Mathmech Life

Abstract

Life at the Mathmech on Vasilyevsky Island 10th Line continued until late at night, and I often stayed there the whole day, like many other students. If lectures ended at 3 o’clock, seminars usually began at six to allow people from other schools to arrive on time for them.

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5.1 The Mathmech Cafeteria

Life at the Mathmech on Vasilyevsky Island 10th Line continued until late at night, and I often stayed there the whole day, like many other students. If lectures ended at 3 o’clock, seminars usually began at six to allow people from other schools to arrive on time for them.

That is why the Mathmech cafeteria could not but play an important role in my life. Due to the fact that I gave away almost all of my scholarship to Mother, I was a bit hard up for pocket money and preferred the department cafeteria to the two more expensive neighboring ones (on Middle Avenue and the 8th Line) where they served better food.

Our cafeteria was on the first floor in the left wing of the building. Shortly before receiving my monthly scholarship the semi-communistic character of the cafeteria became important: you could satisfy your hunger for free because on each table there was sliced bread. What a wonderful taste a chunk of rye bread had with pungent mustard spread on it and strewn with some salt! I am imagining all that now and my mouth is watering, although this type of food did not fit every stomach.

But “Man shall not live on bread alone”,¹ so the Department cafeteria could allow you, without going bankrupt, to treat yourself to beetroot salad or a hard-boiled egg. Even the stewed fruit would cost you kopecks. Quite a few students loved it, and I was one of them. At the same time the meat patties filled with bread were terrible. I clearly remember it.

My student years were flying, the deans, Party secretaries, and their deputies were changing. The atmosphere in the country and Department was changing too. I changed as well. Only our Mathmech cafeteria remained the same.

5.2 Fractional Derivatives

Although even at high school my greatest dream was to come up with something new in mathematics I was unable to do it in those days. But at the beginning of the second year in Mathmech there was a moment when it seemed to me that I made a discovery in mathematical analysis: I invented fractional derivatives. (The next paragraph can be omitted by a lay reader.)

The idea was simple – to define a real parameter dependent operator which coincides with the differentiation of integer order for natural values of the parameter. Omitting the details, I acted in the following way: I expanded a function into Taylor series, replaced the sum with an integral, and the factorial with the Gamma Function. Thus an integral operator was obtained which gave the required result after an inversion.

Photo 5.1

V. M. Babich (before 1967) (© Naukova Dumka, reproduced with kind permission)

Previously I had not come across anything like that.² In answer to my question Professor S. G. Mikhlin sent me to the associate professor V. M. Babich.³ He looked at my calculations and said: “Because of exactly the same fractional derivatives, Liouville nursed a lifelong grudge against Gauss. He invented those derivatives and sent his formulas to the Great Gauss who said the following: ‘Young man, don’t engage in rubbish!’” That was how V. M. Babich immediately destroyed my youthful illusions. As a consolation he added: “If you lived in those days they would believe you were a good mathematician”. These words, to a certain extent, made up for my injured self-respect.

5.3 Something New at Last!

From the second through the fourth year I devoted little attention to the obligatory course program. I would not say no attention at all, and even if I said that you would not believe me as I got a diploma cum laude with only one “B” for “theoretical mechanics” that I hated.

But what can explain my irresponsible behavior that complicated my life and made me nervous, thus causing uncertainty? The answer is simple: I am absolutely unable to work under the lash. I suspect that this trait of mine was inherited by my daughter and poet Gali-Dana Singer. Here is her answer to a question posed in an interview about her life.⁴

“…This is a very interesting observation. Probably this effect is caused by my general inability to study within educational systems. I played truant almost the entire ninth and tenth grades, then dropped out of the Institute… I could continue the list, but its gist is that having escaped from the pedagogical trap at least partially I had voluntarily doomed myself to the fate of “eternal student” who is in search of his lessons in the most unfitting places. But on the other hand, my poems lack exams, and I have lessons incomparably more numerous than teachers”.

I, too, desperately tried to find an occupation that would suit my taste. As I’ve already written above, still in high school I wanted passionately to introduce to mathematics something that was mine and new… In the late fall of 1957 a case like that presented itself at last. In one of the rooms of the Main Building Library there was open access to journals including mathematical ones. When visiting the library I made it a rule to look through them. I could not wait to see what real mathematicians were doing at that time.

One day, turning over the pages of the American journal “Quarterly of Applied Mathematics” of October 1955 I stumbled on a three page article by Hartman and Wintner “On the oscillation criterion of de la Vallée-Poussin”. For second-order ordinary differential equations the authors improved a condition of solvability of the first boundary-value problem, which was proposed in 1929 by the famous Belgian mathematician mentioned in the article’s title. It was a matter of a few minutes to read the article after which I suddenly understood how it was possible to further strengthen de la Vallée-Poussin’s result. The discovery inspired me and I began developing the theme. It turned out sufficiently benign, though not so much from my present day point of view as according to the understanding of V. Maz’ya as a third year student in 1957. This is how my first scientific work appeared, it was even “quasi-published” which I am going to describe a little later.

