Celebratio Mathematica

Friedrich Hirzebruch (uni­ver­sally known as Fritz) died on May 27, 2012, at the age of eighty-four. He was the out­stand­ing Ger­man math­em­atician of the post­war years who helped to re­store math­em­at­ics in his coun­try after the dev­ast­a­tion of the Nazi era. Ap­poin­ted at a very early age to a full pro­fess­or­ship at the Uni­versity of Bonn, he re­mained there for the rest of his very act­ive life and moved the cen­ter of grav­ity of Ger­man math­em­at­ics from the tra­di­tion­al cen­ters of Göttin­gen and Ber­lin to Bonn. The fam­ous “Arbeit­sta­gungs” (more prop­erly Arbeit­sta­gun­gen), which he es­tab­lished in Bonn in 1957, have been run­ning an­nu­ally or bi­an­nu­ally ever since and are a fo­cal point of math­em­at­ics world­wide. They car­ried his per­son­al im­print in their con­tent, at­tend­ance, and style, be­ing al­ways broad, top­ic­al, and in­form­al and do­ing much to edu­cate suc­ceed­ing gen­er­a­tions and to foster cross-fer­til­iz­a­tion. Many new ideas and col­lab­or­a­tions grew out of these en­coun­ters. An­oth­er last­ing con­tri­bu­tion to math­em­at­ic­al re­search in Ger­many and in the world is the Max Planck In­sti­tute for Math­em­at­ics, which he foun­ded, op­er­at­ing on the same lines and cre­at­ing bonds between math­em­aticians from many coun­tries, in­clud­ing those that were oth­er­wise cut off from the in­ter­na­tion­al scene.

Al­though Fritz, giv­en his mul­tiple roles, re­tired sev­er­al times, he re­mained act­ive till the very end and was pre­par­ing to at­tend a con­fer­ence in his hon­or in Po­land when he was struck down.

In this in­tro­duc­tion we will give an over­view of Fritz’s life and of some of his most im­port­ant achieve­ments. More de­tailed ac­counts will then fol­low in the in­di­vidu­al art­icles by the two co­ordin­at­ing ed­it­ors, with the one by Atiyah con­cen­trat­ing on the work in to­po­logy and the years be­fore 1970, and the one by Za­gi­er on the work in num­ber the­ory and the years after 1970. The sub­sequent art­icles by the in­vited con­trib­ut­ors de­scribe fur­ther as­pects of his per­son­al­ity, his sci­entif­ic work, and the role that he played in the math­em­at­ic­al lives of many in­di­vidu­als, or­gan­iz­a­tions, and coun­tries. \[\ast \qquad\qquad \ast \qquad\qquad \ast\]

Friedrich Ernst Peter Hirzebruch was born on Oc­to­ber 17, 1927, in Hamm, Ger­many, to Dr. Fritz Hirzebruch and Martha Hirzebruch (née Holtschmit). His fath­er, who was the head­mas­ter of a sec­ond­ary school and him­self an in­spir­ing teach­er of math­em­at­ics, gave him his first in­tro­duc­tion to the sub­ject — in­clud­ing, when he was nine years-old, the proof that \( \sqrt{2} \) 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 and he was nev­er sent in­to com­bat, be­ing as­signed in­stead to an an­ti­air­craft bat­tery with the task of com­put­ing ar­til­lery tra­ject­or­ies. He was even able to at­tend some sci­entif­ic courses, though when his com­mand­ing of­ficer asked him on one such oc­ca­sion to con­firm that winter and sum­mer are caused by the earth’s vary­ing dis­tance to the sun and Fritz dared to con­tra­dict him, point­ing out that then the sea­sons in Ger­many and Aus­tralia would co­in­cide, he was pun­ished for in­sub­or­din­a­tion. In the fi­nal months of the war the Amer­ic­ans put him in­to a pris­on­er of war camp, and even there he man­aged to do math­em­at­ics (partly on toi­let pa­per, still pre­served today). He was re­leased in Ju­ly 1945 and entered Münster Uni­versity that winter.

