I hesitated a good deal before writing the text that follows: is a person of my age (early fifties) and of my modest mathematical achievements justified in inflicting his memoirs upon the international mathematical community? Three factors helped me overcome my hesitation. First, the fact that my scientific path, between two continents and three cultures, is rather unusual, going, as it were, the “wrong way” — from West to East, rather than from East to West. Secondly, my involvement in a number of activities in Moscow that are practically unknown in the West, although — as I strongly believe — they certainly deserve to be. And finally, will I ever have another opportunity to speak out like this?
It has been a great privilege, and a fascinating experience for me, to come in close contact with several great mathematicians of my time, including Kolmogorov, Gelfand, Maslov, Arnold, Novikov. However, I do not feel ready to concentrate my reminiscences on these (and other) personalities, and this account will mainly be about happenings and atmosphere. It is about people only insofar as they fit (or do not fit) into the atmosphere and participate in the events.
Publ. in: Golden Years of Moscow Mathematics, AMS, Providence, 1993.
I was born in Paris in 1937, in a family of Russian émigrés. On my father's side, I come from Russian nobility that can be traced back to the sixteenth century; however, the family had lost their land by the turn of the century, and my paternal grandfather, Bronislav Sossinsky, was not landed gentry, but a highly qualified and well-known railroad engineer. In contrast, my maternal grandfather, V. M. Chernov, was of peasant stock (his father was born a serf, but emerged as a brilliantly educated politician and rose to become the only democratically elected president of Russia, only to be overthrown (less than 24 hours after his election by the short-lived Constituent Assembly) by Lenin and the Bolsheviks (1917). His wife, my grandmother O. E. Kolbasina-Chernova came from a well-off literary family (her father was a close friend of Ivan Turgeniev), and was also a “professional revolutionary”.
My parents met in Paris. My father, Vladimir Sossinsky, settled there after stops in Constantinople, Shumen (Bulgaria) and Berlin following the defeat of the White Army by the Bolsheviks (he was a very brave cavalry officer). My mother got there via Prague, with her two sisters and her mother, after the latter had been freed from Bolshevik prison in 1923, thanks to Gorki's intervention on her behalf. My parents' main interest in life was Russian literature, hardly a lucrative activity in Paris at the time; my father held a variety of positions, eventually saving enough money to open a tiny printing shop of his own. Our circumstances were very modest, but my parents and their circle of friends cared more about the quality of poetry, often read out loud at their gatherings, than that of the food and wines served there.
When World War II began, my father volunteered for the French Foreign Legion, once again covering himself with glory on the losing side of the better cause, then spent time in the German stalags and (after being freed in 1943) fought in the French Resistance. In 1946 he was granted Soviet citizenship, but his attempt to return to Soviet Russia then was unsuccessful (fortunately so, because practically all Russian emigres who returned at that time were soon sent to Stalin's camps). In 1947 my father got a job as an international civil servant at the UN (he headed the Russian Verbatim Reports section for 13 years) and our family settled in Great Neck, Long Island.
Thus, a timid French-Russian boy of ten, bilingual in French and Russian as far back as he can remember, entered the fifth grade of PS 21 in Great Neck in the fall of 1948, not knowing a word of English. Everyone was very friendly and helpful (typical of the US), and I adapted very quickly to the “American way of life”. Two years later, noticing that I was learning nothing in school (except the English language, which I had picked up in a few months), my parents opted for a French education, and I eventually (1954) graduated (with all kinds of honors) from the Lycee Francais de New York.
My interest in (or should I say fascination with) mathematics began at age 13, when the French curriculum first introduces algebra and geometry. Geometry was my favourite subject, and I began “research” at age 14: I “proved” that Euclidean geometry is contradictory and “showed” that the universe is “closed” in the sense that straight lines “don't have two ends” but are “like very big circles“. I was too shy to communicate my “results” to my teacher (or to other grown-ups), but wrote them up in great detail, in a calligraphic handwriting, and sealed them in an envelope, meant to be opened to the world at large later on, when I would be old enough to be taken seriously. (In my later life I have found that social and emotional timidity combined with intellectual conceit is typical of many mathematicians.) A year later I learned about non-Euclidean geometry, discovered the logical error in my “argument“, and shamefully flushed the torn up shreds of my first mathematical paper down the toilet.
