Leray in T Edelbach - Université de Nantes€¦ · Lieutenant Jean Ville, who had pub-lished, just before the war, an inge-nious elementary proof of von Neu-mann’s minimax theorem

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Dirk Huylebrouck, EditorThe Mathematical Tourist

Leray inEdelbachAnna Maria Sigmund, Peter Michor,and Karl Sigmund

This is a most unlikely place for themathematical tourist to visit. In fact,

it is off-limits for tourists of any kind.Photographing, filming, even drawing,is prohibited by law, as signposts tellyou sternly, and trespassers will bepunished. If they survive at all, that is.Indeed, the signposts also warn you ofLEBENSGEFAHR, meaning mortaldanger. You are in a military zone, andhad better watch out. Don’t step on anymines, and avoid getting shot, says anurgent inner voice.

But this is ridiculous. We are in Aus-tria, after all, with almost sixty years ofpeace and prosperity behind us. No-body wants any trouble. Let’s not getcaught, that’s all.

Welcome to Edelbach, or what isleft of it. The place is not easily foundon a map: it ceased to exist many yearsago, during the darkest days of Aus-trian history. Nobody lives here anylonger. The main road between Viennaand Prague is a couple of miles to thenorth, but it can be neither seen norheard. An eerie silence hangs over theplace. All that remains of the formervillage are a few stone-heaps betweenthickets of fir trees, and a small, aban-doned graveyard. To the north of it, amodern fence surrounds a vast ammu-nition depot. It is very well guarded,and you can be sure, by now, thatbinoculars are fixed on you.

This place was once a camp for pris-oners of war, mostly French officers.An “Offizierslager”—or Oflag for short:the bureaucrats of the Third Reichwere fond of abbreviations. OflagXVIIA was the birthplace of a substan-tial part of algebraic topology. Spectralsequences and the theory of sheaveswere fathered here by an artillery lieu-tenant named Jean Leray, during an in-ternment lasting from July 1940 to May1945 ([Sch 1990][Eke 1999][Gaz 2000]).

In the annals of science one findsseveral examples of first-rate mathe-matical research conducted by prison-ers of war. The Austrian Eduard Helly,for instance, wrote a seminal paper

on functional analysis in the Siberiancamp of Nikolsk-Ussurisk, during WorldWar I; and a century before, the Napo-leonic officer Jean-Victor Poncelet de-veloped projective geometry while inRussian captivity for five years. Thismay sound as if the monastic reclusionand monotonic regularity of confinedlife provided ideal conditions for con-centrating the mind. And indeed, An-dré Weil wrote that “nothing is morefavourable than prison for the abstractsciences” [Weil 1991]. He wrote thiswhile he was in prison, and managed,during his months of captivity, to findsome of his major theorems. But hehad a prison cell for himself, could re-ceive visits from his family, and knewassuredly, to use his words, “captivityfrom its most benign side only.” Thephysical and psychic deprivations ofyears in a POW camp, with its over-crowding, sickness, hunger, and bitingcold, on top of the boredom and un-certainty, were something else: inthese conditions, intense intellectualpursuit must have been a desperatemeans for keeping hold of sanity.

The prisoners of Edelbach foundeda “University in Captivity.” Of the 5,000inmates of the camp, of which a fewhundred were Polish and the restFrench, almost 500 got degrees, andtheir diplomas were all officially con-firmed in France after the war. The factthat Jean Leray had been the director,or recteur, of this impromptu univer-sity must have helped with the Frenchauthorities. His academic credentialswere impressive: he had received hisdoctorate at the élite École Normale

Supérieure in Paris, and had been pro-fessor at the Université de Nancy beforebeing drafted into the war. His jointwork with the Polish mathematicianJuliusz Schauder (later a victim of theHolocaust) developed a topological in-variant to prove the existence of solu-tions of partial-differential equations.This earned him in 1940 the Grand Prix

in mathematics from the Académie des

Sciences de Paris.

© 2005 Springer Science�Business Media, Inc., Volume 27, Number 2, 2005 41

But Leray was not the only distin-guished scientist in the Oflag. Therewas the embryologist Étienne Wolff, byall testimonies a driving force behindthe university, but obliged, for racialreasons, to keep discreetly in the back-ground. Étienne Wolff later becameprofessor at the Collège de France, andmember of the Académie des Sciences

de Paris as well as of the Académie

Française. Another luminary wasFrançois Ellenberger, a future presi-dent of the Société Géologique de

France. The geologists at Oflag XVIIhad to content themselves with thestones they could find in the prisonyard. Their laboratory was an oldkitchen which they could use for a fewhours daily.

