebrate his birthday, to meet with allthe Polish friends and former scholars of MPI. Wewere very happy to prepare this event to thankhim for all that he had done for the BanachCenter, the Institute of Mathematics, and Polishmathematicians. Unfortunately, a few days beforethe symposium started, Fritz had an accident athome and was not able to come. The letter, perhapshis last letter (which can be found on the websiteof the Banach Center) was brought to us by his son,Michael, and his daughter-in-law, Anne Hirzebruch.It was the most meaningful gift which we neverexpected to get—to be in Friedrich Hirzebruch’sgreat mind and soul till the last days of his life.

Jean-Pierre Bourguignon

Some Personal RecollectionsPersonally, over the last forty years, I owe a lotto Friedrich Hirzebruch for his unfailing supportand continuous inspiration. I met him in Bonn in1970 while I was visiting Wilhelm Klingenberg asa very young researcher in differential geometry.It was really during the academic year 1976–77,spent in Bonn with my family as guest of theSonderforschungsbereich 40, that I got to knowhim better.

The Arbeitstagung, a major mathematical eventthat he organized with his Bonn colleagues formore than thirty years, offered each year in June abroad overview of the most exciting mathematicsof the time. It was an exceptional place to meetmathematicians of all sorts, famous and lessfamous, senior or just beginning. Like many youngmathematicians, I benefited a lot from it, directly

Jean-Pierre Bourguignon is professor of mathematics at the

Institut des Hautes Études Scientifiques. His email addressis [email protected].

Friedrich Hirzebruch lecturing on Chern classesat IHÉS, November 17, 2011.

through the new perspectives gained by listeningto the lectures and indirectly through the greatnumber of encounters, some of which had a greatimpact on my professional life.

He was always curious to know what kindof mathematics was on your mind and showedspecial interest in young mathematicians. Alsoworthy of remark was his determinedly proactiveattitude towards women mathematicians at a timewhen gender equality was not given much priority.Several women colleagues consider that they owehim a lot because of his continued support.

The numerous encounters with him that fol-lowed the wonderful year in Bonn gave me ampleopportunity to witness his many talents: as anoutstanding mathematician of course, but also asa remarkably clear lecturer, an efficient commu-nicator, and an exceptionally talented manager.Some of them were quite unexpected for me, suchas accompanying him to a press conference withGerman journalists to discuss the development ofmathematics in his country.

He was a great supporter of the collaboration be-tween the Institut des Hautes Études Scientifiques(IHÉS) and the Max-Planck-Gesellschaft (MPG). Herepresented the MPG on the board of directorsof IHÉS for several years. Both he, as director ofthe Max-Planck-Institut für Mathematik, and SirMichael Atiyah, as founding director of the IsaacNewton Institute in the Mathematical Sciences,endorsed immediately the idea of the EuropeanPost-Doctoral Institute (EPDI) which I proposed inthe fall of 1994 shortly after becoming the directorof IHÉS. Already in 1995 the three institutionscould join forces to get young postdocs to movearound Europe. For the inaugural ceremony in


Bures-sur-Yvette, Hirzebruch gave a very inspiringspeech on the role of institutes in mathematics.

A Very Early and Critical Involvement inEuropean Mathematical AffairsAll along his career, Friedrich Hirzebruch had alot of interactions with Henri Cartan, a dedicatedEuropean very early on: his first interaction wasin relation to Cartan’s efforts to renew contactbetween German and French mathematicians afterthe Second World War. Indeed, as early as November1946, Henri Cartan lectured in the Lorenzenhof inOberwolfach.

In this connection, Friedrich Hirzebruch wrotethe following: “The ‘Association Européenne desEnseignants’ (European Association of Teachers)was founded in Paris in 1956. Henri Cartan waspresident of the French section. As such he took theinitiative to invite participants from eight Europeancountries to a meeting in Paris in October 1960.Emil Artin, Heinrich Behnke and I were the Germanmembers. The second meeting of this committeewas in Düsseldorf in March 1962. As a result, the‘Livret Européen de l’Etudiant’ (European Student’sRecord) was published and distributed by theAssociation. The booklet contained a descriptionof minimal requirements for basic courses. It wassupposed to increase the mobility of studentsfrom one country to another. The professor of oneuniversity would mark in the booklet the contentsof courses attended by the student. The professorat the next university would then be able to advisethe student in which courses to enroll. The bookletwas not used very much.” This was indeed theearly form of the by now well-established “ErasmusProgram”.