5.4 Student Scientific Society (SSS) and Tseitin

After the Picasso story my Komsomol work came to an end. Would Vladimir Maz’ya ever be revisited by the optimistic impulses that drove him to come to the damp basements in the vicinity of the Mathmech? He had to campaign among the residents for voting during elections in exchange for a promise of repairs. Would he, as before, enthusiastically organize cultural outings or sweat over student wall newspapers? The answer is negative. My activity of this type was never renewed afterwards – my character is that of a maximalist, either I completely give myself up to a business or abandon it.

However, rejecting social work entirely was a bad idea because it hampered my chances to enter the graduate division. Fortunately, the Mathmech offered an alternative to the Komsomol bureau called the SSS Council (see the title of this chapter). In it, at that time first violin was played by Gera Tseitin⁵ whom I would describe first of all as a child prodigy. Even his oddities fit this characteristic: for example, he never shook hands with anyone so as to avoid infection.

Gera, just 1 year older than I, was admitted to the Mathmech in 1951 when he was 15 years old. In the fall of 1957 when I, a third-year student, became closer acquainted with him, he already was a second-year graduate student and worked in the theory of algorithms and in constructive mathematics. Gera seemed to me a god – I was sure he knew the whole of mathematics. As a confirmation, having barely leafed through my above-mentioned opus, which I was going to submit to the contest of student works, Tseitin, with good precision, advised me to use Schauder’s⁶ fixed point principle. This fundamental topological theorem was not included in any lecture course of that time and I had not the slightest idea of its existence. But I coped with it quickly and used it. (Tseitin’s role is certified in my paper.)

Gera’s hobby was foreign languages.⁷ He was even studying Chinese seriously, but our first point of contact was Esperanto. It is no secret that this is a simple thing. According to Leo Tolstoy he “after no longer than two hours of studies was able, if not to write, but to read fluently in that language.” Tseitin knew Esperanto to perfection – I don’t know how long it took him to learn it. My achievements were not so impressive, but I, too, after some time, was able to communicate with Gera fluently in Esperanto.

Under the aegis of SSS, Tseitin and I organized the “English Club” which fact is confirmed by a notification in the Department wall newspaper. I preserved it and reproduce it here:

5.4.1 The English Club

Introduction

The English Club is known to have been founded recently by some students of our Department. Of course, the organization of this club as that of any unusual enterprise caused a certain distrust. It is its novelty that accounts for the hesitation which this undertaking has occasioned. The diversity of opinion about the cause shows that this initiative is new, difficult and vital. We write this article with the aim of dispelling all the doubts and indecision by throwing light on the tasks and methods of the English Club.

Our goal

We have only one aim. It is mastering English. What the clubmen want is to prove that English is not a luxury but a means of communication. Most of the pupils graduating from secondary schools cannot speak foreign languages, and all of us are victims of unproductive teaching methods. The members of the club are bent on improving these methods starting to work on a system described by a single word “conversation”.

Rights

Any member of the club may speak Russian with anyone who is not a member yet.

Duties

The duty of a member of the Club is to speak English with other clubmen and not to understand when they speak Russian to him/her.

Methods

The general method is the absence of special methods

We speak English. That’s all!

ENTER THE ENGLISH CLUB!!!

Next an additional text followed:

Announcement

The honorary member of the English Club Mr. A. D. Alexandrov, DSc, Corresponding Member of the USSR Academy of Sciences will deliver a lecture about his impressions from a trip to India.

This wall newspaper was issued in the fall of 1956. Did our Club last for long? Who were the participants? Alas, I don’t remember. All that was left are two yellowed sheets with the just quoted enthusiastic statements and two signatures: Maz’ya, Tseitin.

Today, Grigoriy Samuilovich Tseitin works at Stanford University, and in all probability has mastered English. As to his achievements in mathematics, I cannot ascertain them, being a complete ignoramus in mathematical logic, constructive mathematics, mathematical linguistics.

Photo 5.2

A. D. Alexandrov (about 1967) (© Naukova Dumka, reproduced with kind permission)

5.5 “Quasi-publication” and S. M. Lozinsky

S. M. Lozinsky (1914–1985).

In those years the SSS ran student contests in solving difficult mathematical problems, arranged popular lectures by professors for the students and Mathmech circles for grade school students. Recently, Tolya Slisenko⁸ reminded me of his participation in the Mathmech circle that I headed in the 1957–1958 school year when he was in the tenth grade.

We began to issue the “Student Scientific Journal”, a rotaprint edition. There appeared two booklets, one in May 1958 and another in October 1959. From the technical point of view both issues were quite mediocre, especially the second one. Many pages in it were impossible to read, but it is interesting to examine as a curiosity and a relic of that time.

The first issue of our journal contained two articles. One was “On the least number of multiplications for raising to a given power” by Rafa Valsky, a first-year student, with a supplement by the second-year graduate student Gera Tseitin, and the other was my article “On de la Vallée-Poussin’s criterion”. There was also an attachment presenting a talk by Vladimir Abramovich Rokhlin⁹ with a review of results obtained in geometric topology. At that time the students were not acquainted with this subject. Rokhlin began working at the Mathmech 2 years after his talk, and before him it was N. A. Shanin¹⁰ only who gave a semester-long course on set-theoretical topology.

Photo 5.3

My article in the first issue of the Mathmech Student Scientific Journal (On the criterion of De La Vallée-Poussin) (© Vladimir Maz’ya, private collection)

Photo 5.4

V. A. Rokhlin’s paper (Survey of some results of the geometric topology) (© Vladimir Maz’ya, private collection)

I did not send my first article, just mentioned, to a more solid journal as it was rather naïve. But I received the University’s highest prize of 100 rubles for it at the contest of student scientific works.

