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 jointly^{3} 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 discrete field, including Galois theory); the next school year he lectured on “Discrete Mathematics”, which in his interpretation included a lot of combinatorics, some probability (very little), a bit of logic and a unique cycle of lectures on the theory of algorithms culminating in the algorithmic proof of Gödel's incompleteness theorem (!).

The latter course, which was absolutely absorbing (but was never published), stimulated Kolmogorov to carry out his last great cycle of research (on the complexity of finite binary sequences and the foundations of probability theory). For me, it revived my interest in mathematical logic and was a unique teaching experience: I was not only stimulated from above by Kolmogorov, but also from below, by the strongest group of students I have ever taught. They included Leonid Levin (now at Boston University), A. Zvonkin and Z. Maimin (whose future mathematical careers were shaped by Kolmogorov's high school discrete math course); E. Poletski (now in the US), V. Sklerenko, T. Lukashenko, Yu. Osipov, A. Uglanov (the strongest of the lot, whom a freak accident prevented from realizing his full potential) and A. Enduraev. I still recall the exhilaration I felt before each of the lessons in this class; much later, some of its former students told me how much they also had looked forward to these mutually stimulating classes.

The Kolmogorov school, where I taught (first as a graduate student, then as an assistant and later associate professor at Mekh-Mat) was of course exceptional by the sheer talent and deep interest in physics and math typical of its students. Although my democratic family background has biased me against elite schools, I must admit that I would never had done so well as a teacher, learned so much, enjoyed it all so much if it were not for Kolmogorov and the carefully selected student body of School No. 18.

Another essential feature of the school was the great interest in general culture (music, the arts, literature, sports) that resulted from Kolmogorov's selection of teachers, each of whom, besides being a highly qualified mathematician, had his own *violon d'Ingres* that he would readily put at the disposal of his students. Optional courses in (or “evenings” related to) music, poetry and the arts given by mathematicians and physicists, camping trips and sports events opposing students to teachers, were a normal part of studies. At the school I met some very unusual and interesting people: D. Gordeev (a graduate student of Kolmogorov's who eventually gave up mathematics to become a successful avant-garde painter), E. Gaidukov (who gave up a career as a violinist to do mathematics under Arnold, was active in the dissidence movement, and when things got too hot moved to Khabarovsk), Yu. Kim (the now famous bard and playwright, who taught history and literature at School No. 18, organized very successful — and dangerously anti-establishment — musicals and was forced to resign after cosigning a political letter of protest), A. Zilberman (the best teacher of physics I have ever met, still a graduate student then), the late V. M. Alexeev (a talented research mathematician, a softspoken man of colossal classical erudition), V. A. Skvortsov (another unusual mathematician, for many years the president of the University English Club) and several others, who became my friends.

For several years, especially under the directorship of the late R. A. Ostraya, a well-educated historian, the Kolmogorov school was not only an elitist mathematical institution, but a school where high-level general culture and intellectual freedom were part of the curriculum.

^{2}
It is typical of the informal democratic atmosphere of Mekh-Mat of those times that there nothing unusual about a serious examination being conducted in such a setting.

^{3}
Here again, there was nothing unusual about an established world-class scientist heading a seminar jointly with an obscure graduate student.

The year 1968 was the turning point of many lives, including my own. It was the year of the May barricades in Paris, of draft-card burning and riots on American campuses, of the Prague spring crushed by Russian tanks. For me it was the year that put an end to my hopes and illusions, the year of dramatic events that mark the end of the Mekh-Mat golden era.

These events are described elsewhere in this volume^{4} (see pp. 220-222 of Fuchs article).

Briefly, let me remind the reader that in March 1968, A. S. Esenin-Volpin, a mathematical logician and well-known human-rights activist, was forcibly and illegally placed in a psychiatric institution by the KGB. This shocked the Moscow mathematical community; uncharacteristically, it reacted by writing a (mild) letter of protest known as the “Letter of the 99”, signed by many of the leading mathematicians of the day. The letter was almost immediately published in the West, against the wishes of its authors and cosigners; most observers agree that this was a calculated leak by the KGB, then in the need of *casus belli*. In any case, the letter was the pretext^{5} for a crackdown at the Moscow University math department: the administration at Mekh-Mat and the party leaders were all subsequently replaced by hard-liners. The cosigners were subjected to humiliating public disavowal procedures and severely reprimanded; some eventually lost their jobs.

Another significant development was the organization of systematic anti-Semitic practices at the Mekh-Mat entrance examinations; but much has already been written about this in the West and I will contribute only one episode, to give the reader an idea of what +1 in the statistics can stand for from a personal viewpoint.

