Ilya Piatetski-Shapiro Memorial page » by Andre Toom

Ilya Piatetski-Shapiro in 1963-1973

Passion for Applied Mathematics

André Toom

Andre Toom

In the year 2000 American Mathematical Society published Selected Works of Ilya Piatetski-Shapiro, an internationally renowned mathematician, who not only worked in various areas of mathematics, but succeeded to connect them together and received many deserved prizes during his long life.  However, out of more than eight hundred pages of this weighty volume, the section «Applied Mathematics» occupies only ten pages and this is just from the purely professional point of view because the scientific value of this outstanding mathematician’s applied works is much less than that of his theoretical works.  In addition, his applied works are localized in time: all of them were published between years 1963 and 1973.

Why did a scientist so successful in a bunch of interconnected topics of mathematics spend ten years of his life at an age, most productive for many scientists, on activities, which seem remote from those topics, where he was especially successful, and brought him no prize?  I don’t believe that somebody forced Piatetski-Shapiro to work in applied areas in defiance of his genuine interests.  I am generally convinced that to spend one’s life doing what is not interesting for you or, as one poet said, to “step on the throat of your own song” is silly and useless both in poetry and mathematics.  Ilya Iosifovich (as we called him in Russia) understood this very well because he had the specific wisdom of a creative person.  If during ten years half of his publications were applied works, written in collaboration with specialists in various applied areas, we must conclude that this activity was interesting for him or perhaps even necessary for some reason.  I claim that the history of Piatetski-Shapiro’s interest in applied mathematics is connected with a certain period in the life of the Soviet society, to which we belonged at that time and with a history of several scientific groups, including that one, which he led and to which I belonged.

I became acquainted with Ilya Iosifovich in 1963 when I was yet an undergraduate student of the department of mechanics and mathematics of Moscow University.  He turned his attention to me because I had already received my first result and published my first paper – an estimation of complexity of multiplication of multi-digit numbers (a partial solution of a problem posed by Kolmogorov).  Ilya Iosifovich asked me, what I was going to do in mathematics and I answered that an older friend advised me to work in functional analysis, but I decided not to go there. “Why?” - he asked.  I answered that I could not explain it.  He said: «I also decided not to go into functional analysis, but unlike you, I can explain my decision.”  I asked, why, and he answered: “Because it is not interesting for me”.  Now, when I am writing this memoir more than forty years later, functional analysis is very interesting for me, but there is a time for everything.  I believe that at that time I was right choosing an area far from the central parts of mathematics, because I felt awkward in pure mathematics.  Mathematics at that time (I don’t know about the present time) was taught at the department of mechanics and mathematics as an abstract discipline without any connection with other sciences and this monotonous diet made me sick.  Lectures in physics might help, but they were delivered by Fursov, the dean of the physics department, protégé of the Communist authorities, and his lectures were detestable. That is why I willingly accepted Piatetski-Shapiro’s invitation to his seminar.  The name of his seminar was broad, something like “Mathematical methods in biology” and this broadness attracted me.  Ilya Iosifovich asked me to speak at his seminar first about my publication and then about Turing machine.  Shortly after that, with his advice and support, I accepted a position at the laboratory of mathematical methods in biology directed by Israel Moiseevich Gelfand.

          At that time Piatetski-Shapiro worked in the Institute of Applied Mathematics or IAM for short, in the department headed by Gelfand. This institute was created following Academician Keldysh’s suggestion, to provide mathematical support of the most secret projects.  It was known in academic and governmental circles that when Sakharov faced a mathematical problem, which he could not solve, he went for advice to Gelfand at IAM.

When Gelfand’s young son tragically passed away from an unstudied disease, Gelfand from his personal experience comprehended the importance of medicine and decided to use his position of influence to develop mathematical methods in biology.  In this vein, besides his department in IAM, which “worked on war”, he created and headed a laboratory of mathematical methods in biology in the Moscow University, which, as he supposed, would “work for life”.