I should mention in passing how grateful I am for my French secondary-school mathematical education. The French curriculum then, the result of the pioneering educational ideas of Borel, Hadamard and Poincare (and Felix Klein, although the French don't like to admit that), was very stimulating for creatively-inclined people. Not surprisingly, it led to the postwar Renaissance of French mathematics: Leray, Serre, H. Cartan, Grothendieck, Chevalley, Weil, Dixmier, Dieudonne, A. Borel, Douady, Deligne, Cartier are all products of the system, whereas 30 years of the Bourbakized curriculum seem to have produced no more than one or two mathematicians of the same caliber.
In 1954, having obtained my bachot (the French high-school diploma), I had no doubts that I would study mathematics. My parents could not afford a campus college, so that my basic choice was between Columbia and downtown NYU, the latter winning out (because of the Courant Institute). Unfortunately, I got only a year's worth of credits for my bachot and, what is worse, my faculty advisor at Washington Square College would not let me take Advanced Calculus and other serious math courses, because I didn't have credit for the prerequisites. I was stubborn too, and would not take any “beginner's math“ that I felt I already knew. So in the first semester I took no math at all and besides some general courses, did some English Lit. I was interested in my studies and made the dean's list, but dropped out after one semester. This was 1955, Stalin was two years dead, the “Khrushchev spring” had set in and my father had been allowed to return to Russia for summer vacation. I was planning to continue my education in Europe, either in Moscow or in Paris, the following year.
Our two-month trip to Russia was quite a shock for the family: we saw what the standard of living there was truly like and obtained a first-hand account of the tragedy of Stalin's camps (which until then had been discounted by many leftist intellectuals in the West as “bourgeois propaganda”). It was clear to me personally (my parents did not press me one way or the other) that I needed time to make up my mind about where I would continue my education… and my life.
In the meantime, back in the US with my parents, I returned to NYU to continue my studies.
This time around I convinced my faculty advisor to let me take Advanced Calculus and Differential Equations without any prerequisites. It was a period of anxiety and doubt in my mind; I was undecided about everything, even about doing mathematics — for a while. (One of the options I seriously considered at the time was emigrating to… Iceland, and beginning a life of “isolation, meditation and study”.) The person who got me back on track was John van Heijenoort (who was giving the calculus course I had signed up for), a great teacher and an extraordinary personality, whose varied achievements include a doctorate in Paris in functional analysis, fluent knowledge of many languages (including Russian), the design and construction of the first really operative high-fidelity stereo record player, work on radar systems with Shannon, Wiener and von Neumann during the war and (unbelievable but true) the position of Leon Trotsky's personal secretary at the time of the latter's assassination. Jean van H (as his graduate students liked to call him) showed me many of the most beautiful aspects of mathematics; he introduced me to algebraic topology (via Lipman Bers' superb mimeographed lectures, in the framework of a math honors course) and stimulated my interest in mathematical logic (his field of research at the time), an interest later revived by Kolmogorov, Markov and L. Levin.
I had no trouble adapting to the atmosphere at NYU (although my fairly leftist political views were not too well regarded by some of the faculty in that period of late McCarthyism), I was very successful in the role of American student, not only academically but socially and even athletically (I was cocaptain of the undefeated tennis team in my senior year; the other cocaptain, also a math major, was Herman Gluck, now a distinguished topologist at the University of Pennsylvania). Yet deep down I never really felt American; the strong European cultural heritage that was prevalent in school and family life never let me really adapt to American culture and American lifestyles, except for the superficial behavior patterns which I had easily assumed. I should say, however, that I never shared (and am still very irritated by) the snobbishly superior attitude of some Europeans to American culture, usually the result of their own narrow-mindedness. I got my B.S. from NYU in 1957 with the usual honors (Pi Mu Epsilon, Phi Beta Kappa, honors in Math and English Lit, cum laude); I would have made summa cum laude except for an extraordinarily stupid and biased course of American Government (which did not do justice to the remarkable institutions of the USA), where I was rewarded with a C for making fun of an incompetent instructor. This was a prelude of the problems I would later have with… another course, Communist Party History, in Moscow.