The city and the uni­versity lay in ru­ins and con­di­tions were very dif­fi­cult, with lec­tures be­ing held only at long in­ter­vals, but he had very good teach­ers, es­pe­cially Hein­rich Behnke, from whom he learned the func­tion the­ory of sev­er­al com­plex vari­ables, and Behnke’s as­sist­ant, Karl Stein, a former pu­pil of his fath­er’s. His third teach­er was Heinz Hopf, a Ger­man who had gone to Switzer­land be­fore the Nazi takeover and who in­vited the young Fritz, first to be his house guest and then for one and a half years to be a re­search stu­dent at the ETH in Zürich. Fritz re­turned to Münster with the es­sen­tials of a beau­ti­ful doc­tor­al thes­is about the res­ol­u­tion of cer­tain sin­gu­lar­it­ies in com­plex sur­faces. Already this earli­est work showed the char­ac­ter­ist­ics of all of his math­em­at­ics: el­eg­ance and brev­ity of thought and ex­pos­i­tion, an ef­fort­less syn­thes­is of soph­ist­ic­ated the­or­et­ic­al ideas with in­sights in­spired by non­trivi­al con­crete ex­amples, and the fu­sion of ideas from ana­lys­is, to­po­logy, and num­ber the­ory.

In 1952 came the de­vel­op­ment that was not only to be a turn­ing point in Fritz’s math­em­at­ic­al ca­reer but, as it tran­spired, to have a ma­jor in­flu­ence on the later de­vel­op­ment of math­em­at­ics in Ger­many and in Europe: he was in­vited to the IAS in Prin­ceton, where he re­mained for two years. At the IAS, he came in­to con­tact with many of the most bril­liant math­em­aticians and most ex­cit­ing new ideas of the peri­od and where he made the two dis­cov­er­ies with which his name is most strongly as­so­ci­ated: the Sig­na­ture The­or­em and the Hirzebruch–Riemann–Roch The­or­em. Those years and also the early years in Bonn, when the core of Fritz’s re­search was still in to­po­logy and its ap­plic­a­tions to al­geb­ra­ic geo­metry, will be dis­cussed in de­tail in the con­tri­bu­tion by Mi­chael Atiyah.

This peri­od also in­cluded three ma­jor events in Fritz’s per­son­al life: his mar­riage to Inge Spitzley in Au­gust 1952 just be­fore tak­ing the boat to Prin­ceton, and the birth of his first two chil­dren, Ul­rike (1953) and Bar­bara (1956). His third child, Mi­chael, was born a little later, in 1958. Inge, known to and loved by count­less math­em­aticians, was a big part of everything he built up dur­ing his life. Both of his daugh­ters later stud­ied math­em­at­ics and even­tu­ally worked in re­lated areas (Ul­rike in math­em­at­ic­al pub­lish­ing and Bar­bara as a school­teach­er), while Mi­chael was to be­come a doc­tor. Ul­rike’s con­tri­bu­tion to this art­icle gives us a vivid pic­ture of Fritz as a fath­er.