A lot about their relationship can be learnedfrom reading the letter that Friedrich Hirzebruchwrote in 1994 to Henri Cartan on the occasion ofhis ninetieth birthday.

The European Council of Mathematics (EMC)opened the way to the European MathematicalSociety (EMS). The foundational meeting of the EMSwas held in October 1990 in Madralin and was notan easy affair, as opposite views on the structureof the EMS were presented by some delegations:should it be a federation of societies or a societywith individual members? Friedrich Hirzebruch,who had agreed to be considered as the first EMSpresident, led to success the rather tense meetingbehind closed doors between supporters of theconflicting positions. The next day the new societycould be created with statutes ensuring a goodbalance between individual members and membersocieties, a feature that still remains operationalto this day.

Under Friedrich Hirzebruch’s leadership, theEMS developed successfully. A lot had to be

achieved in a short time to take advantage ofthe dynamics that accompanied the creation ofthe society. Among milestones of his mandate,one can single out the setting up of the firstEuropean Congress of Mathematics in Paris in1992 and laying the groundwork for the Journalof the European Mathematical Society (JEMS) thatwas finally created in 1999.

To my great surprise, he asked me to becomehis successor as EMS president in 1994 to servefor the second term, 1995–98, another great honorthat he bestowed on me.

Final Visit to ParisIn November 2011 Hirzebruch came to IHÉS on theoccasion of a conference in honor of the centenaryof Shiing-Shen Chern, a close friend of his since1953, whom he described as “one of the greatestmathematicians of the 20th century [and] for me afatherly friend whom I owe very much.” He lecturedbrilliantly on Chern classes and was able to meetChern’s daughter, Mae Chern. At the end of hislecture, he told me, “I am afraid that this will bemy last visit to Paris.” It is very sad to remark thathe was indeed right.

Matthias Kreck

After my oral Vordiplom examination in the summerof 1968, I asked Hirzebruch whether I could attendhis seminar. “Of course, but it is rather difficult,”was his reaction. He was right; I was not preparedto follow that seminar. But it gave me the chance tobecome familiar with some of the other studentsand assistants in his group, such as Klaus Jänich,Erich Ossa, and Walter Neumann. A year later Iasked Hirzebruch whether I could write my diplomathesis with him and the answer was the same:“Of course.” Often when he supervised a diplomathesis, he asked one of his assistants to help him,which meant that he gave the assistant the majorpart of the responsibility, but he followed whatwas going on. In my case he asked Klaus Jänich tosupervise me and that was wonderful. At that timeJänich had started to systematically investigateinvariants which share a fundamental propertywith the most important manifold invariants,such as the Euler characteristic and the signature,namely, that they are invariant under cutting andpasting. In this way I became a friend of twoof Hirzebruch’s best friends: the signature andthe Euler characteristic. This has been a lifelongfriendship, with the signature in particular playinga role in many of my papers.

Matthias Kreck is professor of mathematics at the Haus-dorff Center for Mathematics. His email address [email protected].


Around 1969 Jänich received an offer fromRegensburg and asked me to join him there withthe aim of becoming his assistant and to writea Ph.D. with him. I felt very honored and agreed.Soon after my arrival in Regensburg both Jänichand I discovered that we had almost oppositepolitical opinions. This was a great pain for Jänich,so that he felt unable to continue working withme. To resolve the situation he wrote a letter toHirzebruch asking him whether he would be willingto take me back: “Perhaps you can deal with thisyoung man.” The answer: “Of course.” In this way Icame back to Bonn and became Hirzebruch’s Ph.D.student (in parallel with Don Zagier).