Overall, however, I look back with great pleasure to my two-and-a-half years at NYU, where I began to feel myself becoming a mathematician. An instructive episode that sticks in my mind is my interview with Lipman Bers (when I had been recommended by the math department for graduate study at the Courant Institute). After asking me lots of questions about non-Euclidean geometries and his own algebraic topology course (I had no trouble in answering them), Bers asked me what other interests I had in mathematics. I told him that I had read a great deal about complex projective spaces and had thought a good deal about certain questions… but he interrupted me with a lecture about projective geometry being a “finished science”, that had reached a dead end at the turn of the century with the work of Veblen and his school, and that there was nothing to do there anymore. I was very impressed and felt ashamed of my foolishness. Only many years later did I realize that better people than I (Henkin, Gindikin, Penrose), apparently unmindful of similar advice, were about to reopen this remarkable field of modern mathematics and physics.
In the summer of 1957, after a two-month vacation near Moscow, my parents returned to New York, while I, after overcoming a lot of bureaucratic red tape, transferred from NYU to Moscow University (third year) and stayed on in Moscow. It was a tough decision to make; I was aware that I could not expect to leave Russian again in the foreseeable future; my parents were neither supportive nor opposed to my decision; I was clear about the difficulties that lay in store for me, although I did have naïve hopes that the Khrushchev thaw was only a beginning, that he would soon be replaced by a younger, more educated and more liberal man, that the Soviet Union would adopt a non-totalitarian regime, that some form of socialism with a human face would prevail…
My career as a Soviet student at the Mechanics and Mathematics Department (Mekh-Mat) of Moscow University began with a very unfortunate interview with Professor Shirshov, then Deputy Dean of Studies. I politely explained to him that although I spoke fluent Russian, I had never done any mathematics in Russian before, so that I could be expected to have problems with terminology at first, and would he please bear this in mind when asking mathematical questions. Shirshov, however, did everything possible to ignore my request, wording his questions in typical Slavic words when synonyms with Latin roots were available in Russian (e. g., he asked for the definition of an “opredelitel” rather than that of a “determinant”). He concluded the interview by saying that although I had easily answered some difficult questions, I had certain significant lacunas in my mathematical education, I would not be able to follow third-year courses and should begin at the second-year level. This unfortunate decision was crucial to my mathematical life, as will be explained below.
I don't think that the story of how I adapted to Mekh-Mat life (generally quite well) is of much interest. However, I feel that the atmosphere of my undergraduate and postgraduate years there (1957–1964) deserves some description.
Those years, in the unanimous opinion of practically everyone who had the good fortune to be at Mekh-Mat then, were a period when mathematics and mathematicians flourished in a highly stimulating environment. Undoubtedly, the one person most responsible for this state of affairs was the Rector of Moscow University, I. G. Petrovsky. An outstanding mathematician (who headed the chair of differential equations for nearly two decades), Petrovsky will be remembered even more for his honesty, his personal courage and his remarkable ability as an administrator. He began his rectorship in the late Stalin years, managed to concentrate a great deal of power in his own hands (“he has more clout than many Central Committee members, although he's not even in the Party”, a well-informed administrator once told me) and used it to expand and enrich the university in general, but especially the Mechanics and Mathematics Department, the apple of his eye.
As its name indicates, Mekh-Mat is divided into two sections (otdeleniya): mechanics and mathematics. The mathematics section, whose main administrative function is running the graduate math program, was then headed by the distinguished topologist P. S. Alexandrov, who was always fond of and helpful to talented students of mathematics. During his tenure as the head of the otdelenie matematiki, he did his best to ensure that scientific talent and achievement be the prevailing factors in the choice of graduate students, as well as in new appointments to the department. With the powerful help of I. G. Petrovsky, he was often successful in implementing this policy, getting his way in continuous struggles with party bosses and the rank-and-file, especially in the period when N. V. Efimov was the Mekh-Mat dean (1959–1969). An able administrator, a very popular and careful man, Efimov in fact did most of the infighting, in his friendly low-key style, with the party people, and, to my mind, is second only to Petrovsky as the individual most responsible for the golden era of Mekh-Mat.