When Fritz re­turned to Ger­many in 1956 to take up his du­ties at his new chair in Bonn, he had a clear am­bi­tion and a mis­sion: to es­tab­lish a cen­ter that would at­tract math­em­aticians from all over the world. After the First World War, Ger­man math­em­at­ics had been os­tra­cized by the in­ter­na­tion­al com­munity, led by France. This las­ted for many years and em­bittered re­la­tions. For­tu­nately, the 1945 gen­er­a­tion of French math­em­aticians, led by Henri Cartan, was more en­lightened, and pre­war math­em­at­ic­al friend­ships were rap­idly re­newed. The Münster school mem­bers un­der Hein­rich Behnke were wel­comed back in­to the field by Cartan, while Fritz, part of the Behnke team, played a full role in this rap­proche­ment. So with his Prin­ceton con­tacts, in­clud­ing Kodaira from Ja­pan and the new tal­ent emer­ging from Par­is ( Serre, Borel, Grothen­dieck, …), Fritz was well placed to re­in­teg­rate Ger­man math­em­at­ics in­to the world com­munity. Con­tacts with Bri­tain came ini­tially via the Cam­bridge geo­metry school of Hodge and Todd and later the young­er gen­er­a­tion (Atiyah, Adams, Wall, …). The di­vi­sion of Ger­many and, more gen­er­ally, the cold war par­ti­tion of Europe were par­tic­u­larly chal­len­ging, but Fritz spent many years of his life for­ging links between East and West, in­clud­ing not­ably the Ger­man Demo­crat­ic Re­pub­lic and the So­viet Uni­on.

He achieved his goals re­mark­ably quickly. At Bonn Uni­versity he built up the math­em­at­ics de­part­ment to a high level, doub­ling the num­ber of full pro­fess­ors and at­tract­ing people such as Klin­gen­berg, Tits, Brieskorn, and Harder. The Arbeit­sta­gung, which he es­tab­lished in 1957, soon served as a world­wide meet­ing point and at­trac­ted many who would nev­er oth­er­wise have re­turned to Ger­many. But Fritz’s main goal stemmed from his ex­per­i­ence in Prin­ceton: to set up a vis­it­ors’ cen­ter modeled after the IAS. A first at­tempt to cre­ate this as a Max Planck In­sti­tute failed be­cause of the opin­ions of vari­ous ref­er­ees (in­clud­ing Cour­ant and Siegel) that, at least at that time, there were bet­ter ways to use both Fritz’s tal­ents and the tax­pay­ers’ money to fur­ther math­em­at­ic­al re­search in Ger­many. But some ten years later, when the Ger­man Re­search Coun­cil (DFG) set up a new re­search pro­gram for Ger­man uni­versit­ies whose units (called Son­der­forschungs­bereich or “Spe­cial Re­search Do­mains”, ab­bre­vi­ated SFB) would be sup­por­ted for a lim­ited peri­od of time, Fritz presen­ted his ideas of a vis­it­or cen­ter to the de­cision com­mit­tee and came back with two SF­Bs: one (SFB 40, with him­self as Sprech­er or chair­man) for the­or­et­ic­al math­em­at­ics and one (SFB 72, with Rolf Leis and Stefan Hildebrandt) in a more ap­plied dir­ec­tion. With his Son­der­forschungs­bereich, he star­ted the en­vis­aged vis­it­or pro­gram on a lim­ited basis. This turned out to be so suc­cess­ful that when the sup­port ended and Fritz ap­plied for a takeover from the Max Planck So­ci­ety, his pro­ject no longer met with the former re­ser­va­tions, and a per­man­ent Max Planck In­sti­tute for Math­em­at­ics was foun­ded in Bonn in 1981 and has been flour­ish­ing ever since. Through the Arbeit­sta­gung, the Son­der­forschungs­bereich, and fi­nally the Max Planck In­sti­tute, Fritz cre­ated an ex­tens­ive vis­it­or pro­gram that he guided with his many out­stand­ing qual­it­ies: his per­son­al tastes in math­em­at­ics were broad and gen­er­ous — he was no nar­row spe­cial­ist; his in­ter­na­tion­al con­tacts were ex­tens­ive; his ef­fi­ciency be­came le­gendary; and above all, he en­cour­aged an in­form­al and friendly at­mo­sphere, far re­moved from the tra­di­tion­al ri­gid­ity of Ger­man aca­demia.

