Roger Apéry, 14 Nov 1916-18 December, 1994: A Radical Mathematician

By François Apéry ^*
The Mathematical Intelligencer, vol. 18, n° 2, 1996, pp. 54-61

A Spaghetti Lover

His father Georges Apéry, a Greek born in Constantinople in 1887, came to France in 1903 to prepare for studies at the École Nationale Supérieure d’Ingénieurs at Grenoble. Enlisting as a volunteer as a way to obtain French citizenship in 1914, he took part in the Dardanelles campaign in 1915 and was brought back to France aboard a hospital ship after contracting typhoid. Later he was allowed to travel to Rouen, where he married Justine Vander Cruyssen. She had gallicized her family name to Delacroix and, not liking the name Justine, called herself Louise. It was in Rouen that their only child, Roger Apéry, was born on November 14, 1916.

His childhood was spent in Lille until 1926, when the family moved to Paris. Installing the family in a cold-water flat on the rue de la Goutte d’Or in the 18^th arrondissement, his father counted on the situation improving so they could move to better lodgings. But Georges Apéry, like so many others, was a victim of the 1929 economic crisis: he lost his position as an engineer and, being judged too old, was never again able to work in his field. His job was custodian at the Ministère des Anciens Combattants. Louise gave piano lessons here and there, but the hope of better days died out and for the rest of their lives they remained in the run-down apartment, with communal toilet, no gas, and heat only from the old cast-iron stove in the kitchen. Lighting was by lamps until, after the Second World War, Georges Apéry himself installed electricity.

Naturally a young boy would try to find a way out of this environment. Roger, encouraged by his parents, followed the republican path of relying on academic success.

Intellectual, in the sense that all his time was devoted to pursuits of the mind, he would always feel wary and a bit inferior when any problem demanded a technical solution. His first and only experience with manual labor, in a woodworking class in school, ended with the breaking of a board over a fellow student’s head.

A chess lover, he would regularly take on the students at the University of Caen, and in 1966 presided over the newly formed "Alekhine circle". He was proud a few years later (during a mathematical meeting at Antwerp) to be the only winner in a simultaneous match by the champion of the Netherlands.

His romantic life was troubled. He married in 1947 and had three sons, but a tense and bitter home life ended in divorce in 1971; a second marriage in 1972 was followed by a second divorce in 1977. He could not seem to reconcile family life, mathematical research, and political activism.

From the Greek side he inherited a taste for pasta and a more-than-healthy appetite (his grandfather died from a spaghetti-induced indigestion), as well as the habit of drinking strong black coffee with a glass of water on the side.

From his mother came his musical bent: she taught him enough piano that he could contemplate, at 18, a career in music, but his parents made him see the hazards of such a path. He headed instead toward mathematics, for which he showed promise early on.

And finally, from his father he got his love of country, his intransigence in the defense of republican ideas as incarnated by Clemenceau, and maybe a need to always be right. He never did learn how to make concessions, even when it might advance his cause. This and his occasional resistance to the most basic social decorum might account for his nonconformist career – especially when combined with the distracted behavior of the absent-minded professor. (Students at Caen would long tell of the time the concierge interrupted his class because there was a crying child on his motorbike: he had forgotten to take his son out of the baby seat.)

He underwent an operation for cancer of colon: but it was Parkinson’s disease, first diagnosed in 1977, that caused his death. It eroded his motor ability, reduced his ventures outside, kept him from writing or playing the piano, and finally more and more impaired his intellectual faculties. He died on the 18^th of December 1994, in Caen.

Education: You Should Do a Bit of Chemistry

Roger was a solitary, dedicated student at boarding school, bringing home prize after prize. He left Lille’s Lycée Faidherbe in 1926 after skipping two grades, and gained a reputation as a crack student at the Lycée Ledru-Rollin and the Lycée Louis-le-Grand in Paris. He was passionately interested in history and above all mathematics; not so much in languages, as he spoke only a little German and Italian.

At 8 he was interested in the "preuve par neuf"; at 12, in Euclid’s postulate; but at 16 he learned from his former professor that the cross-ratio of the four tangents from a point to a nonsingular plane cubic is a projective invariant of the curve, and his passion for algebraic geometry began.