Eventually, friends and relativesfrom France were permitted to sendbooks. Over the years, Leray receiveda small library from his former teacherHenri Villat [Sch 1990], [Ell 1948].

From eight in the morning to eightin the evening, Barrack 19 housed lec-tures on law and biology, on psychol-ogy and Arab language, on music andmoral theology, on horse-raising (by aPolish fellow-officer, bien sûr!), on

public finances, and on astronomy. Thecourse on probability was given byLieutenant Jean Ville, who had pub-lished, just before the war, an inge-nious elementary proof of von Neu-mann’s minimax theorem [Poll 1989].

Recteur Leray lectured mostly on cal-culus and topology. He had succeededin hiding from the Germans the fact thathe was a leading expert in fluid dynam-ics and mechanics (a mécanicien, as he liked to say). He turned, instead, toalgebraic topology, a field which hedeemed unlikely to spawn war-like ap-plications. This led, first, to some notesin the Comptes Rendus de l’Académie

des Sciences de Paris, and eventually toa three-part work “Algebraic topologytaught in captivity,” which was submit-ted in 1944 to the Journal des Mathé-

matiques Pures et Appliquées, throughthe good offices of Heinz Hopf from neutral Switzerland, who endorsed itenthusiastically. It was published, afterLeray’s release, in 1945 [CRAS 1942][JMPA 1945].

The university’s curriculum showsthat on Sunday nights, the prisonerscould listen to a lecture giving “practi-cal advice for constructing an inex-

pensive house,” before having to returnto their cheerless cold quarters. Thebarracks consisted of two rooms hous-ing 100 inmates each, one small kitchen,and one toilet with eight wash-basins.There was a special building for theshowers: each officer could use ittwice a month. Half of one barrack wasused as a chapel. More than seventy ofthe prisoners were priests, and eachcould say mass daily if he wished. Thecaptives founded a first-rate choir anda theatre group, and soon set up theirown sports stadium, named stade Pé-

tain. The prisoners even managed toproduce, behind the back of theirguards, a documentary film of aboutthirty minutes’ length, entitled Sous le

Manteau (“Beneath the Cloak,” be-cause the camera had always to remainhidden). Three versions of it have sur-vived to this day ([989][Kus 2004]).

As in many other POW camps, thecaptives printed their own newspaper,a weekly called Le Canard . . . en KG.

KG is Wehrmacht shortspeak meaningKriegsgefangener, or prisoner of war,and the French would pronounce it asLe canard encagé (The caged duck), apun referring to the celebrated Le Ca-

42 THE MATHEMATICAL INTELLIGENCER

Fig. 1. Tourists are not exactly welcome in Edelbach nowadays, but what can you expect from an ammunition depot?

nard Enchaîné (The duck in chains),which was, and still is, a hugely popu-lar satirical journal in France. The pris-oners’ version was not permitted tocomment on politics, satirically or oth-erwise: it was filled with harmless car-icatures, theatre bills, sports news,crossword puzzles, and announce-ments of special lectures. Nothingabout the war, or about the conflictsdividing the French community intowhat, with hindsight, was simply theissue of collaboration vs. résistance,but seemed much more confusing atthe time. The Vichy régime tried to fos-ter a network of “hommes de confi-

ance,” but an underground résistance

group, who called themselves themafia, eventually became the dominat-ing force in the camp. For many of theprisoners, the dilemma was whether tobecome a civilian worker in Germany,with a freedom . . . of sorts, or to stickit out behind the barbed wire, in thehope that the legal status of a captiveofficer would protect them from theworst. For Leray, who in 1933 had wit-nessed in Berlin the accession of Hitlerto power, collaboration was never anissue.

When Leray later spoke about Edel-bach, he located it “near Austerlitz, inAustria” [Sch 1990]. Actually, Austerlitzis across the border, in Czechia, and

not really nearby (some 83 kilometresaway). Edelbach is closer to Vienna thanto Austerlitz, but for the defeated Frenchofficers, the thought of being near thesite of the great Napoleonic victory—“àportée de canon d’Austerlitz,” as someliked to say—must have been a solace.At first, they all had hoped to be backin France by the end of 1940. The warseemed over. When this proved an il-lusion, many fell prey to depressionand to homesickness. Leray and hisacademic colleagues used to meetevery evening in the highest, southern-most corner of the camp, and watch,weather permitting, the sunset over “lapetite France.”