Several young mathematicians were enlisted into this laboratory about the same time with me: Nickolas Vasilyev, Andrey Leontovich, Marina Petrovskaya and Olga Stavskaya.  Piatetski-Shapiro, whom Gelfand put in charge of our group, decided to puzzle us by performing computer simulation of some processes similar to biologic ones and stimulate us to study these processes theoretically.  The seminar led by him grew little by little: we were visited by students of Roland Dobrushin, Yasha Sinai and Robert Minlos. Leonid Vaserstein, who always walked by himself, also visited us. Then the next generation appeared: Alexei Vasilyev, Alexander Koganov, Leonid Mityushin. Then came another generation, of which I remember Gregory Galperin best.

I believe that it is impossible to understand events of our lives of that time without remembering political changes in the country. Mass de-stalinization had already taken place and many victims of repressions were acquitted. Anti-semitism decreased. Life became better for all. Passed away the time of the bitter joke at the department of philosophy: “What is thought? It is the shortest distance between two quotations.” The country became more open, some important scientific books were translated, including books on “cybernetics”, the topic which had been forbidden not long before that, in fact on theoretical issues of control and communication.  New interesting possibilities opened for the Soviet scientists.

Gelfand, Piatetski-Shapiro and all of us were very much influenced by the ideas of Michail Lvovich Tzetlin, who had been an engineer and came to mathematics with his own very interesting theme.  It was a kind of game theory, namely games of automata.  Tzetlin correctly observed and convinced all of us that the game theory, as it was developed in the West, made a very strong assumption about unlimited intelligence of the players.  This assumption makes some sense if by players we mean giant corporations, which have their “brain trusts”, capable to analyse the situation at the capitalist market and develop optimal strategies of economic behavior.  However, many human individuals have to play games, which life imposes on them, being very short of information and means of its processing. Tzetlin proposed to consider games, in which the players were simple automata incapable of complicated thinking.  «A small animal in a large world” was his favorite saying.

In the same year 1963, when I started to communicate with Piatetski-Shapiro, he published his first “applied” work on games of automata in collaboration with Gelfand and Tzetlin.  It was a declaration of interest rather than a report of completed results.  Its purpose, as I think, was to attract young scientists to this theme.  Since that year Piatetski-Shapiro appeared as an author in various applied works including determination of the structure of crystals, earthquakes, physiology, cell regulation and development of biological forms – up to year 1973 when his last “applied” work was published; it was a report of modeling of a biological process in collaboration with L.V.Badyin and E.B.Vul.

Why did Piatetski-Shapiro, so successful in abstract mathematics, indulged in this so new for him kind of activity?  Not just because Gelfand, his supervisor at IAM and the university laboratory, ordered him to do it.  I am sure that there was another strong reason - the fruitful idea of connection between mathematical theory and engineering practice, which received at that time a new and original impetus in foreign studies, especially in the works of Norbert Wiener.

At that time Gelfand, Piatetski-Shapiro and Tzetlin gathered around themselves a very productive group of scientists to develop new studies in an area, which had been forbidden quite recently.  This group was large and heterogeneous and it is impossible to present its exact list.  Besides the three “whales” mentioned above, it included at different years  Yu. I. Arshavski,  L. V. Badyin,  M. B. Berkinblit,  V. A. Borovikov,  V. I. Bryzgalov,  A. V. Butrimenko,  L. М. Chailakhian,  S. L. Ginzburg,  V. S. Gurfinkel,  S. А. Kovalyev,  V. I. Krinski,  V. Yu. Krylov,  V. B. Malkin,  А. А. Miliutin,   М. L. Shiк,  V. V. Smolianinov,  V. L. Stefanyuk,  V. I. Varshavski,  V. А. Volkonski,  I. P. Vorontsova,  Е. B. Vul and many others.