He placed only third in the national mathematics Concours Général in 1932, for having written that the absolute value of a sum is the sum of the absolute values. The following year, at the Concours Général for the last year of secondary school, he learned by a leak from the son of a grader that he had the best mark in all Paris. Unfortunately, it turned out when the grades were posted that he had been beaten by a nose by a student from outside Paris: Gustave Choquet. He had to be content with second prize in mathematics and honorable mention in physics.

He received his baccalauréat in mathematics and philosophy in 1933, and entered the taupe of Paul Robert. (Robert later got Apéry’s first scientific article published in the Revue de Mathématiques Spéciales). Too much political activity kept him from getting into the École Normale Supérieure in 1935 ; despite a near-perfect score in mathematics, he ranked only 93^rd.

"Mister Apéry, you should do a bit of chemistry," Robert told him – crucial advice for the competition the following year. The professor in the oral exam said, "What did I give you last year? I hope that this year you know the answer." You bet he did. In analysis, faced with a series to sum, he started by studying the first terms. "You might as well go scratch your balls somewhere else!" cried Jean Favard in his stentorian voice.

(Later, when he was in Favard’s course at the Sorbonne, he found to his surprise that some of his classmates were at the café across the street. "We’re not skipping class," they said; "it’s more comfortable here, and we can hear him just as well.")

He entered rue d’Ulm in 1936 with the second ranking, and, coached by an upperclassman, Raymond Marrot, he came out on top (cacique d’agrégation), having at the same obtained his Diplôme d’Études Supérieures in inversive geometry under Élie Cartan.

The agrégation doesn’t constitute a prime objective for a future researcher; all a good grade does is ensure a good start toward a career in teaching. However, the students at the École Normale Supérieure traditionally treat it as a competition among themselves. The golden boy in 1939 was a golden girl: Jacqueline Ferrand (who as Jacqueline Lelong-Ferrand was to become a well-known geometer). Her reputation for efficiency intimidated the opposition. Marrot, having heard Apéry’s oral entrance exam, knew the kind of fireworks he could produce and knew that this could make the difference, provided he came through with them when it counted. What was needed was motivation to rise to the challenge.

At the time, the students in the École Normale Supérieure liked to divide themselves into two hostile camps: those who attended Mass, the talas (from "ceux qui vont-à-la messe") and those who didn’t, the anti-talas. Apéry was a rabid anti-tala. Marrot said to him, "You’re not going to let a tala take first place, are you?". From that moment on, Apéry put himself in training under his mentor Marrot, and the race was on.

One of the three written tests was in analysis, set by Jean Dieudonné. He recalls, "I was a member of the agrégation jury, for the only time in my life, by the way, and I gave a rather unusual analysis problem. Only two of the papers impressed me with their sense of analysis and precocious maturity very rare among candidates for the agrégation. Those two were by Roger Apéry and Jacqueline Ferrand." And now in the words of the candidate: "I was never so bored with an analysis test. After reading the statement, I told myself that it would be stupid to turn in a blank test form. So I did the first question, the second, and one after the other, until just as the seven hours were up I was finishing the last question. But at no time could I grasp the spirit of the problem." Still, it was good enough that Dieudonné gave him the maximum mark.

To relax after the written exam, his mother took him to dinner: he was served a plate overflowing with whitings, and he downed 37 before declaring himself sated.

At the oral, Apéry got to know Jean Dieudonné, who would remain his friend. Impressed by his aplomb during the algebra oral, Dieudonné exclaimed, "Mister Apéry, you can really juggle determinants!" Then he asked, "Have you read Van der Waerden?". Apéry admitted he had not, but continued, "It is not in the École Normale Supérieure’s library." This was a good enough excuse to save him first place, which he shared with his rival Jacqueline Ferrand. The book appeared in the library the next year.

Italian Algebraic Geometry: I’ve Heard You’re in Germany

He was mobilized in September 1939, taken prisoner of war in June 1940, repatriated with pleurisy in June 1941, and hospitalized until August 1941. During his captivity he stayed in touch with Lucien Godeaux, also with Francesco Severi, from whom he received several articles through the Red Cross. He received for example, the following letter in 1941: "Dear Apéry, I’ve heard you’re in Germany. You should take the opportunity to visit professor X at the university of Göttingen…, signed F. Severi."