Needless to say, the French did notmerely bemoan their fate. Some triedto change it. The prison guards becameexperts at discovering tunnel en-trances beneath the barracks. Theywere so good at it that they overlookeda tunnel entrance which was out in theopen, right under their noses. It wasthrough this 90-meters-long tunnel thaton the nights of September 17 and 18,1943, no fewer than 132 prisoners de-camped. It was the greatest escapefrom a POW camp in World War II, andits story is almost unknown [Kus 2004].

The prisoners had established anopen-air theatre, called Théâtre de la

Verdure. They were allowed to deco-

rate it with twigs and greenery, hidingit partially from the guard towers. Be-cause delegates of the InternationalRed Cross had found that the camplacked protection against Allied airraids, the POWs were told to dig a fewtrenches, and were even provided withshovels and wheel-barrows. Under aplank bridging one of the dug-outs, theystarted burrowing in earnest. The tun-nel grew quickly, by almost a metre perday, although water kept flooding in.After some time, ventilation became aproblem: through a hose made from tincans, fresh air had to be pumped intothe gallery, which was less than twofeet wide and three feet high. In paral-lel, a tailor shop produced civilianclothes, and the printing press pre-pared maps and forged documents.Canned food was hoarded in hiddendepots.

The first group left on a Saturdaynight. Their escape went unnoticedduring Sunday, because some of theguards were on holiday. The secondgroup left on the following night. Mostof the runaways hoped to pass forFrench civilians, of whom there weremany working in Germany at that time.The first escapees were arrested andreturned to the camp by the policeeven before the break-out was discov-ered by the military guards. Eventually,


Fig. 2. Lieutenant Jean Leray, POW, became the rector of the “University in Captivity.” The picture on the right shows him with his Edelbach

colleagues. Some would later join him at the Sorbonne or the Collège de France [Gaz 2000].

only two fugitives managed to reachFrance.

Soon after, a panel of agitated Ger-man officers, including several gener-als, visited the Oflag, where they werefilmed surreptitiously by the Frenchprisoners. The commission decided toplay down the escape—it did not showthe Wehrmacht in a favourable light.The prisoners were sternly told thatthey should not try it again. Handbillswere distributed warning that “break-ing out is no longer a sport” and thatdeath zones were waiting for the run-aways. Half a year later, 76 British fly-ers escaped from Oflag III Luft inSagan. This time, the Wehrmacht couldno longer keep it a secret from Hitlerand Himmler. Only three of the fugi-tives reached England; 50 were shot.

During the five years that Lerayspent in Oflag XVII, battles raged fromone end of Europe to the other, at notime touching Edelbach. Nevertheless,the booming of great guns and the an-gry buzz of Stukas could be heard at alltimes by the inmates of the camp. In-deed, Oflag XVII was located within an evacuated zone, strictly off-limitsfor civilians, the Truppenübungsplatz

Döllersheim. This was the largest mil-itary training ground in central Europe,twenty kilometres in diameter, largerthan the dukedom of Liechtenstein. A few months after the Anschluss,Hitler’s annexation of Austria in 1938,the German army had taken over theground. Forty-five villages with morethan seven thousand inhabitants werehastily evacuated, and huge mechanisedforces rattled across the fields, takinglittle notice of the fact that the harvestwas not yet in. The Wehrmacht had tolive up to its new and as yet untesteddoctrine of the Blitzkrieg. The barrackswhich Leray and his fellow-prisonerswere soon to use were erected origi-nally to house the first German soldiersclaiming the exercise grounds. Verysoon, the Truppenübungsplatz provedan ideal stepping-stone for the armieswhich were assembling to invade anddismember nearby Czechoslovakia, inspring 1939, and for preparing the as-sault on Poland during the followingsummer months [Poll 1989].