It was Professor S. M. Lozinsky who recommended it, head of the Chair of Mathematical Analysis in 1956–1960. At the end of his response it was written: “The results can be used as an excellent diploma thesis in any of the chairs: analysis or ordinary differential equations”.

Photo 5.5

S. M. Lozinsky (about 1967) (© Naukova Dumka, reproduced with kind permission)

Sergey Mikhaylovich was the son of Mikhail Leonidovich Lozinsky¹¹ who had died 2 years before the events described here. Mikhail Leonidovich was a remarkable poet and translator of whom I heard significantly later. After his death, his son Sergey Mikhaylovich, a man of high humanitarian culture, wrote magnificent reminiscences describing his father’s work on the translation of “La Divina Commedia” by Dante Alighieri.¹² His attitude to students was respectful but a little dry, with, I would say, a condescending smile, although barely visible. He was especially demanding with regard to formatting scientific works and even gave a lecture to the students on rules of writing papers. By the way, in his commentary to my article he included a compliment: “The work is written much better than you can expect from a third-year student.”

Photo 5.6

S. M. Lozinsky’s review of my first student scientific work (© Vladimir Maz’ya, private collection)

In order to inform me of his remarks on my paper S. M. Lozinsky invited me to his home. So, I came to his apartment on Kamennoostrovsky Avenue which was my first visit to a professor’s home. He took me to a huge study with bookcases lining all the walls from floor to ceiling. That was a library belonging to at least three generations – Sergey Mikhaylovich’s grandfather, a defense attorney was a famous bibliophile. Having entered the study I saw, on the right side of the door, a huge mahogany desk covered with green cloth and a big table lamp on it having a green shade. I have forever retained a reverential memory of that study. A real professor should perform his creative work in such a temple!

In 1960 S. M. Lozinsky left the University, which was his second job.¹³ In that year, working in more than one place was forbidden by Khrushchev who thus struck a blow against the material well-being of the scientific and technical elite. The Mathmech, too, suffered a notable damage having lost Lozinsky, a person of high moral standards. In the same year, the great mathematicians Y. V. Linnik,¹⁴ L. V. Kantorovich¹⁵ and A. D. Alexandrov quit the chairs they headed: “Probability Theory”, “Computational Methods”, and “Geometry” respectively. Besides, G. M. Fichtenholz passed away in 1959. In this way, the Mathmech grew poorer literally during a year.

5.6 The Mathmech Choir

Photo 5.7

Honorary diploma for singing in choir (Leningrad S. M. Kirov Palace of Culture. Issued to V. Maz’ya, participant of the choir of the Mathematics and Mechanics Department at LSU, for active participation in the VI Traditional “Celebration of Songs” at the S. M. Kirov Palace of Culture. Directorate, 1957) (© Vladimir Maz’ya, private collection)

Here is a document that characterizes another aspect of my life when I was a third-year student. This is a certificate issued to

Maz’ya V.

participant of the Leningrad University Mathmech Department Choir

who took an active part in the VI Traditional

“SONG FESTIVAL”

at the S. M. Kirov Palace of Culture

on Saturday, May 11, 1957

Of course a Department choir was not like the Big University choir conducted by G. M. Sandler and famous in the whole country. Not everyone was admitted to it. In our Matmech choir I was a baritone. My voice quality was modest, but if only you knew what a felicity it was to participate in polyphonic singing! It grips me at the heart even now: “On the little river, on a steep bank Marusenka washed her white feet…” or “Dear little wind, take my boat to the Kurland girl…”

5.7 My Doubts and S. G. Mikhlin’s Advice

Coming back to mathematics. In spite of my locally limited success in the work on the de la Vallée-Poussin criterium, I thought of myself as lagging behind. For instance, Yura Burago, a year older than I, had chosen geometry long ago. He came to the lectures and attended the seminar headed by A. D. Alexandrov. Rudik Yasnogorodsky, a student of the same year as mine, took part in Professor D. K. Faddeev’s algebra seminar. Both of them generously shared their impressions with me. At that time Yura lived on Pravda Street and we went home in the same direction. So, when we walked home on foot from the 10th Line he talked about the geometric theorems he had learned at the seminar. I understood not too much because I grasp unfamiliar mathematics by ear quite poorly. In general neither algebra, nor geometry attracted me.

The third year started. At that time I was solving problems from the two volumes by G. Pólya and G. Szegö¹⁶ and tried to cope with the magnificent book “Inequalities” by Hardy, Polya, and Littlewood. But because of a lack of definite plans for the future I began feeling nervous.

Photo 5.8

S. G. Mikhlin in early 1970s (© Vladimir Maz’ya, private collection)

At that time, unexpectedly, I was helped by Professor S. G. Mikhlin. On a rainy autumn day I happened to stand with him at a streetcar stop on Middle Avenue next to the 10th Line. I felt it was improper to speak first, but Mikhlin asked me: “Which specialization would you like to choose?” Having answered that no decision had been made yet, I added that my hesitation concerned the choice among analysis, complex function theory, and ordinary differential equations. Then he gave me advice: “Take mathematical physics. With this choice you would be able to research into anything: partial differential equations, ordinary differential equations, and even number theory.”