He was lucky enough to get from Georges Bruhat, director of the École Normale Supérieure, a research fellowship at the CNRS ; then in 1943, Élie Cartan, who had intervened with the authorities to get him repatriated, offered him a post as assistant at the Sorbonne. He again took up his mathematical production that had been interrupted by the war; he wrote his doctoral thesis in algebraic geometry under Paul Dubreil and René Garnier in 1947, and became the youngest Maître de Conférences in France that year, at Rennes.

In 1948, he gave the Cours Peccot at the Collège de France on the theme of "Algebraic geometry and ideals". He became professor at Caen in 1949. His specialty, from algebraic geometry over the complex field, gradually slid to algebraic geometry over the rationals, and then toward number theory.

His work between 1939 and 1948 on the Italian algebraic geometry in the tradition of F. Severi and L. Godeaux led to his thesis, in which he gave a theory of ideals in the framework of graded commutative rings without zero divisors and applied it to the notion of liaison among algebraic varieties. He proved in particular the theorem of liaison among curves which states that every space curve of the first kind in P³ is equivalent modulo liaison to a complex intersection. (This theorem was rediscovered later by F. Gaeta and was generalized by C. Peskine and L. Szpiro in 1974 with improvements by A. Prabhakar Rao in 1979.) He also generalized a Van der Waerden result about the order of intersection of a surface with a variety of codimension 2, and constructed a variety of degree 22 in P¹³ for which the sections are canonical curves of genus 12, in contradiction to a claim of Fano.

He never got on board the bandwagon of schemes that Grothendieck launched in the early sixties; but in summer 1964, at the Séminaire de mathématiques supérieures in Montréal, Dieudonné, who was giving an exposition of the algebraic geometry of schemes, called on Apéry to give a simultaneous translation of all the results into classical language.

In 1945 he proved the following result in algebraic topology about the neighborhood of a curve lying on an algebraic surface: If neither the algebraic curve nor the algebraic surface has singular points, then the boundary of an appropriate neighborhood of the curve is a Seifert fiber space whose first Betti number is that of the curve and whose torsion coefficient is the degree of the divisor defined by the curve.

Diophantic: I Said Nontrivial

His turn toward arithmetic during the fifties was manifested notably in his study of the Diophantine equation x² + A = pⁿ, where A is a given positive integer and p is a prime. He demonstrated that, except in the case p = 2, A = 7 treated by Ramanujan and Nagell, there are at most two solutions. In 1963, Helmut Hasse, a former student of Hensel, worried that he didn’t understand the proof, although it was based on Skolem’s p-adic method. Yielding to Charles Pisot’s pleas, Apéry answered Hasse, citing his Zahlentheorie. It was hard for him to suppress the feelings he had toward this man who had come during the occupation, in a Navy uniform, to ask the collaboration of French mathematicians with the Zentralblatt für Mathematik, and who had seemed when they met in 1960 to have retained pre-1945 loyalties, to judge by his remark, "I don’t like the city of Caen, it’s where the invasion started."

In 1974, Apéry presented at the Journées arithmétiques in Bordeaux a result on a Mordell conjecture concerning the existence of nontrivial rational solutions to the equation y² = px⁴ + 1, where p is a prime congruent to 5 modulo 8. He was able to prove by a descent argument that the group of this curve of genus 1 is either of rank 1 or of rank 0, depending on whether the equation pX⁴ – 4Y⁴ = Z² admits nontrivial integral solutions or not.

The proudest moment of his career was his proving, at more than 60 years of age, the irrationality of z (3). Having noticed that the procedures developed for summing divergent series in the heyday of complex function theory were convergence accelerators, he applied them to construct sequences of rational numbers for which the rapidity of convergence implies the irrationality of their limit. This method worked for logarithms of rationals greater than 1; it worked for z (2); and, applied to the series

which he obtained from the diagonal of a number table due to Ramanujan, it allowed him to show the irrationality of z (3) in 1977. H. Cohen showed that the method also applies to p /3 Ö 3, and M. Prevost was able to extract from that the irrationality of the sum of the reciprocals of the Fibonacci numbers.

Apéry, in the manner of Diophantus, Fermat, Euler, Kronecker, or Ramanujan, had a feeling of personal friendship toward particular numbers, although the book Diophantic which he was dedicating to them was never finished.