The fact that both the father and themother of Adolf Hitler had been born inthe region, which was so suddenly andruthlessly evacuated, gave rise to spec-

ulations. One of the closest as-sociates of the Führer, Hans Frank,would later write, in his death cell inNuremberg, that Hitler intended there-by to erase all traces of his origins[Frank 1953]. He reported that thesetraces could reveal a dark secret, theshame and scandal of the Third Reich:namely, that Hitler had a Jewish grand-father. This rumour, which had beenwidespread in Nazi Germany and stillfinds adherents today, has been de-bunked by scores of historians since.Hitler’s father had been born out ofwedlock, as Alois Schicklgruber, andwas later to change his name, but theFührer was far too powerful to have feltthreatened by slurs concerning his an-cestry. In fact, when the villages aroundDöllersheim were evacuated, all churcharchives were properly stored. They arepreserved to this day. For years, theWehrmacht had been looking for a king-sized training ground to accommodateits frantic growth, and to manoeuvrewith its new weapons, whose rangewould not fit into existing exercise ar-eas. The Waldviertel (or woods district),with its poor soil and its sparse, lowlypopulation was perfectly suited: a hilly


Fig. 3. The curriculum of the Université en Captivité [Poll 1989]. As Leray later said, “students had no other distraction than their studies.

They had little to eat, and little to keep warm; but they were courageous.” [Sch 1990]

plateau, some 600 metres above sealevel, with long, bitterly cold winters,and no reputation for hospitality.

It is clear that Hitler had no emo-tional ties to the Waldviertel. The prop-aganda from the Goebbels ministry hadhailed it as the Ahnengau, the cradle ofthe ancestors, and the humble dwellersof the tiny hamlet of Grosspoppen, ledby their inn-keeper, had conferred hon-orary citizenship on Hitler in 1932,when he was a rising young politicianand demagogue. In return, they first gotscolded by the authorities of LowerAustria (who pointed out that the ac-tion was legally void because Hitlerwas no longer an Austrian citizen),then frowned upon by the Vienneseregime, which was engaged in a hope-less struggle against illegal Nazis, andfinally, right after the revels of the an-nexation, expelled from their landwithout further ado. No account wastaken of the fact that 220 out of the 220citizens of Grosspoppen had voted forthe Anschluss. In fact their hamlet,which obstructed a planned artilleryrange, was the first to become men-

schenrein (the callous Nazi expressionfor “evacuated”) and be knocked down.

A fortunate few were compensatedwith hastily built ersatz farms, not toofar away. Others were given provi-

sional quarters and the promise of asettlement after the war. In 1942, allevacuees were offered a special re-duction on a richly produced coffee-table book, Die alte Heimat, completewith pictures of their empty villages, andHitler’s family tree as a keepsake [Heim1942]. In the ensuing years, Nazi au-thorities had other things on their minds.Eventually, the district of Lower Austriawas occupied by the Red Army, whichcould find good use for the vast trainingopportunities filled with bunkers andartillery ranges. By 1955, the Allied oc-cupation troops left Austria, but theevacuated region was not returned toits former dwellers. They had beenscattered all over the district and werefar too weak to succeed in their de-mands for a return. The small new Aus-trian army managed to keep the over-sized training grounds for itself. Thoseabandoned houses which were stillstanding, after the years of Nazi and Soviet occupation, including Edelbach,were now flattened in a remarkablyshort time. The Austrian army had in-herited an amazing amount of ammuni-tion, and made a point of spending it lav-ishly by shelling the empty settlements.Today only the church of Döllersheimsurvives: its spire serves as a conve-nient mark for ranging artillery sights.

But during Leray’s years of intern-ment he was daily faced with the vacant houses of a seemingly intact,menschenfrei Edelbach behind thebarbed-wire fence. The chimneys didnot smoke and the doors never opened.The window-panes had been replacedby planks. A poem on the front page ofthe Canard en KG, with the title Le vil-lage ignoré, describes the mute bell-tower of the deserted hamlet, and thesilence broken only by the wind [Poll1989]. And while the Nazi picture bookacknowledges that when Edelbach hadto be cleared out, some left it with ableeding heart, the captive French poetimagines how his heart, far from bleed-ing, “jumps with joy on the day, knownonly to destiny,” when he is releasedand the forsaken village vanishes be-hind the firs.