Using this opportunity I told S. G. Mikhlin of my suffering from doubts concerning my mathematical illiteracy, to which he answered: “I never studied anything just for the sake of erudition. And I don’t advise you to do it either. Choose a problem and study the materials related to it. Then you would look upon your studies from your own standpoint, and your knowledge would grow like a snowball.”

Solomon Grigoryevich was a wise man. In five minutes he described a strategy for the whole of my further scientific work. I was a new person when leaving the streetcar stop compared to the one who had come there!

5.8 A Few Words About Mikhlin

S. G. Mikhlin knew that I had grown up without a father, and, I would say, he looked after me in a fatherly way for many years. He often invited me to his place, talked about his life and answered most diverse questions. It was from him that I heard, still being a student, that Lenin was no less cruel a killer than Stalin, that concentration camps were first created under Lenin’s rule in the Soviet Russia. S. G. Mikhlin meant the Party and Administration University officials when he told me: “They just have power, but we have theorems. Therefore we are stronger!”

He himself, son of a melamed,¹⁷ resembled a rabbi in many of his features. When, in 1978, I told him that I was going to marry Tatiana, his former graduate student, his comment was: “How come you did not notice her earlier?!”

He explained to me the rules of scientific ethics, in particular the importance of referring to your predecessors irrespective of your sympathies, antipathies, and considerations of profitability.¹⁸

A convinced atheist, S. G. Mikhlin knew the Pentateuch and, by the way, reproached Thomas Mann for his exceedingly audacious handling of Torah in “Joseph and His Brothers”.¹⁹ Mikhlin liked neither the latter novel nor M. Bulgakov’s “Master and Margarita”. On the other hand, I reveled in both books, but did not dare oppose his opinion because I had not read the Bible yet, and probably had not even seen it. My sources for biblical history were the Hermitage museum and the popular book by Zenon Kosidowski “The biblical legends”, published in Russia in 1963. (Note that in those years it was very difficult to find a Bible, which was not for sale in book stores). In general it was difficult to argue with S. G. Mikhlin on humanitarian themes because of his confidence in his opinions, erudition and strength of argumentation.

But if we return to mathematics I can say that he did not offer me directly a single problem and that I myself thought up the themes of my diploma thesis and both dissertations. If I happened to develop his subject area I did it on my own volition. In this sense he was my teacher to no greater extent than, let’s say, S. L. Sobolev²⁰ or O. A. Ladyzhenskaya.²¹ But my work is characterized by a certain peculiarity that I undoubtedly inherited from S.G. Mikhlin. The thing is that on a large scale S. G. Mikhlin divided his research into “works”, each of them consisting of articles, and, as a rule resulted in writing a book. In the book he collected and regularized the results of his “work” considering it his duty. I got an impression that Mikhlin began his “work” impelled not so much by his own curiosity as by lofty objective ideas about the usefulness of the corresponding theory for the development of mathematics and its applications. Of course scientific curiosity played its part too, but so to say secondarily. The aspect of sportsmanship in mathematics was exceedingly alien to Mikhlin’s creativity. I confess that I don’t share this attitude with him.

Solomon Grigoryevich Mikhlin always considered S. L. Sobolev a great mathematician. The latter, who had studied with Mikhlin in the same group, always called Mikhlin by his diminutive name Zyama. Mikhlin’s other idol was Hadamard,²² and Mikhlin proudly told me that someone found a resemblance between him and the famous French mathematician.

Mikhlin’s monographs and textbooks are remarkable from the point of view of pedagogics, especially those devoted to variational methods and different classes of integral equations. Their style and accessibility to poorly prepared readers made Mikhlin famous in the world of engineers, which was a rare achievement for a mathematician.

S. G. Mikhlin’s highest accomplishment was creation of the theory of multidimensional singular integral equations which was presented in a large article, published already in the 1936 edition of “Matematicheskiy Sbornik”. Mikhlin introduced the notion of a “symbol” of a singular integral operator which made him the forerunner of a fundamental area in mathematical analysis of the twentieth century, known as the theory of pseudodifferential operators.

However, some influential Leningrad experts on partial differential equations did not admit that Mikhlin’s subject area was a part of the “main trend” and this attitude nagged the life out of him. He was never nominated to the position of corresponding member of the USSR Academy of Sciences. Now, when you read the names of some inhabitants of the “temple of Soviet science” of that epoch, they sound ridiculous.

But it’s time now to change the subject and add several other features to the portrait of an extraordinary person to whom I owe so much.

S. G. Mikhlin had an inherent sense of humor. He roared with laughter at the compositions of the “Oberiuts”²³ which in my time were accessible only through “Samizdat”.²⁴ He remembered by heart “Plisch und Plum” translated by D. Kharms from the German poem by Wilhelm Busch, and many poems by Edward Lear translated by S. Marshak, such as “The Cat and the Owl”, “In the country of the Jumblies”, “The Pobble who has no toes” and others.

Mikhlin never came to concerts, saying only that he perceived music as noise.

Self-critically he said that he lacked capabilities for foreign languages, although I happened to hear him speaking German and French.

He never prompted answers to poor achievers among the graduate students, and liked to repeat after Ilf and Petrov: “Saving drowning people is their own problem”.