It must be difficult for Western mathematicians to understand how, in a totalitarian society, scientific achievement as the main criterion of success in institution is something absolutely unusual. The usual criterion a scientific at the time in Russia, as in almost all other places of science, was politics or ideology, not scientific truth. The phrase partiinost nauki,1 coined by a party functionary claiming to be a philosopher, was a guide to action in such cause célèbres as the Lysenko case in biology, the official banning of cybernetics (as a “bourgeois pseudo-science”), of psychoanalysis (as another “capitalist fraudulent science”), of mathematical methods in economics (as “inapplicable in principle”) and of sociology (as such), as well as the attempts to denounce “Mach-inspired” physics (including all the work of Einstein, Plank, Bohr, etc.), stopped only by the development of the H-bomb. With this as background, it is remarkable that Mekh-Mat, until the end of 1968, was a unique place, an oasis, a haven where the objective value of one's research work was one's best asset. This was understood and accepted by most students and teachers; it was an essential feature of the atmosphere at Mekh-Mat at the time. To the list of those primarily responsible for this state of affairs (Petrovsky, Alexandrov, Efimov) mentioned above, I think the name of Kolmogorov should be added: although he held no important administrative position (except for a brief tenure as dean), he symbolized the total scientific involvement, the intellectual probity viewed by many of us as the ideal for a mathematician.
This being said, I would like to describe more specifically what was going on then in my own field — topology — at Mekh-Mat. The year 1957 was a great one for this topic, with a number of striking results demonstrating the effectiveness of algebraic methods in the classical geometric problems of topology and confirming the central role played by algebraic topology in all of mathematics. Yet at Moscow University, one of the world's leading mathematical centers, there wasn't a single working algebraic topologist of serious international stature! P. S. Alexandrov had moved into pure abstract topology, A. N. Kolmogorov's interest in the field had been short-lived, L. S. Pontryagin had left topology for optimal control theory, M. M. Postnikov had stopped doing or publishing original research work, V. A. Rokhlin was just getting settled in Leningrad. The only competent person teaching the subject was V. G. Boltyanski, but his brilliant lectures struck me as being somewhat superficial, spoiled as I was by Lipman Bers' fundamental lecture course.
What I didn't know then was that during this period some hard-working Moscow U. students had actually begun teaching each other algebraic topology (the hard way: from recent original research papers). I didn't know about this for a good reason — this was not an official course or seminar, no one supervised their activity, and while I was in my second year, they were in their third — the year I should have been in, if it weren't for the disastrous interview with Shirshov…
The names of these people are now well known in the mathematical world: S. P. Novikov, G. N. Tyurina, D. B. Fuchs, A. M. Vinogradov were the most active; D. Anosov and V. I. Arnold, a bit older, were also frequent participants. I'm sure that I would have become a member of that exclusive circle had I known about it, and my mathematical life may have evolved along different lines. At least, that's what I like to think: it is always nice to have someone other than oneself to blame for one's missed opportunities.