The day known only to destiny wasApril 17, 1945. The camp had to beevacuated because the Red Army wasperilously close. The Wehrmacht wasby now out of gas and lorries. TheBlitzkrieg days were over. The prison-ers had to march, carrying their be-longings on their backs. Some of theguards used bicycles, and their officerssat on underfed horses. The trek aimedfor Linz, some 128 kilometers away tothe west. The group covered, on aver-


Fig. 4. Notes from captivity. KG Jean Leray reports, in this Comptes Rendus note from 1942, that in his present condition, he is unable to

guarantee the originality of his results [Gaz 2000].

age, less than ten kilometres a day, anddwindled rapidly in size. The marchingcolumn was long, the forest dense. Un-derfed François Ellenberger schleppeda rucksack half his own weight: he hadinsisted on taking along his volumi-nous mineralogical notes, a hand-madetelescope, and his rock samples, someof which had come from the tunnel. Hestill found the strength to sketch thelines of the hills in his notebook, and

the interiors of rural chapels. The pris-oners had to look after their own food;some managed to get it from old wivesand barefoot children, in exchange forsoap, which they had produced in theircamp. By May 10, the column had beenreduced by half. This was the day theWehrmacht surrendered.

After his liberation, Jean Leray be-came professor, first at the Universityof Paris (which had appointed him in

1942), and then, in 1947, at the presti-gious Collège de France. In 1953 he waselected to the Académie des Sciences

de Paris (which had made him a cor-responding member in 1944). He wasshowered with prizes: among them, theprix Ormoy in 1950, the Feltrinelli prizein 1971, the Lomonosov gold medal in1988 (jointly with Sobolev), and in1979, the Wolf prize, jointly with AndréWeil (who, incidentally, had also been


Fig. 5. No open university, but a closed universe of 440 � 530 meters. The camp, and campus, of Oflag XVII housed some 5,000 prisoners.

Today, the barracks are gone: in their place one finds concrete, earth-covered ammunition dumps. The village of Edelbach is a rubble of

stones covered by a dense forest.

Fig. 6. A room with a view. The barracks were originally built for the Wehrmacht soldiers claiming the grounds. Fences and watchtowers

were added later [Poll 1989].

a candidate for that same chair at theCollège de France). In an obituary writ-ten for Nature, Ivar Ekeland calledLeray “the first modern analyst,” andcompared him with Weil, “the firstmodern algebraist” [Eke 1999].

The parallels, which also werestressed by Jean-Michel Kantor [Gaz2000], are indeed intriguing: the twomen share their year of birth, 1906, andtheir year of death, 1998. They bothwere among the very select few to at-tend the École Normale Supérieure,and both did some of their best workin prison. But the differences are evenmore striking. Weil followed his

dharma (that is to say, he was a con-scientious objector) and therefore tookhair-raising risks to avoid waging waragainst Hitler. Leray served as a patri-otic officer and remained stolidly at hispost to the end, during the swift Ger-man assault and throughout the pro-tracted years of confinement. WhereasWeil studied abstract algebraic struc-tures and shunned anything even re-motely smacking of applications orphysical intuition, Leray was deeplysteeped in physics and geometry. Thismakes all the more remarkable the factthat he switched to algebraic topologyin the prison camp, and laid the basis

for what soon became a main item onBourbaki’s menu, although he had leftthe Bourbaki group in 1935.

Changing direction seems to haveposed no problem for Leray. “The es-sential characteristic of my publica-tions is their diversity,” he later said,simply. “It was my interest in mechan-ics that obliged me to give new devel-opments to mathematical analysis andalgebraic topology” [Sch 1990]. Indeed,Leray had been interested in topologyeven before the war, but as a toolrather than as an end in itself. The ho-motopy invariant now known as theLeray-Schauder degree was created in


Fig. 7. Underground film. These clandestine stills show the escape tunnel, also known as métro pour la liberté. The clandestine movie Sous

le drapeau, shot in real time, is a French alternative to Hollywood’s The Great Escape [Corr 1954].

Fig. 8. Cold feet and frosty advice. Unaware of being filmed, a Wehrmacht delegation decided to keep the news of the escape under wraps.

But posters warned the French that henceforth, s’évader n’est plus un sport.

order to prove the existence of solu-tions to non-linear partial-differentialequations. Such equations, particularlythose which stemmed from mathemat-

ness of solutions of the initial-valueproblem for the three-dimensionalNavier-Stokes equations for incom-pressible fluids. He showed, in partic-


ical physics, were at the centre ofLeray’s work. In 1936, he published atruly pioneering paper investigatingthe existence, uniqueness, and smooth-

Fig. 9. The church of Edelbach, in an already deserted village. The poem laments that in the humble church, no bell ever rings. In 1957 the

church was flattened by Austrian artillery.