During a mathematician’s dinner in Kingston, Canada, in 1979, the conversation turned to Fermat’s last theorem, and Enrico Bombieri proposed a problem: to show that the equation

where n ³ 3

has no nontrivial solution. Apéry lift the table and came back at breakfast with the solution n = 3, x = 10, y = 16, z = 17. Bombiery replied stiffly, "I said nontrivial."

Political Commitment: Molière Was Right Not to Like Priests

Parallel to his mathematical work, Apéry was deep in politics from his youth, under the banner of Radicalism.

While in second form at Louis-le-Grand, he boarded with the Fathers of the Catholic école Bossuet. The Abbot, searching Apéry’s papers while he was absent, found a tract containing the words, "Molière, who was right not to like priests…". This "was right" earned him a detention from the director, and the experience confirmed him as an "anti-tala", whose anticlericalism made him right at home in the Radical Party.

If his political consciousness solidified quite early around laicism, his political activity dates from the riots of February 1934, when he joined the Camille Pelletan Radical Party. In 1936, the Camille Pelletan Radical Party got two seats classified "miscellaneous left" in the Legislative Assembly of the Front Populaire, which he actively supported.

It wasn’t easy to be anticlerical at the rue d’Ulm before 1940; his literary friend Robert Escarpit, who secretly shared his convictions, envied this great mangeur de curés who dared to bill himself as the Pope of Radical Socialism. His classmates would welcome him mockingly with "Ave, ave, ave Apéry" to the tune of "Ave Maria", and when he tried to sign them up in the Amicale Républicaine de l’école Normale which he had just founded, they made fun of him by signing other people’s name.

In 1938, Apéry signed the petition of édouard Herriot against the Munich accords, and sent back his membership card in the Radical Party to that other édouard, Daladier.

On his return from the prisoner of war camp in 1941, Apéry plunged back into political activity under the influence of his friend Marrot, a Communist. This despite the disillusionment with the Communist Party over the Non-agression Treaty between Nazi Germany and the Soviet Union in 1939, which had ended all his hopes for the Front Populaire. He signed on to the Louis-Fernand-Marty network under the pseudonym Arthur Morin. He became director of the Front National, a resistance movement at the école Normale Supérieure, when his predecessor Marc Zamansky, was arrested and deported to Mauthausen. Apéry organized a protest march against the arrest of Georges Bruhat and Jean Baillou; he took part in the demonstration against the forced wearing of the yellow star; he distributed clandestine press, such as Le Courrier du Peuple (the underground incarnation of Le Jacobin, organ of the Young Socialist Radicals of the Seine), which made its first appearance on his return from the stalag; he sent men to the underground, manufactured false identity papers, and transported arms.

Danger was ever-present, and his courage sometimes crossed over into foolhardiness, as when he insulted a French adjudant in German uniform in Drancy and threatened him with court-martial, all under the eye of a German officer, who luckily didn’t speak French. Anyway, Le Courrier du Peuple did not camouflage its message with any aesopian formulas: "To work, people of France! With bomb and sabotage, destroy the German war effort! Rather than crying over the bombing victims, the people should throw themselves into resistance and beat the Royal Air Force to the punch in destroying German factories and materials." This in November 1943.

When the Gestapo arrested a student who was absent from rue d’Ulm during the night of August 4^th, 1944, they undertook a systematic search of the premises. Apéry, who had been making false identity papers in his room, hurriedly burned all the compromising documents. The Gestapo took Mrs. Bruhat and Mrs. Baillou hostage, to exchange for their husbands the next day; the cleaning woman, not realizing the implications, was complaining loudly about the ashes left in Apéry’s room. Bruhat and Baillou were deported, and Bruhat perished at Buchenwald.

It’s a miracle that Apéry survived all this without a scratch. His all-or-nothing personality, coupled with a distracted, absent-minded manner, made a combination inconsistent with the caution for which those times called. He felt them breaking down his neck several times, yet he took childlike amusement in episodes like the time he was stopped by the Gestapo on the Boul’mich with a long package wrapped in newspaper under his arm. "It’s a gun?" "No, it’s a leg." It was Marrot’s prosthesis which he was taking to get repaired.

Apéry received the Croix de Combattant Volontaire, as had his father after the First World War.