As things actually evolved, I had enough intuition to feel that the illustrious head of the Moscow topological school, P. S. Alexandrov, who had noticed me and was apparently willing to be my scientific advisor, no longer really understood the best work being done in his field, and that his narrowing research interests were not mainstream mathematics any more. But I was not bright enough to understand on my own what topology I should be learning. I spent most of the 1957/58 academic year in the Mekh-Mat library: I knew most of the mathematics being taught in the second-year courses and could cut practically all the lectures, only attending the exercise groups and, of course, Communist Party History, a subject (as I had been warned) not to be taken lightly. My scientific adviser during that year was A. S. Parkhomenko, a dedicated (although totally blind) teacher and fairly knowledgeable point-set topologist (along the lines of the Polish school). Under his guidance, instead of teaching myself the really important topics (spectral sequences, homotopy theory, fiber spaces, etc.), I learned practically all there was to know about two-dimensional geometric topology, as well as a lot of three-dimensional topology (e.g., R. H. Bing's work), and wrote my first serious research paper, proving an old conjecture claiming that a certain class of continuous mappings cannot raise a continuum's dimension. Parkhomenko made me write and rewrite the proof until every detail was clear, so that he had no doubts about its correctness. However, when I reported the result at P. S. Alexandrov's seminar, it transpired that there was a counterexample to the theorem by R. D. Anderson. The reader can easily imagine my bewilderment and embarrassment, as well as my subsequent despair. (Actually, the argument in the proof was entirely correct but used an erroneous lemma due to Rozhanskaya, published without proof in the Doklady; I learned the hard way — don't use other people's lemmas unless you known how to prove them!) My consternation did not lead me to abandon the topic, however. A year later I proved a general theorem describing monotone open maps of plane continua, based on a technically very difficulty (but correct!) construction by L. V. Keldysh, who by then had become my scientific adviser.
Let me say a few words about the late Lyudmila Keldysh, my teacher, a person who to my mind is a striking example of dedication to science, courage and intellectual honesty. She was herself a pupil of N. N. Luzin, along with A. N. Kolmogorov, P. S. Alexandrov, N. K. Bari, M. A. Lavrentiev and P. S. Novikov (her husband). She was the only one of his pupils to remain true to him, even in the ominous year of 1937 when considerable pressure was applied on her to denounce Luzin publicly in the framework of an official campaign against him. (This fascinating topic is mostly terra incognita; it is unknown who was behind this campaign, or why it was aborted without developing into a political purge of mathematics and mathematicians, as could logically have been expected.)
L. V. Keldysh came from a large, close-knit family. Her father, Vsevolod Keldysh, was a military engineer who rose to the rank of general in the Tsarist years but succeeded in adapting to Bolshevik rule; her brother, Mstislav Keldysh, is known for his work in the theory of functions of a complex variable, but more for his long tenure as the President of the USSR Academy of Sciences; of her five children, four became scientists, two of them outstanding ones (Leonid Keldysh in semiconductor physics, and Serge Novikov).
At the time, she was in her late fifties and occupied a senior research position at the Steklov Mathematical Institute of the Academy of Sciences (MIAS); the image of her that most often comes to mind is that of Lyudmila Vsevolodna sitting behind her desk in her modest office at MIAS (where we met on Tuesday mornings, almost every week, for many years) commenting on something I would be writing on the blackboard…
Although L. V. Keldysh, a geometric topologist, was unable to keep up with the rapidly expanding field of algebraic topology, she encouraged her students — in contrast to P. S. Alexandrov — to learn a lot of algebra and algebraic topology, and in fact was most insistent about this. Her pupils at the time included A. V. Chernavskii, M. E. Shtan'ko and L. V. Sandrakova. We did not take our teacher's instructions lightly, and in the late fifties and early sixties organized extremely intensive informal schools (usually at somebody's dacha out of town), where we taught each other a lot of algebra and algebraic topology, mainly under the leadership of Alexei Chernavskii, who became a good friend.
Looking back, I must say that these years at Mekh-Mat were extremely rewarding, not only because of my youthful enthusiasm and naïve political expectations, but also because of the extraordinary feeling of kinship with people whose main interest in life was serving science, from the established older generation of mathematicians (Kolmogorov, Alexandrov, Petrovsky, Markov, Menshov, Gelfand) to my own (the generation of 1937, as I like to call it), whose talents flourished early (Anosov, Arnold, Kirillov, Fuchs, Tyurina, Sinai, Manin, Novikov) in the stimulating atmosphere of the Mekh-Mat of the sixties.