My Ph.D. time was difficult. Hirzebruch hadstarted his fundamental work on Hilbert modularsurfaces and gave a course about it. He asked meto take notes (which he used to write his beautifulEnseignement survey article). He didn’t give me aspecific problem to work on. Instead he pointed atcertain invariants occurring in connection with theresolution of singularities, which are some sort ofsignature defects, and suggested that I investigatethem further. He had proven some relations toL-functions and had a rather general conjecturewhich he mentioned. This was all too hard forme and I became rather frustrated. After about ayear, I decided to give up and to study Protestanttheology. When I told this to Hirzebruch, he lookedunhappy but said, “Of course, I understand,” atthe same time clearly thinking what would be thebest for me in this situation. He suggested that weshould meet in the near future again and look atwhat I had done so far. “Perhaps what you haveproven so far is already enough for a Ph.D.,” he said.This was extremely kind, since I had not done verymuch. We met, and he suggested certain thingswhich I could realistically work out, and we agreedon a plan which would lead to my Ph.D. within ayear.

I defended my Ph.D. in July 1972 and immedi-ately started an intensive three-month course inHebrew, which is a prerequisite to study theology.After this course I went to Hirzebruch to finally saygood-bye and thank him for all his support. He wasvery friendly and asked how the Hebrew coursewas going and whether I was looking forwardto starting my theological education. Then hecontinued: “I have just lost another assistant whobecame a professor at another university. Wouldyou be willing to be my assistant? Of course, I knowthat you need most of your time for theology, butI will give you enough free time. I also know thatyou don’t want to do mathematical research; thisis not necessary. I need your help with supervisingseminars, other students, courses, examinations,etc. Do you agree?”

Princeton, 1967, in the house of Louise andMarston Morse. Inge and Fritz Hirzebruch withtheir children, Ulrike, Barbara, and Michael.

I found this extremely generous and agreed.Whether he had an ulterior motive I really don’tknow. But in any case, his offer bore fruit. Aftermore than two years of not thinking about mathe-matical research, I found myself thinking about amathematical problem in my theology courses. Idid not tell this to Hirzebruch; this was just formy own personal fun. I did not even consult theliterature to find out what was known. Within a fewweeks I could solve half of the cases and told this toHirzebruch. He looked rather skeptical and asked,“Do you know that this is a well-known problemwhich had been attacked by mathematicians likeThom, Browder, and Sullivan?” I had no idea andimmediately said that my solution must be wrong.“Why? Let me hear,” was his answer. I explained theidea to him, and he said, “This has a chance; writeit up in detail. And if you can do the (much harder)remaining cases, this is your Habilitationsschrift.”

Based on this result, in 1976 I received an offer asprofessor at Wuppertal University, even before myHabilitation was formally finished. This was alsothe time when I had to begin my final examinationsin theology. I asked my friends:,“What should I do?The devil is tempting me.” Their reaction was, “Ifyou have not learned more about the devil, youbetter go back to mathematics.”

When I told Hirzebruch that I would accept theoffer from Wuppertal, I could see from his facethat he was pleased. But his reaction also made itclear that if I had decided to stay in theology, hewould have respected this equally.

Whenever I think about mathematics I aminfluenced by my teacher, Friedrich Hirzebruch.My strongest impression is of the enthusiasm


Inge and Fritz Hirzebruch on the boat trip toRemagen, Mathematische Arbeitstagung, 2007.

for mathematics he lived; this was a relaxed andfriendly enthusiasm. He was always open fordiscussions with me, either about questions I hador to share his mathematical ideas. I was impressedby his clear thinking and writing, his ability tobring different mathematical areas together, andhis deep insights. And also by his always visiblehumanism, both within and outside mathematics.I am so grateful that I had the chance to haveclose contact with Hirzebruch and that he alwaystreated me like his friend. I will never forget him.

Ulrike Schmickler-Hirzebruch

Since my childhood I have always been convincedthat I had a very special father. I am the eldest ofthree children and was born in 1953 (my fatherwas twenty-five years old) in Princeton, New Jersey,where my father spent a productive time at theInstitute for Advanced Study from 1952 to 1954.These were the formative years of his mathematicalcareer but also of his family life. My father metmy mother, Inge, also a mathematics student, forthe first time in Münster in 1947. After a fewoccasions where they saw each other again, itbecame clear that they should get married, whichthey decided on in 1952 before my father leftGermany for Princeton. He arrived in Hoboken,NJ, in August 1952 with the Holland-Americaoceanliner Ryndam; my mother came over inNovember on the Maasdam, and a few days laterthey married. With a settled married life, my fathercould be “free” to concentrate on his scientificwork.