His speech was logical and aphoristic, although not every one of his statements, as I understood later, was true to fact, e. g. “in a joint authorship one contributes his talent and the other his labor”, or “You don’t quit the Leningrad University, they carry out your body from it”. I myself violated the latter postulate by leaving the suffocating atmosphere of Leningrad University in 1986 – and never regretted that action.

In July of 1981, S. G. Mikhlin was elected a foreign member of the Italian National Academy (Accademia Nazionale dei Lincei²⁵). When he was not allowed to travel to Italy to receive the title, the Italian mathematician Gaetano Fichera and his wife brought to Leningrad the small gold lynx – a badge of Academician. They handed it over to Mikhlin in his apartment on October 17, 1981. Tanya and I were the only guests at that “ceremony”.

Towards the end of his life Solomon Grigoryevich Mikhlin dreamed of leaving for Israel but, for family reasons, could not achieve that dream.

5.9 In the Fourth Year

From the fall of 1958 I was permitted to have a “flexible schedule” at the Mathmech and practically stopped attending lectures. After Mikhlin advised me that I should study in accordance with a concrete problem, I decided to find a multidimensional extension of my one-dimensional results of the de la Valée Poussin type, considering the solutions of second-order elliptic partial differential equations. The Russian translation of C. Miranda’s²⁶ monograph published at that time gave a vast picture of the theory of these equations, and I soon became familiar with the book from cover to cover. In a short time I was significantly advanced and the new results with regard to unique solvability of the Dirichlet problem for linear and quasi-linear elliptic equations were presented in the second issue of our Student Scientific Journal. A brief statement of my results became my first real publication. It was a note in the “Doklady of the Academy of Sciences” submitted by Academician V. I. Smirnov on July 3, 1959.²⁷ The detailed proofs never went beyond the Student Journal as I did not value that work too highly.

Photo 5.9

My first “real” publication, 1959 (V. G. Maz’ya. On the solvability of Dirichlet problem for elliptic equations)

At that time and later two thick red volumes of “40 years of Mathematics in the USSR” were extremely useful to me. The first contains very well written reviews in all the areas of mathematics, and the second one is a collection of references. Those books allowed me to be in the know of all that was done by Soviet mathematicians in the theory of functions, functional analysis, and the theory of differential and integral equations. I went to the library and looked through the articles I was interested in. For example in this way I found out what O. A. Ladyzhenskaya had had time to do. A regular student would experience difficulties when listening to her lectures in the special course on second-order partial differential equations, but I attended the lectures fully prepared and understood every nuance, which resulted in nothing but pleasure on my part. O. A. Ladyzhenskaya only spoke about her own works and I had already read them! I did not like the special courses given by S. G. Mikhlin on multidimensional integral equations and M. S. Birman²⁸ on operator theory, but in the future their subject matter influenced me no less than Ladyzhenskaya’s special course.

The library just mentioned above belonged to LOMI (see Footnote 33 in Chap. 4) and was located on 24 Fontanka River not far from Marat Street. Starting from the fourth year I began going there regularly. The library had a huge stock of journals, a remarkable display of arriving supplies was renewed every 10 days, and close contacts with the Library of the Academy of Sciences allowed readers to be informed of the mathematical literature from the whole world. Customer service was fast, professional, and benevolent. I never stopped admiring the enthusiasm of the library workers who were ready to help everyone! I came there once or twice a week without fail, looked for new releases in the theory of partial differential equations, read, and took notes. After some time I began thinking that I knew all that had been done in the field.

In the spring of 1959 success came my way. The story started in the beginning of my second year at the Mathmech. By that time I had been keeping notebooks where I wrote down mathematical questions, thoughts linked to the materials I had read or my own ideas, and references to articles and books. I still have all of these notebooks. So, in the first one of them I see the words: “to study the growth of functions by the behavior of their level surfaces.” This was a vague presentiment of a fundamental idea that I was lucky enough to come across in the early part of the fifth year. This is how it happened.

I, like everybody else who graduated from the Mathmech specializing in mathematics, had a diploma which certified that its bearer was “a mathematician and teacher of mathematics in high school”. The last reference was not just words; in the fall-winter semester of the fourth year we had lectures on pedagogics and had field training in high schools. Each one of us had to conduct a lesson in a senior grade. At the same time other students from our group were present in class and afterwards discussed the lesson, speaking of its negative and positive features.

Once I was sitting at a desk in the last row during one such lesson, and, trying not to be bored, thought about a certain inequality between the norms of a function and its gradient. The above-mentioned idea of studying functions suddenly exhibited concrete outlines. That was a flash of inspiration! Unexpectedly I understood that I had just obtained a new proof of S. L. Sobolev’s classical theorem. Besides, the great inward rejoicing I felt at that moment showed something that my subconsciousness had already perceived: it was far from over. Indeed, within a few days it became clear to me that the case in question was not just a new proof of a well-known fact, but a powerful, and in a certain sense, comprehensive approach to an important area of functional analysis. This is of course not the place to explain the mathematical essence of this development.²⁹

Psychologically, the decisive factor was that, having once dived to the maximum depth, I liked the taste of doing so and subliminally made it my sine qua non to continue in the same manner. In a certain sense, that field training high school lesson determined the level of my future scientific work. I always try to cope with a mathematical problem to the very end.