Fig. 10. Forty years after. This stone commemorates a visit in 1985 by some former inmates of the Oflag. The French prisoners had their own

graveyard in Edelbach, complete with funeral statue.

ular, that non-stationary solutions forsmooth initial data remain smooth fora finite time only; beyond this, theymay only be continued in a weak sense(giving rise to what are called weak so-lutions nowadays). Leray called suchsolutions turbulent, thereby suggest-ing that the onset of turbulence iscaused by the breakdown of smooth-ness. He certainly had good reasonsnot to wish the Germans to learn of hiswork. It is interesting to speculatewhat he would have done if he hadbeen given an opportunity to do scien-tific work for the Allies.

As it was, he “turned his minor intohis major interest” and started workingon algebraic topology as an end in it-self—Weil-style, as it were. He workedin great, but not total scientific isola-tion, avoiding contacts with Germanmathematicians. Apart from somereprints provided by Heinz Hopf, fromneutral Switzerland, Leray was cut offfrom ongoing research, in particularfrom contemporary, related work byEilenberg and Steenrod, and had tostart from scratch.

As Armand Borel later wrote,Leray’s original concepts, based on alanguage of his own making, have beenstrongly modified or have not survived[BHL 2000]. Leray’s aim was to createsomething similar to differential forms,keeping their multiplicative algebraicstructure, but in a purely topologicalframework. His cohomology was simi-lar to that created by Cech, and his re-sults did not, as Borel wrote, “seem togo drastically beyond those of main-stream algebraic topology.” But the in-tention behind them was different:Leray aimed at studying, not only thetopology of a space, but the topologyof a representation, i.e., topological in-variants for continuous maps. He tookas starting point his notes on a courseby Élie Cartan on differential forms,published in 1935 [Cart 1935]. Heaimed to understand cohomology(which he persistently called homol-ogy) in a way similar to the de Rhamcohomology, with its multiplicativestructure.

From his work with Schauder onfixed-point theorems, he was used tothe relative viewpoint. He consideredmappings between two spaces as

the basic object. This was a lastingachievement. The Leray-Serre spectralsequence of a filtration is still in gen-eral use today. Grothendieck wouldalso stress the importance of the rela-tive point of view in algebraic topology[Jack 2004].

Soon after his release, Leray found away to define cohomology with respectto sheaves, and introduced the spectralsequence of a continuous map, whichrelates the cohomology of the domainto that of the range and of the fibre. Hisoriginal ideas, intended to be as generalas possible, were still not generalenough, however, for three youngFrenchmen named Henri Cartan, Jean-Louis Koszul, and Jean-Pierre Serre.They extended his concepts to obtainspectacular applications to analyticspaces and algebraic geometry. In thelate forties, the development became al-most breathless [Gaz 2000]. The twoFields medallists of 1954, Serre and Ko-daira, both based their work on Leray’ssheaves and spectral sequences.

In the hands of Cartan and Oka,sheaves became an essential tool forthe theory of several complex vari-ables. Weil used sheaf cohomology andspectral sequences on real manifoldsto give a lucid proof of de Rham’s the-orem, generalising the Mayer-Vietorissequence from an open cover of twosets to one of infinitely many sets.Godement wrote the definitive treat-ment of sheaves and their cohomol-ogy for algebraic topology. Serre andGrothendieck adapted the notion ofsheaves for algebraic geometry. Eventhe (still unfinished) theory of motivesconcerns a category of sheaves. Thecentral problem, on which Voevodskymade some recent inroads, is to findenough injective resolutions for coho-mology to work. With the papers of Ko-daira and Spencer, and the Habilita-tionsschrift of Hirzebruch [Hirz 1956],sheaf cohomology crossed the Frenchborders. Sato used complex analyticsheaf cohomology to define hyper-functions as generalised boundary val-ues of holomorphic functions, and in-vestigated microlocal analysis on thecotangent bundle. Sato’s microfunc-tions are more powerful than Hörman-der’s wave-front sets, which in turnwere inspired by Maslov. Later, Leray

would devote a whole book to the roleof Planck’s constant in mathematics,again in an attempt to understandMaslov [Ler 1981].

Leray’s concept of spectral se-quences appeared first as a compli-cated set of relations among variouscohomologies of double complexes.They allowed Leray to compute the co-homology of compact Lie groups andflag manifolds. Serre used spectral se-quences, already in their modern form,to determine the dimensions in whichthe higher homotopy groups of the n-sphere are not finite, namely n and2n-1. Massey made spectral sequencesmore easily accessible via the notion ofexact couples.