It is worth noting that almost all the members were not mathematicians in the exact sense.  Piatetski-Shapiro (excluding Gelfand who took the median position of an arbiter) is almost the only one mathematician in this list with a level of abstract thinking allowing him to maintain rigorous and complicated arguments.  Thus Ilya Iosifovich had to represent almost alone one of the two sides in Wiener’s idea of connection between theory and application. He had to counter-balance fertile but non-rigorous creative work of all the others, among whom even some nihilism towards mathematical rigorousness might be heard sometimes. At a seminar devoted to games of automata the speaker admitted that one of his theorems was not proved. “It is good that it is not proved!”– shouted somebody in the public. Ilya Iosifovich, who told me about this case, added:  “Of course, there is nothing good about it.”

Besides all that, the scientists, who united themselves around Gelfand to study problems of control and communication, perceived their work as a contribution to democratization of the country.  Officially, all members of this group solved different specific problems, remote from politics, but due Wiener we understood that all parts of the world were interconnected.  For example, solving problems of physiology of nervous system to help invalids, Tzetlin and his co-members in the physiological seminar had in mind very broad analogies, which they held back from their technical reports.

Piatetski-Shapiro and Tzetlin spent many hours together. Discussing their ideas, they well complemented each other.  Piatetski-Shapiro’s contribution was high professionalism in rigorous abstract mathematical reasoning.  Tzetlin contributed his experience as an engineer and in addition he told gorgeous parables.  Our common dream was to turn these parables into rigorous mathematics.  This is one of his essays (published in his book).

«For example, I found out with astonishment that the work of prisoners is more expensive than work of free people, although prisoners are fed much worse, dressed worse and the number of their working hours is not less.  This is not just because the prisoners’ productivity is less; the point is that a prisoner needs to be fed and dressed and guarded.  I am treated in the following way: they pay me salary twice a month, then I pass it to my wife and this is enough for my administration to be sure that I am provided with food, dress, shoes, that I shall not be kidnapped and so on.  Administration does not need to think when I need new shoes or bed-sheets or what to do with my children etc.»

It is appropriate to compare the ideas of Gelfand and his colleagues with ideas of their rivals – Academician Victor Glushkov and his collaborators.  Yet in the beginning of the 60-s Glushkov, supported by the Academy of Sciences of USSR, proposed to the government to develop a National Automatized System of Administration of Economy.  According to his estimations, this goal could be achieved in 15-20 years with an investment of 20 billions of Soviet rubles.  If this system were realized, it would push to the extreme centralization of the soviet economy, which already was excessive, and after all the expenditures it would be undermined by the collapse of the Soviet rule.  Perhaps, Soviet bureaucrats had some presentiment of this sort, because in spite of the kinship of Glushkov’s idea with their mentality, he did not receive their support on that scale on which he expected it.

What about Gelfand’s group, it did not offer any radical solutions of economic problems, but rather emphasized inefficiency of small-minded, petty control and advised to appreciate and develop independent actions and self-rule.

When it became clear to us that we could not realize our dream of tuning parables into rigorous scientific knowledge due to lack of various kinds of resources, we turned our attention to another, more realistic goal: to develop a mathematical theory of processes with local interaction, bearing in mind its future application to natural sciences.

It was Piatetski-Shapiro who aroused our interest in random processes with local interaction!  At that time only a few scientists studied such systems.  Only John von Neumann’s auto-reproducing robot, which lived on checkered paper, came many years ahead of us.  All the further studies in this area started to be published only in the sixties and at first only in the deterministic case.  The first probabilistic process with local interaction, namely Glauber Dynamics appeared in 1963 – the same year, when we started our work, and it was only one concrete system in a concrete physical context.

To me processes with local interaction seemed at first so natural, so obligatory for study that I thought that they should already have been investigated and that it was sufficient go to a library and burrow around the theory of partial differential equations to find a reference book in which all these processes should be already described and classified.

That I was wrong was easy to conclude from that well-known fact that Turing machines are easy to imitate by deterministic cellular automata, which are special cases of probabilistic cellular automata , and therefore to describe their behavior certainly is not easier than to describe behavior of all Turing machines, which is pretty hard to put it mildly.  Perhaps, this was the reason why Ilya Iosifovich had invited me to give a talk on Turing machines at his seminar – to make me aware of this connection?