After about 1970 the main thrust of Fritz’s math­em­at­ic­al work slowly moved from pure to­po­logy and al­geb­ra­ic geo­metry to the con­nec­tions of these do­mains with num­ber the­ory. They will be dis­cussed in more de­tail in the con­tri­bu­tion by Don Za­gi­er. Dur­ing these years he also be­came more and more act­ive and in­flu­en­tial in the de­vel­op­ment of math­em­at­ics, both na­tion­ally and in­ter­na­tion­ally. These activ­it­ies, which will be de­scribed in more de­tail later, in­cluded most not­ably his un­flag­ging ef­forts to build up re­la­tions with the coun­tries of the East­ern Bloc, his many con­tri­bu­tions to re­hab­il­it­at­ing Ger­many’s im­age after the years of the Third Reich and to cre­at­ing new sci­entif­ic and hu­man bonds with Is­rael, his two pres­id­en­cies of the Ger­man Math­em­at­ic­al Uni­on, and his roles as first pres­id­ent of the European Math­em­at­ic­al So­ci­ety (de­scribed by Bour­guignon) and as hon­or­ary pres­id­ent of the In­ter­na­tion­al Con­gress of Math­em­aticians in Ber­lin in 1998.

Dur­ing all the years be­fore the Iron Cur­tain fell, Fritz in­defatig­ably kept up con­tacts with math­em­aticians in the East­ern Bloc, no mat­ter how much ef­fort this re­quired and how un­avail­ing it seemed. Rus­si­an math­em­aticians were al­ways in­vited to the Arbeit­sta­gung, though only once — per­haps be­cause of an over­sight by the Rus­si­an bur­eau­cracy? — were some of them al­lowed to come: in 1967, Anosov, Man­in, Post­nikov, Sha­far­ev­ich, and Ven­kov took part, and all of them gave a talk. But these ef­forts were not in vain, be­cause, as we learned later, the yearly in­vit­a­tions to come to Ger­many, even when they had to re­main un­answered, of­ten helped their re­cip­i­ents by demon­strat­ing to the au­thor­it­ies their vis­ib­il­ity in the West. Fritz him­self was seen quite pos­it­ively by those same au­thor­it­ies and in 1988 was elec­ted as a for­eign mem­ber of the Academy of Sci­ences of the USSR. After 1990, of course, many more pos­sib­il­it­ies of ex­change opened up, and the MPIM today is nev­er without some Rus­si­an con­ver­sa­tion in its cor­ridors. An­oth­er East­ern coun­try that he be­came deeply in­volved with was Po­land. His con­tri­bu­tions here, in par­tic­u­lar in con­nec­tion with the Stefan Banach In­ter­na­tion­al Math­em­at­ic­al Cen­ter in Warsaw, are re­coun­ted by Stan­isław Janeczko.

By a co­in­cid­ence that seemed willed by fate, Fritz was elec­ted twice to the pres­id­ency of the DMV (Ger­man Math­em­at­ic­al So­ci­ety) at key mo­ments in the his­tory of post­war Ger­many and post­war Ger­man math­em­aticians: in 1961 when the Ber­lin Wall was built, and again in 1989–90 when it fell. The sep­ar­a­tion of Ger­many in­to two blocks fell in the middle of his first term, and he solved the prob­lem of the in­ab­il­ity of the East Ger­man math­em­aticians to cross in­to West Ber­lin by re­peat­ing in its en­tirety the first DMV meet­ing that he chaired after the sep­ar­a­tion. But of course such make­shift meas­ures could not last, and soon the DMV was split in­to a new East Ger­man branch (MG­DDR \( = \) Math­em­at­ische Gesell­schaft der DDR) that for al­most three dec­ades was no longer of­fi­cially con­nec­ted with the West Ger­man one. When the polit­ic­al world changed again and the two halves of Ger­many were re­united, Fritz was able to preside over the re­uni­fic­a­tion also of the Math­em­atiker-Ver­ein­i­gung and to en­sure that the trans­ition took place in a spir­it of re­con­struc­tion rather than of re­crim­in­a­tion or re­tali­ation.