In the summer of 1969 (or was it 1970? — I have kept no written records), still teaching at Boarding School No. 18 and training the students for entrance examinations, I had spoken privately to each of the Jewish students, described what lay in store for them at the exams, and explained how they should prepare themselves for these. The most talented one of the lot, a boy by the name of Kogan, a year younger than his classmates, did not take my advice seriously enough, and was easy prey for the young enthusiastic anti-Semitic thugs at the oral math exam. “I've learned my lesson,” he told me later, “but I'll be back again next July.” And he was (being too young for the draft that year), highly trained this time, matured and sobered by his first experience in the world at large, ready to fight any odds. His results—written math: 5, oral math: 5 (surviving four hours of olympiad-level questions), oral physics: 5 (where the two examiners were also out to get him). That left the Russian literature essay, where even a passing grade would be enough to get him in; it was clear to him, and to me, that Kogan had beaten the system. We were wrong; the philology department examiners failed him (Kogan had been an A-student in literature, there were no spelling, grammar or stylistic mistakes in the essay, but it was given a “2” (=F): for "not clarifying the topic"). I have never seen Kogan since. (Thank God. What would I have said to him?) For the first time I asked myself the question: what moral right did I have, as a teacher, to be, if not an accomplice to, then a passive observer of such practices?

The following years gave me several occasions to ask myself this question again: In 1971, when the Mekh-Mat Party Bureau decided to forbid me to teach at School No. 18; I was the second to go, after A. Zilberman, in a clean-up campaign that eliminated, one after another, the people Kolmogorov had handpicked as teachers for the school and who had contributed to creating its unique spirit; in 1972, when the most talented person I have ever had working under my personal guidance, a straight-A student named Isaac Lapitski, was deprived of doing graduate work by the Komsomol leaders, who would not give him a *positive kharakteristika;*^{6} in 1973, when my teacher, Lyudmila Keldysh (a cosigner of the Letter of the 99) had been tricked into early retirement (she had agreed to retire so as to free her research position at the Steklov Institute for A. V. Chernavskii, but the second part of the bargain — hiring Chernavskii — was never kept by the Institute's authorities); finally, in 1974, when two other students doing research under me had refused to take part in the so-called elections to the Supreme Soviet and had been thrown out of the University, a fact actually reported by the BBC; I had managed to convince them to act “reasonably”, the incident had been smoothed over and they were almost immediately reinstated, but of course they were not allowed to do graduate work, although neither was Jewish and both were clearly Ph.D. material. In this episode I particularly felt the ambiguity of my own ethical position — was I right in counselling them to retreat from their stand “for their own good”?

Two other happenings, not directly related to mathematics, finally led to my cutting the Gordian knot. The first was a very strongly worded letter of protest about the forced exile of Solzhenitsin that I had written jointly with my friend Maxim Khomyakov and that we had circulated in *samizdat*; this made my chances for reelection to my associate professorship, due in 1975, pretty uncertain. The second concerned something called the “University of Marxism-Leninism”, a two-year course in political indoctrination for faculty members. In my ten years at the Section of Topology, I had managed, under various pretexts, to avoid taking it, but it was made clear to me that this year was it: I must either take it, or leave the Section.

This was the fourth of March, 1974. I was very depressed and tired. My wife had been in the hospital for several months, my mother was very ill, I had two children to take care of, an overwhelming workload at the University (besides my own courses, I was standing in for M. M. Postnikov, in the hospital for heart troubles, in a new untested third-year course on Differentiable Manifolds). I was ready to capitulate, and I took the application form for the Marxism-Leninism University with the intention of filling it in. I read the mimeographed text: “Professor [blank] kindly requests the Party Bureau of the Department to recommend him for a two-year course at…” That proved to be too much: to write that it was I, I who was “kindly requesting to be recommended…” I tore up the application form, took out a blank piece of paper and wrote my resignation from the department.

A few days later, Yu. M. Smirnov, a professor at my former Section, tried to convince me to take the resignation back; I refused. Then the chairman, P. S. Alexandrov, invited me to his apartment to discuss the situation. This was a memorable talk, practically a monologue, that began with the following remarkable opening gambit:

“Alyosha” he said, using the informal diminutive of my name, “traditionally, we intellectuals of the Russian nobility have always placed our duty to our fatherland (he used the archaic *otchizna* rather than the typically Soviet *rodina* for homeland) above our personal interests and feelings. A Russian nobleman does not leave a sinking ship — he fights to keep it afloat. It is people like Kolmogorov, like you and me, who have made this department into the unique scientific oasis that you know. Even in the Stalin years, we have always done all we could, swallowing our pride if need be…”

I had known that Alexandrov's parents were small-landed gentry, but was not aware that he knew about my own origins, and had expected anything but an appeal to values that fifty years of Bolshevik rule were supposed to have eradicated, especially in a careful and successful establishment scientist. But the divulgence of this hidden facet of Alexandrov's personality and the implicit flattery in his monologue were not enough to make me change my mind. Although I did promise to give the remaining lectures that semester in the Differentiable Manifolds course, something no one else at the Topology Section appeared to be able to do, I confirmed my resignation.