Several years later George Kurdyumov and Nickolas Petri appeared among us with proofs of algorithmic unsolvability of two very natural problems from our area. Then Leonid Mityushin and me made a similar contribution. All this showed to me still more clearly that my idea of looking for a reference book, which would answer all the questions about such processes, was doomed to failure.  I had to admit that Piatetski-Shapiro’s approach, which consisted in a study of particular cases, was the only possible one in that area.

The first one of these particular cases was the random process, which later became known as Stavskaya process because it was Olga Stavskaya  (under Ilya Iosifovich’s leadership, of course) who performed a computer simulation of this process and showed its most interesting property – phase transition at a critical value of parameter, which was strictly between zero and one.  This property, in fact non-ergodicity for small values of parameter, was proved using two different methods by Mikhail Shnirman (Sinai’s student) and me.  Both proofs were published in 1968, the same year when the result of Stavskaya’s simulation was published.  This success moved our seminar, led by Piatetski-Shapiro, on a par with the two great Moscow seminars devoted to mathematical problems of statistical physics, namely Dobrushin’s seminar in the Institute of Problems of Information Transmission and Sinai’s seminar  at the department of mechanics and mathematics of Moscow University.  It turned out that all this time we were doing mathematical physics without knowing this, like Monsieur Jourdain who spoke prose without knowing this. This is where we found our proper place in the world! The Journal of Statistical Physics continues to publish my articles on random processes with local interaction till now.

In addition, it became clear to us that we played the role of Jourdain in another way also.  In the very beginning of our work we noticed that our random processes were easy to approximate by more simple ones, obtained from them by mixing all the components randomly at every time step.  First we treated this approximation as a mistake, but  Lev Rozonoer, who once came to our seminar, explained to us that this approximation was widely known in physics and therefore its using is respectable enough, even though nobody could estimate the resulting error in the limit behavior.  It is this method of approximation, which plays an important role in the only applied work included in Piatetski-Shapiro’s Selected Works published by AMS. The same method provided a good approximation for Stavskaya process that is it suggested existence of phase transition, although, of course, it did not provide exact critical value and could not substitute a rigorous proof.  I became interested in Stavskaya process because it certainly was oriented to some natural phenomena (if not biological, then at least physical) and it was clear that it could be generalized in various ways.  Other members of our group became interested in other problems posed by Piatetski-Shapiro.  Andrey Leontovich was so carried away by mathematical modeling of biological form development that he works with it till now.

In spite of all his scientific merits, Piatetski-Shapiro was a poor teacher. He could not explain and collaborated successfully only with those who understood him without explanations.  This happens to great scientists.  It was said about Kolmogorov that when he spoke to students, only professors understood him and when he spoke to professors, only Khinchin understood him, and Khinchin passed away in 1959…  A student of Norbert Wiener remembered that when delivering a lecture, he turned his back to the audience, wrote something on the board, then erase it murmuring something like “This was very wrong,” then started again – and this alternation of writing and erasing might continue till the end of the hour. Gregory Galperin described Piatetski-Shapiro’s lectures as follows: «He would run hither and thither in front of the board, every time poking into one and the same spot on the board and then, when a cone of chalk accumulated at that point, he would turn to the audience and ask: «Is it clear?»

So what was so attractive about Ilya Iosifovich for rather fastidious young mathematicians including myself?  I think that first of all it was his mentality and scale of values of a creative person. He never carped, never piled up trifles, but found and posed interesting key problems, which seemed easy at first sight, but in reality were deeply non-trivial, and insisted on solving them.  He turned us into an excellent scientific team – everyone got a place according to his or her interests and possibilities, every member contributed to the common cause.   