5.10 The Virgin Soil

In August, between the fourth and fifth years, a large group of Mathmech students was sent to the virgin soil in the Kokchetav Region. I went because of enthusiasm and did not regret it afterwards. We lived in the middle of the steppe in huts we made ourselves. Above us was the cloudless bowl of the sky, and in the evenings – strikingly beautiful sunsets. Water was brought to us in barrels from the closest settlement, and food was prepared right in the field by students on duty. Wheat was planted in gigantic areas. That is why, in spite of low crop productivity, grain was harvested in great amounts. Some of us helped the combine drivers, but we mainly worked on the threshing floor, loading grain or shoveling it to prevent it from rotting. Closer to the fall we built sheds, installed casings and filled them with adobe. By the end of the day we naturally were worn out, but after supper we sat around a campfire, sang songs, told jokes, and baked potatoes.

We were young and the physical exertion did not seem intolerable. What I cannot remember without disgust was the self-made toilet. It was a shallow pit (the ground was too hard) that was quickly filled; it was covered with boards all dirty with crap. Around it there buzzed myriads of dung flies and the stench was unbearable. There were no walls, and not too far away another toilet was set up with the same design, it was for women. We did not go to our toilet without making sure the ladies’ analogue was unoccupied and vice versa.

Your humble servant composed the lyrics to the tune of “I remember that Vaninsky port”.³⁰

You shall willy-nilly get crazy

While taking the fork and the spade.

Damn Virgin Soil, we’re too lazy –

For with death our work will be paid.

We live without food, without sleeping

By the will of Nikita Khrushchev.

Oh, why Virgin Soil are you keeping

Poor students until they die-off?

And so on in the same pessimistic vein.

When I was still at home I decided to spend all my free time at the Virgin Soil in studying French and got myself a textbook and some French texts. I woke up two hours before everybody else got up, learned new words, and read in French. My first book was a French translation of “Sherlock Holmes” which I had recently read in English; it was of great help. I did not have a dictionary except for the small one given at the end of the textbook. Therefore I read the same pages repeatedly, moved forward in the text, then returned and read it again until a more or less tolerable understanding was achieved. As a result I practically did not have unknown words. After Conan Doyle I switched to Maupassant and mastered “Bel Ami” in a similar way.

That linguistic activity adorned the monotonous existence at the Virgin Soil which unexpectedly took longer than planned. Because all transportation was used to carry grain urgently, we were kept in the steppe until the first frosts came. When we were at last leaving, the water in the barrels began to be covered with a thin layer of ice.

Back at the Mathmech I enrolled in a French elective class for beginners as I had never had a teacher of French. The class instructor on the one hand was terrified by my pronunciation and on the other was surprised by the fluency of my translation. This situation remained in the future: I am shy to speak French but can read fluently without a dictionary.

5.11 In My Fifth Year

All during the fifth year I enjoyed freedom and did whatever I wanted. From the elliptic equations I moved to the parabolic ones and, relying on a certain idea of my “elliptic” work, quickly obtained something interesting: a necessary and sufficient condition for the validity of the maximum principle in an arbitrary Banach space.³¹

Photo 5.10

M. A. Krasnoselsky (Berlin, 1995) (© Vladimir Maz’ya, private collection)

At that time Professor Mark Alexandrovich Krasnoselsky³² came to Leningrad and gave a talk at Vladimir Ivanovich Smirnov’s seminar. In those years Krasnoselsky was a recognized head of the Voronezh mathematical school.

Photo 5.11

Article on semigroups (© Vladimir Maz’ya, private collection)

S. G. Mikhlin showed Krasnoselsky my theorem and the latter told him that a similar one had just been proved by Pavel Sobolevsky,³³ his graduate student. It was suggested that Sobolevsky and I should write an article as coauthors which was duly carried out. The expanded version of that article became a basis of my diploma thesis.³⁴ And completely different things with which I constantly was busy as well (they originated from the memorable lesson during the high school field training in the fourth year) were left to be used in my Candidate dissertation³⁵ on S. G. Mikhlin’s advice.

Photo 5.12

P. E. Sobolevsky (about 1967) (© Naukova Dumka, reproduced with kind permission)

Photo 5.13

I. Y. Bakelman (about 1967) (© Naukova Dumka, reproduced with kind permission)

5.12 Bakelman’s Special Course

Among other impressions of my last year at the Mathmech (1959–1960) I would like to single out the special course offered by I. Y. Bakelman.³⁶ It was devoted to the geometric methods of investigating elliptic equations. He described the results of his doctorate dissertation³⁷ that had just been defended at the Pedagogical Institute where he worked. Although the title of my first paper in DAN USSR³⁸ looks similar to Bakelman’s dissertation, our results and methods have nothing to do with each other.

Of course my theorems were new, and their proof was based on some methodological finds, but speaking seriously, my work was traditional. I was still in the process of learning while trying to cope with technical difficulties. The modest experience accumulated by me caused my interest in Bakelman’s course and allowed me, as Mikhlin taught me, to see it in my own way.