And I became — as I then liked to joke when talking to friends — the only unemployed mathematician in the Soviet Union.

I recall the mixed feelings I experienced visiting the department, to work in the library or attend a seminar, in the months immediately following my resignation, the perverse pleasure I derived in refusing to shake hands with the most odious people there. And a strange form of solidarity from other faculty members: after furtively looking around, somebody (at times people whom I hardly knew) would walk up to me and give me a prolonged, silent handshake.

Looking back and trying to analyse how the Mekh-Mat we knew and loved was being destroyed, I think that the process (in which anti-Semitism was only one of the aspects) can be understood only in terms of Bettelheim's remarkable analysis of situations of stress in Hitler's camps. The basic premise — to achieve automatic obedience — can be made good by systematically denying people their sense of dignity, their personal identity, by teaching them never to “stick out”. By humiliating a student or a professor, forcing him to dig potatoes out of the mud by hand (as “voluntary” help for a local collective farm), making him hypocritically repeat, in public, obvious political lies about the system, the system succeeds in making this person lose his sense of self-respect. Then he becomes manageable. Talented people — who tend to be unpredictable and more difficult to control — are eliminated. They are flunked at the entrance exams, not recommended for graduate work, not given positions at the department, unless their sense of self-respect is broken and they can prove their docility. What the hard-line administrators wanted were good, competent, solid, stolid, servile mathematicians. And that's what they now have — there are very few world-class mathematicians holding a full-time position at Mekh-Mat today, while there were dozens and dozens in 1968.

^{4}
Golden Years of Moscow Mathematics, AMS, Providence, 1993.

^{5}
A pretext, but not the cause: the Communist Party was strengthening its ideological authority, the “party line in science” principle was being widely reasserted, and any oasis for talented and honest scientists could not be tolerated by the totalitarian nature of the system.

^{6}
An untranslatable expression from the Bolshevik lexicon, standing for a kind of certificate of political and social docility.

After my resignation from the University, I did not start looking for another position, feeling that I could make a decent living for my family and myself as a free-lance translator of mathematical books. However, my family and friendswere worried about my future: they felt that the law about *tuneyadstvo* (a kind of vagrancy law), calling for internal exile of unemployed residents of Moscow and other large cities, most generously applied to dissenters, could be applied to me.

Several months later, I finally agreed to look for an official position, and after some unsuccessful attempts (which all followed the same scenario — I would be enthusiastically recommended by future colleagues, but at the last minute the personnel department would suddenly declare that the position in question was no longer available), I got a job as a mathematics editor at the popular science magazine *Kvant*. It was an underpaid and demanding position. A. N. Kolmogorov had strongly recommended me for it, and Isaac Kikoin had used all his political clout in the Academy of Sciences (which publishes the magazine) to get me in. I should perhaps mention that during subsequent years I made several attempts to find a better-paid and less energy-consuming position, and they failed, all along the lines of the scenario described above.

*Kvant* magazine was another accomplishment of the liberal sixties. It was founded in 1969 by the same tandem (Kolmogorov–Kikoin) as School No. 18, with the assistance of Petrovsky, in the framework of the Academy of Sciences. Like the Kolmogorov school, its aim was to provide gifted high-school students living in rural areas or in small towns with stimulating materials in physics and mathematics that they would not be likely to get from their teachers.^{7} Its circulation, despite the high level of sophistication of most of the articles, started at 100,000, rose to the incredible level of 370,000 during the math-physics boom (early seventies), dropped to well under 200,000 by 1977 (when the boom was over), rose and levelled of at around that figure in the 1980s. Most of the articles are not written by high-school teachers, but by research mathematicians with an interest in education (many of them former Olympiad winners, later involved in the organization of Olympiads). Systematic contributors included Kolmogorov, Arnold, Kirillov, Fuchs, Gelfand, Rokhlin, Pontryagin, Migdal, Kikoin, Frank-Kamenetski. A very important feature of the magazine was the problem-solving section, a permanent contest where our high-school student readers competed in the number of correctly solved problems (five math problems and five in physics were proposed in each monthly issue). The mathematics problem section, from the very beginning, was headed by N. B. Vassiliev; it still is today, and Kolya, a soft-spoken, very musical person, still remains one of the champion “elementary problem solvers” of our times.