There was something about Ilya Iosifovich which made him similar to an ancient Jewish sage.  His great merit was his attention which he paid to everyone of us and his ability to support and to reprove without boredom.  There was a time in my life when my private life was complicated (first marriage, birth of a son, divorce, second marriage, birth of the second son) and I produced nothing in several years.  Ilya Iosifovich treated my scientific barrenness in a very tactful way, being sure that it was temporary and that I yet would prove my worth – and I proved it by obtaining a result for which I received a prize of the Moscow mathematical society.  There was a time when Nickolas Vasilyev lost his level in research.  Ilya Iosifovich reacted thus: “Nick, you have three big faults. You are very modest, very polite and you understand everything very quickly.”  This hit the nail – our friend Nick really had all these three qualities and all the three impeded his research. Nickolas did much for mathematical olympiads and it was well-known that if he could not solve a problem in five minutes, this problem was not for olympiad – too hard for that. But for a big success in mathematical research Nickolas lacked obstinacy.

In the second half of the sixties the studies of games of automata faded away. One reason of it was that Tzetlin, the main generator of ideas in this area, deceased in 1966 and Gelfand claimed that without Tzetlin he could not continue studies in this area. But in reality there was another reason to cut off studies in this direction: political pressure and ideological tension increased in all the country. This was a case when politics influenced mathematics.

Let me remind some events of those years.  In 1966 the writers Yuli Daniel and Andrei Sinyavski were put to jail for “anti-soviet” writings; in 1967 Svetlana Alilueva, Stalin’s daughter received political asylum in USA; in the same year Alexander Solzhenitsyn sent an open letter to the fourth National Congress of Soviet Writers demanding the abolition of censorship;  in the same year the Six-Day War burst out in the Middle East; in 1968 the Soviet troops occupied Czechoslovakia.  In all of these events the Soviet government played an infamous role.  Liberalization of the Soviet regime was dead for a long time.  In the Moscow University this retreat was  aggravated by the death in 1973 of the famous Ivan Georgievich Petrovsky, who had been defending science from silly but dangerous attacks for many years.  Parables could not be tolerated any more as openly declared themes of research.  Drawing of broad analogies in Wiener’s spirit decayed not for lack of thinkers, but for oppressive political situation in the country.

At the same time we, members of the crew collected and led by Piatetski-Shapiro, at full speed studied random processes with local interaction, which we called uniform random media at that time. In our first publications we hoped to use these abstract processes to model biological structures, for example nervous tissues.  However, it was becoming still clearer to us all the time that biological processes were too complicated to study using our simple models.  In this connection Piatetski-Shapiro still more often consulted with Dobrushin, Sinai and Minlos, who had been successfully developing mathematical models of statistical physics for many years.

Of course, human qualities and human relations played an important role in our research.  Mutual help was common in our group.  We had no quarrels about priority.  Ilya Iosifovich served as a model of honesty and generosity for us.  He never included his name into articles of his students if he had not made a concrete contribution.

The only source of quarrels was ...  Israel Moiseevich Gelfand.  Ilya Iosifovich said half jokingly that Gelfand collaborated with Tzetlin by the “method of scandals”.  Tzetlin was a complacent man, easy to deal with, but Gelfand created tensions even where it was possible to manage without them.

Gelfand quarreled with Piatetski-Shapiro also, but the latter almost never complained to us except one case.  As soon as our work with Ilya Iosifovich began to bring fruits - that is publications - and it became clear to us that we found a “bonanza,” Gelfand decided to appropriate it – to make an appearance that it was he who suggested this theme to us and that all our research was a fulfillment of his wise design.  It is natural that Piatetski-Shapiro objected because he was the founder of this direction without any doubt.  Then Gelfand played a bureaucratic trump: he reminded all of us that he was the chief of the laboratory and from that formal point of view we, young mathematicians, were his subordinates, not those of Piatetski-Shapiro.  He behaved as a typical soviet tyrant official corrupted by Stalin’s rule.  But tyrant officials are stupid, while Gelfand is talented!  What is the use of others’ services for him if he has plenty of his own? How a scientific genius and an administrative extortioner could coexist in one person? For me it is an enigma till now.