The methods invented by I. Y Bakelman turned out useful for further development of the theory of nonlinear partial differential equations. The evaluation of a solution of an elliptic equation with a nonzero right-hand side, independent of the continuity moduli of the coefficients, proved to be the most important. In Russia they call it Alexandrov’s maximum principle, and in the West – the Alexandrov–Bakelman–Pucci maximum principle.³⁹

Near the blackboard, I. Y. Bakelman behaved in a temperamental way, laughed loudly, enjoying the theorems, and in general acting like a big child. In spite of the difference in age and position we at once befriended each other. Sometimes we returned home together after his lecture. Once he suggested that I develop the theory of the following nonlinear equation: the sum of squares of all second order derivatives of a solution is equal to one. Unfortunately, I was not interested. I don’t know whether this problem attracted anyone’s attention afterwards.

Bakelman left for America in 1979, worked for 2 years at the University of Minnesota and after that got a professor’s tenure in Texas. He died in a car accident on August 30, 1992. In 1994, the Springer publishers issued his fundamental monograph Convex Analysis and Nonlinear Geometric Elliptic Equations.

5.13 Job Placement

At the very beginning of my fifth year at the Mathmekh, S. G. Mikhlin and V. I. Smirnov discussed the question of my future. The prospect of Graduate School was more or less real, but was it possible to be given a position in the University? Evidently it was a difficult question because of the so-called “fifth item”.⁴⁰ V. I. Smirnov willingly supported the idea and promised to help. But the task proved to be difficult even for Academician Smirnov. He fought the system for several months and at the end of the school year had to acknowledge defeat.

At that time, the Mathmech Research Institute at the Leningrad University was headed by S. V. Vallander,⁴¹ a specialist in aerohydrodynamics who by the end of the war was a flag navigator of an air regiment. He was not only a scientist but also a strong administrator and Party official (in different years he was elected a member of the Leningrad City Party Committee, was the dean of the Mathmech, and deputy rector of the Leningrad University). According to S. G. Mikhlin, S. V. Vallander admired V. I. Smirnov. So, he personally was not expected to cause any problems.

For many years V. I. Smirnov’s presence at the Department purified its atmosphere and in general he exerted great influence in the University. In 1956, V. I. Smirnov “pushed” M. S. Birman through and he was accepted to the Physics Department in spite of his Jewish nationality. When fighting the university administrators Smirnov said: “Either you take Birman or I am leaving the Leningrad University”.

No doubt in my case the difficulties emerged in the Main Building which contained the Party Committee and the powerful deputy rector on personnel affairs Sergey Ivanovich Katkalo, “a specialist on the Jewish question”.

I myself did not regard the Graduate School as a dramatic perspective, an attitude unlike S. G. Michlin’s who thought about the future. He knew that after finishing Graduate School it would be impossible for me to stay at the University. Literally a week before the job placement procedure V. I. Smirnov, with a conspiratorial look, took S. G. Mikhlin and me to an empty classroom, and, chuckling, said that everything was all right and I would be accepted. He warned me: “You would be given a choice: Graduate School or Scientific Research Institute of Mathematics and Mechanics at the Leningrad University (NIIMM). Don’t take it into your head to choose the Graduate School.”

In this way I entered the NIIMM or, more exactly, became a junior researcher at the Mathematical Physics Laboratory headed by Modest Mikhaylovich Smirnov.⁴² Although not a prominent mathematician, he was an exceptionally well-disposed person. In 1968 I was transferred to the laboratory of computation methods under S. G. Mikhlin. During all the years I stayed in the NIIMM I did whatever I wanted. “Maz’ya works for himself” were the words with which the administration branded me, but they were not quite correct. For many years I gave lectures to students and supervised those working on their diploma theses. Here is an excerpt from my character reference of that time:

In different years V. G. Maz’ya gave courses without pay. They included: a course of mathematical physics at the Mathmech Department, courses of mathematical analysis at the Economic and Geographic Departments of the Leningrad University, as well as various special courses such as ‘Embedding theorems for function spaces’ and ‘Additional chapters to the theory of elliptic equations’. Together with S. G. Mikhlin he headed the seminar of the Chair of Mathematical Physics on the theory of general partial differential equations.

The just mentioned seminar began in early March 1967 and continued every Tuesday on 10th Line. The first was V. A. Solonnikov’s⁴³ talk on elliptic systems. Other speakers included: M. A. Krasnoselsky, I. T. Gohberg, A. I. Koshelev, N. N. Uraltseva, G. Anger, G. M. Vainikko, A. Langenbach, G. Wildenhain, L. I. Hedberg, M. I. Freidlin, I. B. Simonenko, S. Prössdorf, N. L. Vasilevsky, G. S. Litvinchuk, M. I. Vishik, O. A. Oleinik, and B. Silbermann. I think that not everyone is mentioned, but do not remember those who were omitted.

5.14 Siegfried

Photo 5.14

Siegfried Prössdorf (in 1990) (© Vladimir Maz’ya, private collection)

What was self-evident to the school children and University students of my young years may require an explanation today. In the Soviet Union, our generation was brought up in compliance with “the spirit of proletarian internationalism”. This spirit forbade one to feel hatred based on someone’s nationality. Hatred based on class consciousness was OK, but hatred of a nationality was a taboo. Even though my father died in the war against the Germans, and that, as a result of that war, I lost many of my relatives, I never had enmity against the Germans as such. Only the fascists were guilty of all the horrors of the war.

Thus, there was nothing unnatural in the fact that in my young years Siegfried Prössdorf⁴⁴ became one of my close friends. I got acquainted with him in 1958 at the beginning of my fourth year at the Mathmech when he enrolled in first-year studies, though he was just 1 year younger than I.