Two other persons I became friends with at Kvant were L. Makar-Limanov (a sharp mathematician and strong personality, now at Wayne State University) and Yu. Shikhanovich (a mathematical logician, under whom I had taught some calculus part-time at the Structural Linguistics Department of Moscow University in my graduate-student days). The latter was always very active in the human-rights movement, came to *Kvant* after being released from a psychiatric ward (where he was forcibly placed for his political activities) and stopped working at *Kvant* only when he was arrested by the KGB in 1982. He got a heavy sentence at his “trial” (a farcical imitation of justice) and was only freed in the political amnesty that marked the beginning of *perestroika*. It is characteristic of the people at *Kvant* that the authorities were not able to get any one of us to testify against him.

Generally, during the drab and totalitarian years of the Brezhnev and post-Brezhnev era, *Kvant*, under the protection of I. Kikoin, remained a strange little islet of liberalism, where a closeknit group of underpaid physicists and mathematicians were doing an unheralded job of making first-rate math and physics accessible to thousands and thousands of high-school students. Fifteen or twenty years later, the yearly lists of prizewinners of *Kvant*'s math context read like a Who's Who in Soviet research mathematics. A number of these people have told me that if it weren't for *Kvant* and Math Olympiads they would never had gone in for the subject. Especially rewarding were the cases (not too frequent, to be sure) when the first clumsy, but promising, problem-solving efforts of a teenager from the middle of nowhere would gradually evolve in technique and sophistication, and suddenly yield a totally unexpected solution to one of our more difficult problems…

The Western reader can get an idea of what *Kvant* was like by looking at any issue of *Quantum*, a magazine based on back issues of *Kvant* and now published by the National Science Teacher Association in Washington, D. C.

An important happening, which saved *Kvant* from the sad fate of many achievements of the sixties, was the failure of a takeover bid, undertaken by L. S. Pontryagin in 1980, when he tried to wrest control of the journal away from Kikoin and Kolmogorov. I was present at the decisive proceedings that took place at the Steklov Institute; I will always remember the tragic and odious figure of L. S. Pontryagin, not as the great mathematician that he once was, but as an old man nervously clicking the beads of his rosary and lashing out at Kikoin, Kolmogorov and even at me (he described an article about Conway numbers in *Kvant* that I had written jointly with A. Kirillov and I. Klumova as an extreme case of the “wrong kind of mathematics” that Kolmogorov and his entourage were inflicting on innocent school children). The takeover bid failed, because Pontryagin's cronies had not done their homework properly: the Mathematical Section of the Academy did not have any legal authority to control the magazine (which depends directly on the Academy's Presidium), and Pontryagin's virulent attacks (supported by the anti-Semitic remarks of I. M. Vinogradov) were simply ignored by Kikoin.

My eleven years at Kvant were not often as arresting — much of the editorial work was routine, and although I did a lot of creative rewriting, the job was very time-consuming and often depressing. I was doing less and less research of my own, not attending any seminars (except Sacha Vinogradov's seminar on symmetries of PDEs, an interesting topic then, about which, however, I published nothing). I felt rather isolated mathematically: L. V. Keldysh had died, A. V. Chernavskii had moved away from mathematics to neurobiological applications, I had no contacts with the West and I had given up on my own work on the applications of the homology theory of Zeeman's tolerance spaces to approximate solutions of differential equations.

I. M. Gelfand, V. P. Maslov and — surprisingly — P. S. Alexandrov tried to get me a job where I could do more mathematics, but failed. I am particularly grateful to P. S. Alexandrov, who offered me A. S. Parkhomenko's associate professorship (which I probably would not have accepted anyway) after the latter died, but this position disappeared into thin air when my name was proposed for it.

At *Kvant*, at least I felt I was doing something useful and had no qualms about the underlying ethics of my position.

^{7}
In the Soviet Union educational levels generally decrease proportionally as one moves away from urban centers.

This short-lived unofficial institution has been known under various names, e. g. ironically as “People's University”, or as “the Jewish seminar”. I prefer to call it by the name of its founder, B. A. Muchnik, an alumnus of (and Ph. D. at) Mekh-Mat, who decided that bright students flunked at the Mekh-Mat entrance exams (for being Jewish, part Jewish, or too smart) should have a chance to get as good a mathematical education as the successful applicants. Another person involved in the organization was V. Senderov, a young dissident, who had taught at an elite mathematical school in Moscow (No. 2) before being dismissed, but was still politically very active.