Gelfand’s ambitions had consequences – they created tension in the laboratory and spoiled our relations with him forever.  It became clear for us that we could not work as before. Piatetski-Shapiro started to reflect on where should he be and what should he do. In 1974 his first wife with their son emigrated to Israel and he started to say that he would like to see his son.  In those conditions it meant emigration. At that time we could not even dream of traveling at will and coming back as all normal people do in other countries.

Soon Ilya Iosifovich applied for emigration to Israel.  Since he worked at IAM, he might have access to secret documents (although in his case it was a mere formality), in result of which he was not allowed to go immediately and became a “refusnik” for a year or two.  We, his students, visited him and wrote with him a survey summing up the ten years of our collaboration.  To make this survey publishable in USSR, Piatetski-Shapiro removed his name from the list of authors, because in Soviet conditions it would hinder publication.  However, publication in USSR did not occur anyway.  Instead the survey was published in England.  Now that book is a rarity.

In 1976 Piatetski-Shapiro was released from USSR.  From time to time he sent me postcards from various places.  Here are texts of some of them.

«Dear Andrei!  Thank you for the copy of “Functional analysis”, which I received from you.  Lately I visited Rome, that is stayed there for two days on my way back from America.  It was very interesting.  Unlike Paris, which I have visited many times and for days, this visit to Rome was practically the first one for me.  Now I plan to spend 4 months in Tel Aviv and then go to Yale for 4 months.  Write to me how your life is going. I. P-Sh

«Dear Andrei!  Thank you very much for the “Func. analysis”, which I received from you lately.  Every time it is a pleasure for me to receive this journal. On my way from America I stopped in Rome for two days.  I shall stay here in Tel Aviv till the end of December. Then I shall start to travel again to Germany, California, New Haven and in the end of April back here. My best regards to everybody. Yours I.»

«Dear Andrei!  It was a pleasure for me to receive your letter.  Please find enclosed my photo with Lenya in Itaca.  The car is rented, not mine.  Itaca is a wonderfully beautiful place.  At the same time I am sending you three books: Impressionism, Raphael, Degas.  Now write to Tel Aviv.  My best regards to you all.  Your I. I.»

«Dear Andrei!  Your letter is very interesting.  I see that your life is very intensive.  I suppose, mine also is.  In the end of December I go for 10 days to Paris and Germany.  In Germany I organize a conference  together with one mathematician who was born in Ireland and now is a professor in Gőttingen.  Then I come back here for a week and then I go for three months to America (Yale).  I am very glad that you revitalized our old seminar.  However, I am concentrated on automorphic functions now.  What is my interest in life?  Work, it was my main interest always and also I have a daughter, 5 ½ years old. Next year she will go to school.  Write to Yale.  Yours I.I.»

Sometimes Ilya Iosifovich sent me letters, but rarely.  I brought these letters to our laboratory, my colleagues read them and passed them to their friends outside the laboratory, so that these letters were read by all who had collaborated with Ilya Iosifovich or was acquainted with  him.  I did not receive them back.  I remember the photo mentioned in the letter, but I don’t have it now.  Lenya mentioned in the letter is Leonid Vaserstein. We joked that this photo showed one spy passing secret documents to another, because Lenya wore dark glasses. The joke played up the gap, which was rapidly increasing at that time, between the healthy reality, which rushed ahead, and old sick rigid Soviet myths, which lagged far behind and were serving an object of jokes still more all the time.

However, since Piatetski-Shapiro’s emigration our scientific collaboration ceased.  Communication with emigrants was forbidden in USSR.  It was forbidden even to refer to emigrants’ publications.  Sometimes this restriction led to funny consequences.  When Piatetski-Shapiro emigrated, Olga Stavskaya still had no degree, although she had done more that anybody else for computer simulations on those awkward machines which were available at that time.  Dobrushin and Sinai considered it their duty to help her.  They served as official opponents and chose Leningrad as the place of defense to get away from the Moscow intrigues.  According to the requirement of authorities, the name of Piatetski-Shapiro was not mentioned at the defense, which made impression that it was our nice friend Olga who founded a new scientific school.  The well-warned members of the scientific council kept straight faces and did not ask inappropriate questions.  The defense was successful.