Siegfried came from the GDR⁴⁵ with a rather weak knowledge of Russian, but soon began speaking at a normal speed and had a barely noticeable accent similar to the Balts’ pronunciation. We did not meet too frequently when he was a junior student but undoubtedly took a liking to each other. I learned from him that he also had grown up without a father who died at the Eastern front in the beginning of the war.

In 1963 Siegfried defended his diploma thesis under the guidance of S. G. Mikhlin, after which he left for Leipzig. There he received permission to enroll in the Leningrad University Graduate School – S. G. Mikhlin’s recommendation enjoyed great prestige among the GDR mathematicians. Siegfried was a graduate student for 3 years.

As if it was yesterday I remember our first meeting at V. I. Smirnov’s seminar in the fall of 1963. After that we began seeing each other quite regularly either at S. G. Mikhlin’s seminar or at his home. In those years I was in close contact with S. G. Mikhlin, who was Siegfried’s scientific advisor. The topic of his future dissertation was going to be one-dimensional singular integral equations with degenerating symbol, and I was interested in a similar subject matter but with respect to the fundamentally different multidimensional case. As a result, Siegfried and I had a mathematical theme to discuss.

The life in the dormitory on Detskaya Street where he spent 7 years was not very comfortable, to say nothing of the irregular diet in the canteen. The latter factor might have been the cause of Siegfried’s ulcer for which he was operated in Leningrad. After surgery he was nursed into recuperation by his wife Roswitha who was a student of a Leningrad medical institute at that time. Because the surgeons removed half of his stomach he never felt healthy afterwards. Nevertheless it was impossible to suspect this problem when looking at him. An elegant and handsome man, he looked the same the whole of his life.

After he defended his Candidate dissertation he returned home and our encounters became infrequent. We only saw each other when he and Roswitha came to Leningrad. As for me, I was not allowed to go abroad on business trips while I worked at the University. But private visits to the “People’s Democracies”⁴⁶ were restricted less drastically. So, when the Prössdorfs invited me and my wife Tatiana to pay them a visit we were let out. We spent 2 weeks in the GDR giving talks in Berlin and other cities, which was forbidden to do for those travelling on a private visit. In the good old days before the Internet, information on our criminal activity did not filter into Leningrad and we got away with it.

We took delight in the trip. In addition to the museums and places of interest the life in the GDR seemed to us flourishing and liberal unlike under Brezhnev in the Soviet Union. They had (who could imagine!) a multiparty system; their stores, both selling food and manufactured goods could not be compared to ours with their near-empty shelves. Among the stores and workshops we even saw privately owned ones where they started to serve you the moment you opened the door and treated you as their good friends. In one of the little shoe shops they selected soft shoes for me produced by the firm “Salamander” which saved me for several years from the pain in a deformed joint of a toe that I inherited from my mother. To put it shortly, we felt we had found a paradise. By the way our Deutsch grew by leaps and bounds, especially Tatiana’s.

On July 27, their wedding anniversary, Siegfried, in his new blue Citroen, took Roswitha, Tatiana, and me to the town of Caputh 6 km from Potsdam.⁴⁷ He wanted to show us Einstein’s summer house where the great physicist spent a lot of time in 1929–1932. Unfortunately the gate was closed and had a notice “Astrophysical Laboratory”. The owner of a neighboring cottage (we soon found out that he possessed an electric goods shop in Magdeburg) having noted the four of us on the street invited us to his garden where he, lying naked in the swimming pool, treated us to some wine. Later, as we forgot the name of that bon vivant, Siegfried called him “Zweistein”. On that day we descended to the lake and swam; then rented a boat, and, caught in the downpour with thunder and lightning, got drenched to the skin and moored to a tiny restaurant. There, the ladies warmed themselves with Cognac, but Siegfried and I abstained. He – as the driver, and I – out of solidarity. On our way to Potsdam we saw perennial trees blown down by gusty winds. We were lucky to have found shelter.

We had other meetings: in the USSR, Germany, and Sweden. For example, we went together to Schwarzwald and Bodensee in June, 1997, and discussed plans of joint research work.

For the last time we saw each other in February 1998 in Darmstadt at a conference dedicated to our mutual friend Erhard Meister.⁴⁸ On our way back, driving in a car, we noticed a stork’s nest in tree branches, a sign that brings happiness according to a German popular belief. But, alas, the belief did not come true. Soon Siegfried was killed by cancer. He died on July 19, 1998 before he was 60.

Prössdorf was a warm and very kind person. My Italian friend, the mathematician Paolo Emilio Ricci, once in talking to me called him “a real gentleman”. This characteristic, sounding unusual for a Soviet citizen, suited Siegfried perfectly.

I do not intend to describe his life and work.⁴⁹ But I would like to mention that in 1980 for his mathematical research Prössdorf was awarded the GDR National Prize. In Berlin he headed a laboratory at the Mathematical Institute of the GDR Academy of Sciences.

At the end of the 1970s and beginning of the 1980s he used his influence to help publish my articles and books in the GDR when it became difficult to do it in the USSR. For example thanks to him the Leipzig publishing house Teubner Verlag printed a three book issue in German of my first version of “Sobolev Spaces”, which had been declined by the Soviet “Nauka” publishers with no explanations.⁵⁰