For sev­er­al years after the wall fell, he traveled nearly every week to Ber­lin, where he had the task of help­ing the nearly two hun­dred math­em­aticians of the pre­vi­ous Karl Wei­er­strass In­sti­tute of the East Ger­man Academy of Sci­ence to find new po­s­i­tions. The in­di­vidu­al cases were very dis­sim­il­ar, and the solu­tions he came up with were var­ied. The cases where no ad­equate solu­tion could be found haunted him, and he some­times spoke to his friends at the in­sti­tute of the sor­row he felt, but in the vast ma­jor­ity of cases pro­vi­sion­al or per­man­ent po­s­i­tions could be set up, wheth­er in tem­por­ary Max Planck Work­ing Groups, in per­man­ent new in­sti­tutes that he helped es­tab­lish, or in schools or uni­versit­ies in Ger­many or abroad. His con­tacts with the GDR dur­ing its years of isol­a­tion and the re­spect in which he was held on both sides of the pre­vi­ous di­vid­ing line made him ef­fect­ive in this role in a way that no one else could have been, and his achieve­ments, though little known to out­siders, were re­ceived with enorm­ous grat­it­ude by the people in­volved.

Of the many oth­er coun­tries with which Fritz built up or main­tained in­tens­ive con­tacts, two must be men­tioned in­di­vidu­ally. One is Ja­pan, which Fritz vis­ited many times, start­ing in 1972, and from which a huge num­ber of vis­it­ors came, first to the Son­der­forschungs­bereich and later to the Max Planck In­sti­tute, at a peri­od when the pos­sib­il­it­ies of sci­entif­ic in­ter­change between Ja­pan and Europe were still severely lim­ited. His con­tri­bu­tions are de­scribed in de­tail by Kenji Ueno. The oth­er is Is­rael, which is dealt with by Mina Teich­er. Fritz al­most nev­er men­tioned overtly, but very clearly also nev­er for­got, what Ger­many had done in the years of the Third Reich, and a leit­mot­if of much of what he did in his life was to help rees­tab­lish an im­age of the coun­try that would be char­ac­ter­ized by de­cency and tol­er­ance.

Not sur­pris­ingly, Fritz was showered with many dis­tinc­tions of every ima­gin­able kind. His first half dozen hon­or­ary doc­tor­ates came roughly at the same times and with the same fre­quency as his grand­chil­dren, and he used to make jokes about this on­go­ing race, but with only three chil­dren, the com­pet­i­tion was an un­equal one and the doc­tor­ates fi­nally won 15:6. He was a full or as­so­ci­ate mem­ber of more than twenty academies, in sev­er­al of which he was sci­en­tific­ally act­ive, and also of the or­der “Pour le Mérite”, which has as its mem­bers Ger­many’s most dis­tin­guished sci­ent­ists, writers, and artists and to which he was par­tic­u­larly at­tached. Among the many prizes that he re­ceived the most not­able were the Ja­pan­ese Seki Prize, usu­ally giv­en to in­sti­tu­tions and which he re­ceived for his role in de­vel­op­ing the con­tact between Ja­pan­ese and non-Ja­pan­ese math­em­aticians; the Lob­achevsky Prize and the Lomono­sov Medal from the USSR; the Pol­ish Stefan Banach Medal; the Ger­man Krupp Prize and Georg Can­tor Medal; and, most im­port­ant of all, the Is­raeli Wolf Prize, which he re­ceived in 1998 and which, quite apart from its im­mense prestige, had a huge sym­bol­ic sig­ni­fic­ance. At one point the hon­ors were ar­riv­ing so thick and fast that his sec­ret­ary once fam­ously re­marked, after check­ing his morn­ing’s mail, “Wir haben heute keine Ehrungen bekom­men!” (“We didn’t get any hon­ors today!”)

That “we” some­how char­ac­ter­izes in two let­ters what was so ex­cep­tion­al about Fritz and the way he made those around him feel. We hope that the art­icles col­lec­ted here will con­vey to those who did not know him some feel­ing for this ex­traordin­ary per­son­al­ity.