Leray himself, after 1950, returned topartial-differential equations. He stud-ied the Cauchy problem, its connectionwith multidimensional complex analy-sis, residue theory on complex mani-folds, and integral representations. Al-gebraic topology became a tool againfor Jean Leray. The interlude whichhad begun in the POW camp of Edel-bach, as a kind of camouflage, wasover. But generations of pure mathe-maticians would exploit the ideaswhich had germinated in Oflag XVIIA.

Acknowledgements

We thank Hofrat Dr. AndreasKusternig for a wealth of informationon Oflag XVIIA. Jean-Michel Kantor,Reinhard Siegmund-Schultze, and Han-nelore Brandt provided considerablehelp in preparing this article.

REFERENCES

[Eke 1999] Ekeland, I.: Jean Leray (1906–

1998), Nature 397, 482.

[Gaz 2000] Gazette des mathématiciens, Sup-

plément au no. 84, 1–88 is entirely dedicated

to Jean Leray, and includes articles by J. M.

Kantor, Y. Choquet-Bruhat, J. Y. Chemin, H.

Miller, J. Serrin, R. Siegmund-Schultze, A.

Yger, C. Houzel, and P. Malliavin.

[Sch 1990] Schmidt, M.: Hommes de science,

Hermann, Paris.

[Weil 1991] Weil, A.: Souvenirs d’apprentis-

sage, Birkhäuser, Basel.

[Ell 1948] Ellenberger, F.: La géologie à l’Oflag

XVIIA, Annales Scientifiques de Franche-

Comté 3, 21–24.

[Corr 1954] Corre, M.: Défense de photogra-

phier, reportage photographique clandestin


sur la vie d’un camp de prisonniers francais,

Oflag XVII A(Autriche)

[Poll 1989] Polleross, F.: 1938 Davor—Danach:

Beiträge zur Zeitgeschichte des Waldviertels,

Horn-Krems.

[CRAS 1942] Leray, J.: Comptes rendus de

l’Académie des Sciences de Paris 214,

781–783, 839–841, 897–899, 938–940.

[JMPA 1945] Leray, J.: Course d’algèbre

topologique enseigné en captivité, J. Math.

Pures Appl. 24, 95–167, 169–199, 201–248.

[Kus 2003] Kusternig, A.: Grosse Flucht aus

dem Oflag XVIIA, Niederösterreich Perspek-

tiven 3, 22–25.

[Fr 1953] Frank, H.: Im Angesicht des Galgens,

München/Gräfelfing.

[Heim 1942] Die alte Heimat: Sudetendeutsche

Verlagsdruckerei, Berlin.

[BHL 2000] Borel, A., Henkin, G.M., and Lax,

P.D.: Jean Leray (1906–1998), Notices AMS

47, 350–359.

[Cart 1935] Cartan, E.: La méthode des repères

mobiles, la théorie des groupes continus et

les éspaces généralisés, Notes written by J.

Leray, Hermann, Paris.

[Jack 2004] Jackson, A.: As if summoned from

the void, the life of Alexandre Grothendieck, No-

tices of the AMS 51, 1038–1056, 1196–1212.

[Hirz 1956] Hirzebruch, F.: Neue topologische

Methoden in der algebraischen Geometrie,

Ergebnisse 9, Springer-Verlag, Berlin.

[Ler 1981] Leray, J.: Langrangian analysis and

quantum mechanics, MIT Press, Mass.


ANNA MARIA SIGMUND


Anna Maria Sigmund is a historian, whose

best-selling Die Frauen der Nazis has been

translated into twenty-one languages so

far. The reader will have inferred that she

is married to a mathematician.

A U T H O R S

PETER MICHOR

Fakultät für Mathematik

Universität Wien

1090 Vienna

Austria


Peter Michor, a differential-geometer, is a

former director of ESI: the International

Erwin Schrödinger Institute for Mathemati-

cal Physics, a center for intellectual ex-

change between East and West and

around the world. He is a proponent of af-

fordable electronic publications.

KARL SIGMUND

Fakultät für Mathematik

Universität Wien

1090 Vienna

Austria


Karl Sigmund has protected Intelligencer

readers against the danger of forgetting

about Austria, by contributing a series of

compelling articles about its history. In his

spare time he is a game-theorist and pro-

fessor.

Documents

Leray in T Edelbach - Université de Nantes€¦ · Lieutenant Jean Ville, who had pub-lished, just before the war, an inge-nious elementary proof of von Neu-mann’s minimax theorem