Our love of mathematics was not, for most of us, only an escape from the tough realities of a totalitarian society, but part of a common outlook, characterized by anti-establishment political views and by great interest in the artistic and literary life of the times and in active sports (especially mountain hiking, camping, canoeing, cross-country and downhill skiing). My own involvement in these sports activities got me acquainted with many interesting Mekh-Mat people informally, in particular V. Arnold, Dmitry Fuchs (who became a lifelong friend), A. Kirillov, A. Kushnirenko, Maxim Khomyakov (my closest friend), Galya Tyurina (who became Fuchs' wife), N. Svetlova (who married Galya's brother Andrei, and later A. I. Solzhenitsin) and Marina Orlova (who became my wife).
My own academic career was proceeding successfully. By the time I graduated (1961) I had written three research papers and was recommended for post-graduate work in the topology section. There was a hitch, however, at the State examinations: in “Party History”, I got a “4” (=B) instead of the “5” (=A) de facto required of applicants for post-graduate work; however, P. S. Alexandrov's influence and his high regard for my research work (although I was not “his” pupil) were enough to get me through. After three years of graduate work (and a thesis on multidimensional knots) I was offered an assistant professorship in Alexandrov's topology section. I was in a situation similar to tenure track in the US, at the best university in the USSR, at the time one of the best research centers in mathematics in the world. Nothing (or very little) forewarned of the troubles to come.
1 This phrase is basically untranslatable in English; it means something like “political orientation of science” and implies that there is no “abstract scientific truth” and that science is socially biased, the only correct science being communist-inspired.
In 1962, when I was still a graduate student, P. S. Alexandrov recommended me to A. N. Kolmogorov as a possible teacher at the Moscow Specialized School No. 18, a boarding school for talented out-of-town students interested in mathematics and physics, that he had recently founded with the help of I. G. Petrovsky and Isaac Kikoin, the well-known H-bomb physicist. I had attended several of A. N. Kolmogorov's lectures (on the foundations of probability theory and self-reproducing automata) before then, and although he had the reputation of being an extremely abstruse and bewildering lecturer, I had had no trouble in following them. I had also seen him once at the famous “topological picnics” organized by P. S. Alexandrov, and recall that he was listening attentively while the latter was questioning me during an oral exam on homology theory.2 But I had never spoken to the man before.
My first impression when I did speak to him for the first time (in his small office at School No. 18) confirmed the feeling that I had experienced during his lectures: that he and I were “on the same wavelength”. Apparently, Kolmogorov was impressed by my taste for the synthetic approach, based on transformation groups, in teaching geometry which I had learned in the French lycée, and which he himself was developing in his own high-school geometry textbook. I was recruited to teach some exercise classes in calculus (following Kolmogorov's lectures on the subject) and to jointly3 head an optional seminar with him (outside of regular class hours) in geometrical problems. That seminar was a fascinating teaching experience: the dozen students or so who stayed on to the end have all become research scientists since then; the most famous one (although not our best problem-solver) was Yu. Matyasevich.
As a lecturer, Kolmogorov had a strong tendency to overestimate the possibilities of his listeners and did not like to repeat anything (including the formulations of the main definitions and theorems), but the contents of his lectures were remarkably to the point and always bore the imprint of his original mind. His lectures for high-school students were easy for professional mathematicians (from the graduate level up) to follow, and thus students who had trouble understanding the material would later get a clear explanation from the instructors, who always sat in on the lectures and took notes. One of Kolmogorov's pedagogical principles (for teaching math to bright students) was that quite difficult material can be presented provided it is specific (concrete rather than abstract), related to our intuition of the physical world, and given motivation (e. g., its usefulness for solving real-life problems should be stressed). In proclaiming and implementing this principle, Kolmogorov was swimming against the current: Bourbakization, “new math”, was in the process of flooding high-school curricula worldwide. I am stressing this point because, paradoxically, Kolmogorov was later accused (in particular in an underhand political campaign headed by L. S. Pontryagin that sought to denigrate his contribution to mathematical education) of ruining the secondary school curriculum by introducing abstract, set-theoretic and “Semitic mathematics” in place of traditional, applications-oriented “Russian math”.
In the next semester, the topic of Kolmogorov's lecture course changed from calculus to algebra (examples of algebraic structures, polynomial algebra over a