The Communists discredited themselves in his eyes after 1948 by defending the Soviet endorsement of Lysenko’s biology. Privately, and in public speeches in 1952, he warned against the imposition of a Marxist line on mathematics in France.

For Mendès-France Against de Gaulle: Thank You for Your Testimony

In his political activity in the 1950s, as he put it himself, "My place is in the party of Ledru-Rollin, Clemenceau, Herriot, and Mendès-France, who embody the Jacobin tradition." He was a founder, at Mendès-France’s invitation, of Les Cahiers de la République; the name was his suggestion. As a candidate on the Mendesist ticket for the legislative elections of January 1956, he had to face up the strong-arm tactics of the bouilleurs de crus, the local farmers who distilled their own eau-de-vie, who disrupted his meetings and attempted to intimidate the agricultural workers present.

Apéry supported Mendès-France faithfully in internal party quarrels and against the attacks of the Gaullists and the Communists. He came before the tribunal of the Mouvement de la Paix in 1954 to lambaste as "pacifists" the opponents of Mendès-France’s Indochina policy.

On June 11, 1957, paratroopers under General Massu in Algiers arrested a Communist activist, Maurice Audin, a French mathematics assistant at the University of Algiers. A military report of 25 June reported him escaped. He was never seen again. Thus began the Audin Case. His family and friends doubted that he had ever escaped and suspected that he had died under torture. Intellectuals led by Laurent Schwartz formed the Comité Audin to publicize the case and demand response from the authorities.

Apéry was in charge of the Audin Committee in Calvados and broadcast the work of Pierre Vidal-Naquet on the case; he got a motion passed on it at the Radical Congress in 1957. Later, he left the Audin Committee when he thought it was being used to defame French policy as a whole.

He led an active campaign in Calvados for republican legality against the coup de force of General de Gaulle in 1958. In the preparatory maneuvers for the election of the President of the Republic, Apéry achieved a reconciliation of the fractions of the Union des Forces Démocratiques (The U.D.F. candidate, nominated by him, was the well-known mathematician Albert Châtelet.) They wanted to prove that there was something between de Gaulle and the Communists: that something was 8.4% of the vote.

He remained loyal to Mendès-France’s vision of North Africa within the French Union. Sent to Algeria on a fact-finding mission as a reserve lieutenant in 1959, he reported to General de Gaulle on the faults of the colonial system but refused to characterize colonialism as simply an "action of exploiters in league with torturers."

When a group of officers complained of the antinationalistic character of the French university and preached disobedience, Apéry’s radical heart beat anew, and he reminded the military of its constitutional duty of submission to civilian rule. "Thank you for your testimony", was the General’s lapidary response.
He would never forgive de Gaulle for his military manner of returning to power, or for abandoning Algeria in the worst of conditions in 1962.

After years of responsible positions in the Radical Party, he quit it in 1969, feeling that the republican spirit had expired in it. He ended, indeed, all involvement in electoral politics; with the departure of General de Gaulle, he felt, the Republic was no longer threatened.

A Budding Intuitionist

His vision of mathematics was individualistic like his political philosophy, rebellious to all orthodoxy.

He was a constructivist. Formalism, with the Hilbert school as champion, succeeded by Nicolas Bourbaki, he saw as an a priori philosophy based on a metaphysic of the infinite, and not corresponding to the practice of the working mathematician. Practicing what he preached, he declined Dieudonné’s invitation to join Bourbaki. Later, at a congress of philosophy, Dieudonné called him a budding intuitionist.

He led the scientific philosophy circle of the école Normale Supérieure starting in 1944. He regularly faced off against Dieudonné in such forums, tenaciously defending a pragmatism close to that of Poincaré, Borel, or Denjoy (although distancing himself from Brouwer) and refusing the idealism of Bourbaki. He collaborated on the journal Dialectica of Ferdinand Gonseth, becoming a member of the editorial committee in 1952 and an advisor to the director in 1966.

The dominance of Bourbaki meant marginalization for the anti-Bourbakiste. Not being in sympathy even with all the other marginalized, Apéry eventually found himself nearly isolated.

He remained able to offer stout defense to potential victims of the fashionable ideology, as with the "Halberstadt question", named for the algebraist who in the view of his patron Marc Krasner was also threatened with ostracism. He supported the call "for freedom in mathematics" by Krasner and Chevalley in 1982, in recrudescence of the "Halberstadt question" on the editorial board of the Comptes Rendus de l’Académie des Sciences de Paris.