What about Piatetski-Shapiro, after his emigration he did not work in computer simulation or processes with local interaction.  In his new surrounding he did not feel urge for this work and stopped it.

However, the group created by him continued to work in the same vein for several years.  We even did more: we started to conduct regular meetings in Puschino, an academic community outside Moscow, where several biological institutes were located, and published several collections of  articles based on these meetings.  However, I.M.Gelfand, the chief of our laboratory, excluded our studies of random processes from the official plan of research.  This was his revenge for our independence.  Luckily, due to the general chaos, we succeeded to continue our work, to keep our positions and even sometimes to get advancements. 

Under normal conditions it would be natural for Gelfand to become our leader after Piatetski-Shapiro’s departure. We would not object to this if he were a little more polite. The problem was his desire to pose as our spiritual leader after years of neglect and complete absence of advice.

In 1989 I left the Soviet Union also.  Having come to USA in 1990, I visited Ilya Iosifovich in New Haven.  By this time he already had moved from Israel to USA and worked at Yale University. However, Israel always remained his dream.  He immediately advised me to go to Israel as if I could substitute him there.  He added: «But they must promise you a university position for sure ».  I was desperately looking for a position at that time and liked this idea.  However I warned him that I was not a Jew according to Galaha, to which Ilya Iosifovich proudly replied that it had no importance (which hardly was true).  At that time he was preparing to go to Israel to receive the prestigious Wolf Prize and was sure that he would be able to help me, but he could not.  This is not astonishing because Israel was overcrowded with scientists from Russia and Eastern Europe at that time.  As one of my friends said in broken Russian, “in Israel doctors of sciences sweep streets”.

In the same year Eugene Borisovich Dynkin invited me to Cornell to give a lecture and just to talk. He was collecting materials on the history of soviet mathematics and recorded our conversation.  By chance he used the same reel on which his conversation with Piatetski-Shapiro had been recorded, so I heard some part of that record.  Dynkin, who knew much about Soviet mathematicians, asked Piatetski-Shapiro: “Is it true that in some articles, where I. M. Gelfand is named among authors, his contribution is close to zero?” Ilya Iosifovichs answer astonished me. He started to shield our former chief: «There was some contribution of Gelfand always, sometimes purely human».  What made Ilya Iosifovich bend the truth?  Gratitude to an older colleague who helped him in hard Soviet conditions?  Unwillingness to wash his dirty linen in public?  I never asked him.  After his departure Piatetski-Shapiro lost interest in the laboratory. He avoided to remember the past.

My last meeting with Ilya Iosifovich was in Princeton, where he was invited as a visitor. My wife, my daughter and me came to see him and his family.  He already was very sick and weak, but his intellect was perfectly productive. We have some photos made by his wife Edit at that day.

Ilya Piatetski-Shapiro, Andre Toom at Princeton Ilya Piatetski-Shapiro and Andre Toom, playing chess in Princeton, 1999

Remembering now the years of our collaboration in Moscow University, I come to conclusion that Piatetski-Shapiro’s interest in applied mathematics was not accidental. It was a collective attempt made by many congenial scientists to connect theoretical and applied mathematics for the sake of their mutual enrichment. In Russia this attempt failed as a whole due to unfavorable political climate, although there were many specific successes. But it does not mean that it failed in general. This attempt stimulated scientific work of our colleagues in other countries.  I suggest that it contributed to the recent upsurge of collaboration of science and engineering in Israel.

Ilya Piatetski-Shapiro left a trace in the lives of his students.  Let me say about myself: my life would be different, less interesting if I did not meet him as an undergraduate and if he did not invite me to his seminar.

During the last ten years I work in a Brazilian university.  I already have raised enough students to create a small school led by me.  In Russia it was impossible, in USA it did not happen, but in Brazil it became a reality.  Ilya Iosifovich’s treatment of young scientists and his methods of leadership serve me as a model in my work.

April-May 2009