I learned about the school from A. M. Vinogradov in 1978 and taught there from the very beginning with him and several of his pupils. Later the “staff” was joined by D. B. Fuchs, Andrei Zelevinsky, two graduate students (Alexander Shen and Arkady Vaintrob, both talented and dedicated teachers) and others. I taught the “Higher Algebra” course and (if I remember correctly) an introductory course on “Analysis on Manifolds”, usually twice a week in the late afternoon.

The “enrollment” was typically some 50 or 60 people (about half as much were later of each year), the classes, first held in Bella's cramped apartment, we later made almost official as optional seminars at the Gubkin Oil Gas institute on Leninsky Prospekt. The students were also learning math at the official institutions where they had been accepted, so our curriculum was not traditional — we tried to teach basic modern mathematics in a novel way. Apparently some of the courses were attractive to more than just first-year students; even some from Mekh-Mat actually attended.

Overall, the enterprise was probably doomed to failure for pedagogical reasons — a small group of teachers could hardly hope to play the role of a real university. But we tried, during classes and in talks outside of class hours, to give some glamour and purpose to the study of mathematics to people deprived of a stimulating environment.

Some of our students did eventually go on to do research in the field, under Arnold and Fuchs in particular. Some of the names I remember are (unfortunately, I have not saved any written records): V. Ginzburg, B. Shapiro, S. Rogov, V. Etkin, B. Kanevski, F. Malikov, A. Lifshitz, A. Gokhberg, M. Fradkin; they have all become mathematicians, except for Fradkin who is a physicist.

Bella Muchnik University ended its existence under very dramatic circumstances, in the winter of 1982/83. V. Senderov was arrested, as well as two students (one was soon released). Several students were interviewed by the KGB. Classes ceased of their own accord. Soon afterwards, Bella Muchnik, returning home very late in the evening, was killed by a truck; the police never found the hit-and-run driver; few of us believed it was an accident. Classes never reopened.

And yet, the combined efforts of the KGB and the party bureaucrats at the Academy of Sciences, at Moscow University and in the Ministry of Education did not succeed in destroying the unique spirit of the mathematical community that had flourished at Mekh-Mat in the sixties. To be sure, mathematical schools were closed down, but new ones opened, where people like Shen, Vaintrop, B. M. Davidovich continued to teach. The All-Union Olympiad, taken over by bureaucrats from the Ministry of Education, had been transformed from a stimulating meeting place for talented students and dedicated research scientists into a training polygon for future performers in the politically prestigious International Olympiad, but the old olympiad traditions were preserved thanks to N. N. Konstantinov and his unofficial “Tournament of Towns”. *Kvant* went on doing its remarkable job. School No. 18, no longer the unique place that I have described, still continued to produce future first-rate mathematicians. Leading mathematicians who did not have (or gave up) full-time positions at Mekh-Mat (e. g., Gelfand, Arnold, Manin, Sinai, Novikov, Fuchs), kept their seminars there going, and the Russian tradition of research work gravitating around large, close-knit groups of mathematicians of various generations, each headed by its own *maestro*, is still very much alive.

Things changed radically after 1985 with the advent of *perestroika*. It has certainly changed my own life completely. I have been allowed to travel abroad (after 30 years!), I finally have the kind of job a mathematician can only hope for (I hold a research position at V. P. Maslov's Applied Mathematics Section at MIEM, the Moscow Electronic Machine Design Institute, where I also do a little teaching), I am happy in my family life (my second wife, E. N. Efimova, shares my world outlook and attitude about mathematics, being herself an alumnus of the Mekh-Mat of the late sixties), and I feel I can now make my contribution to mathematics and to the mathematical community without restricting myself to marginal activities, without having to avoid mainstream mathematical life.

My main regret, however, is not that all of this didn't happen sooner. It has to do with my concern, shared by many Russian mathematicians of my generation, and clearly expressed by my friends O. Ya. Viro and Yu. S. Ilyashenko at the ICM in Kyoto, that our failing economy, together with the freedom of movement granted by *perestroika*, will lead to an unprecedented brain drain and the drying up of the Russian mathematical school. When my generation leaves the active mathematical scene, there will be no one here to teach talented students, since all the best people of the next generations will have emigrated.

I can only hope that this discouraging prediction will prove to be as accurate as my naïve optimism in the late fifties or my hopeless pessimism of the late seventies.