He felt close to Marc Krasner in philosophy and also shared with him an affinity for cats, love of the Balkans, and an immoderate taste for the pleasures of the table. One memorable excursion for bouillabaisse at a harbor restaurant in Nice after an international congress session was led by Krasner. He ordered appetizers and the royale for all 10 mathematicians; one by one they dropped out, on all sorts of lame excuses like wanting not to miss the afternoon lectures; the only ones to finish gargantuan feast were Krasner and Apéry.

At the end of the sixties, he returned to the attack on Cantor’s set theory, plumping for category theory as better suited to the needs of mathematics. Facing André Revuz in a radio debate in 1972, he attacked the Lichnérowicz teaching reforms, denying that the stagnation that had preceded gave any justification for abandoning geometric reasoning and enshrining Cantorian set theory. The reforms passed; he worried that 20 years later there would be a backslash in public opinion against mathematics, a prophecy that unfortunately came true.

The instigators of the Lichnérowicz reform insisted on loyalty to their program and tried to brand any opposition to it as reactionary, which only hardened Apéry’s position and deepened his isolation in the community. It went so far that at the Journées Arithmétiques de Marseille in 1978, his lecture on the irrationality of z (3) was greeted with doubt, disbelief, and then disorder. Its recognition at the Helsinki Congress would finally erase this humiliation.

Apéry a péri

At the école Normale Supérieure he was ever the merry-maker, always ready for a canular, the practical jokes for which the normaliens are noted.

There he had in Marrot the big brother he had missed in childhood. Marrot supported his iconoclasm. Marrot spurred him to reenter politics after the liberation: "You are at the age where one chooses to become bourgeois or not." Marrot was his best man. Then in 1948, a gas worker forgot to replace a cap after a routine check, and Raymond Marrot, newly named Maître de Conférences in Bordeaux, was asphyxiated along with his mother. The blow to Apéry was apparent to all.

In the fifties, vacationing in Naples, he asked in a café if there was a well-known mathematician in the city. He was directed to the legendary Renato Cacciopoli. Descended from anarchists and aristocrats (he was a great-nephew of Bakunin), he was a militant communist who moved in literary circles while professor of mathematics at Padua and then at Naples. Apéry got Cacciopoli’s address and went to introduce himself: "Sono matematico francese…" But Cacciopoli interrupted in impeccable French and invited him for espresso as only the Italians can make it – and for piano four hands, with Cacciopoli singing the Verdi arias at earsplitting volume. Their friendship would last until Cacciopoli’s suicide in 1959.

Apéry was a child of the patriotic and meritocratic left. He would never understand the intellectuals who profess egalitarianism in education and yet look down on technical education, which gives the most underprivileged children the possibility of a job that their family’s means couldn’t get them. His own children got technical educations.

He classed the leftists of 1968 with the Hitler Youth. He opposed the antielitist program – elimination of awards, tenured jobs, and faculties – as a replacement of the values of study and work by nepotism and favoritism. Regrouping old friends from the Resistance, he led the group known as "the Thirty-four" opposing the resignation of the Assemblée de la Faculté des Sciences in Caen; he joined colleagues from other regions and from other disciplines in a national appeal of June 1969 against the degradation of the universities. To him, this was upholding his lifelong political convictions, although to some he now looked as reactionary as he had been progressive.

Resistance against Nazism under the Occupation, and resistance against leftism – for both reasons President Pompidou named him Chevalier de l’Ordre National de la Légion d’Honneur in December 1970. He received the decoration from the hands of Jean Dieudonné.

Resentments endured, and he found himself alone in his field. With the same hearty resolution he had stood up for free thinking against clericalism, for radicalism against the Right in 1934, for Resistance against National Socialism in 1940, for the Republic against Gaullism in 1958, for constructivism against Bourbakism, for the university against leftism in 1968. The graffiti artists, however, had the last word: Apéry a péri.

* François Apéry was born in 1950 and studied at the École Normale Supérieure in Cachan. He has been Maître de Conférences in mathematics at the Université de Haute-Alsace in Mulhouse since 1987. The editor is pleased we prevailed on Dr. Apéry to offer this reminiscence of his father