There are a lot of letters from my parents totheir parents in Germany—in those days it was the

Ulrike Schmickler-Hirzebruch is senior editor at ViewegVerlag. Her email address is [email protected].

only way to communicate. I was amused when Irecently read in a birthday letter from my fatherto his father (Dec. 12, 1953):

…The last days were mathematically veryexciting, so exciting that I almost did noteat anything. I worked during the nightuntil 4:30, once until 6:30. Results cameout that I had already wanted to prove forquite some time, but that had seemed tobe very difficult.…(I assume that Ulrike getsthe credit. They say here that babies, aslong as they are less than 6 months old,know everything about mathematics, butthey can’t tell it.)

Starting from 1956, my father was a professorin Bonn. In 1957 he organized the first annualinternational meeting Mathematische Arbeitsta-gung. My siblings and I got to know this groupof mathematicians in our early childhood on theboat trips and at the parties held in our apartment.The Arbeitstagung was a special event also for us.My father’s enthusiasm was contagious, and hislove for mathematics included his mathematicalfriends.

We were “infected” at an early age by mathemat-ical problems: for example, on our Sunday walksin the forest: “Just think of a number, multiplyit by 4, add 10…and so forth. And what is yourresult?” Then he told every one of us the individualchosen number. Also at lunch he often surprisedus with beautiful simple problems, for example,with the trick of how to quickly multiply twonumbers between 10 and 20 (he had learned thistrick from his father, a math teacher and directorof a secondary school) and easy ways to constructmagic squares. This and much more happenedwithout any prior planning.

My father generally accepted what we did andhow we did it. Without much consideration, Idecided to study mathematics, and later so did mysister, Barbara. Because of the pleasure it gave me,I accepted the challenge, all the more so since atthat time there were only a few female studentsin the mathematics diploma program in Bonn. Mysister, Barbara, wanted to be a math teacher. Mybrother, Michael, wanted to study medicine, whichin later years would be very helpful for my father.

As a father he had authority through hisreliability, personal credibility, and his familiarsmile as a sign of his helpfulness and warmthtowards us. My father gave us cautious advice;sometimes, maybe, he was a bit too cautious. Hewas extremely balanced.

From my student days, I remember that myfather lent me some books that I saw on thebookshelves of his study. I could take them to myhome to work with them. There was discipline: he


would keep track accurately, and his name waswritten in the book; but from the expression inhis face I knew if I liked the book, I would own itsomeday. An arrow from his name to “Ulrike” thenindicated this transfer. For my father, books alsohad been important in his youth when he was luckyto learn from the math books his father owned.

I also remember some Sundays when I cameto my parents’ home an hour before lunchtime.I extremely enjoyed this hour with my fatherin his study. With a light hand he could showme the relationship between mathematical ob-jects and beautiful concrete examples. A smallhint was enough; I immediately understood hisexplanations—the combination of teacher andfather was wonderful.

Later, in my professional life, I met my fathera couple of times at mathematics meetings. In afew of his lectures, he deliberately included—justfor me, and unnoticeable to others—a phrase oran expression with which he wanted to tease me alittle.

My father had three children and six grandchil-dren. He took each of us seriously. We miss hisloving nature and how he could clearly expressindividual good wishes for us. His nice simplewords, enriched with caring humor, always hit thenail on the head.

My parents were a good team for almost sixtyyears. They complemented each other well. Mymother encouraged my father in his organizationalefforts, and she joined him on his ways, espe-cially, whenever she could, on trips and visits tomathematical places.

My mother sometimes refers to the statementby Mephistopheles in Goethe’s Faust II: “How aremerit and luck linked together.” His personalactions and a few good coincidences (for instance,being at the right time at the right place) shapedmy father’s life. We are very grateful that my fatherhad the good fortune to bring his life to a fullend and that, shortly before his death, he was stillable to give two mathematical lectures, which hadalways been such a great pleasure for him.

Note: All photos are courtesy of either theHirzebruch family or the Max Planck Institute.

Friedrich Hirzebruch’s grave on PoppeldorferFriedhof in Bonn.


Friedrich Hirzebruch (1927–2012) · 2014. 6. 26. · then the seasons in Germany and Australia would coincide, he was punished for insubordination. In the ﬁnal months of the war

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