傳播數學知識．促進數學教育

Interview with Prof. Bogdan Bojarski

**Interview Editorial Consultant:** Tai-Ping Liu

**Interviewers:** Tai-Ping Liu (TPL), Fon-Che Liu (FCL)

**Interviewee:** Bogdan Bojarski (BB)

**Date:** November 30th, 2006

**Venue:** Institute of Mathematics, Academia Sinica

Prof. Bogdan Bojarski (1931 – 2018) was born in Błaszki
, Poland. He was a Polish mathematician of the generation raised after the end of World War II. At the age of 20, he received his master degree from the University of Lodz in Poland; then he was sent to study at Moscow State University, where Kolmogorov, Menshov and Sobolev profoundly influenced him. Prof. Bojarski has made outstanding contributions to singular integral equations, equations of mathematical physics and real analysis; he is now a member of the Polish Academy of Sciences, and is the director of the Institute of mathematics there.

Besides, he has devoted himself to promote international mathematical research. He has made effort to hold mathematical activities in a castle in Bedlewo near Poznan, and now many international conferences take place there every year.

TPL: We have a Chinese magazine and we interview distinguished visitors like you, so let us chat from the basics first, ok?

BB: Ok.

TPL: There is a long tradition of mathematics research in Poland.

BB: Maybe the tradition of modern mathematics in Poland should be traced back to the beginning of the last (i.e. XX) century or two last decades of the XIX century. Actually, there were two main sources of modern Polish mathematics. One was France with H. Lebesgue and other French mathematicians in complex function theory, set theory and general topology, P. Montel, E. Borel, M. Fréchet …, also, even somehow earlier, E. Picard, H. Poincaré (Geometry, Analysis, Differential equations) influenced a group of Polish mathematicians studying in France, which started and formed a Polish mathematical school. So that is in Lebesgue’s school and in Montel’s complex analysis. But there were also at the beginning of the twentieth century, last century, very good and fruitful contacts with Moscow mathematicians. So, for instance, W. Sierpiński, whose role in the development of Polish mathematics was crucial, was very closely related to N. Luzin and the Moscow school. An active group of talented young Polish mathematicians arose around Sierpiński. N. Luzin, A. Kolmogorov and also P. Aleksandrov from the beginning were engaged in active fruitful contacts with that group. And so, in Poland, after Zygmunt Janiszewski…

FCL: Not Anthony Zygmund.

BB: No, no, no.

FCL: This Zygmunt is a first name.

BB: Janiszewski, a very young mathematician, after returning from his studies in France, got the idea to organize a new mathematical journal “Fundamenta Mathematicae”. Together with a group around him a program of development of Polish mathematics, called later the Janiszewski Program, was proposed. The principal new concept of the program was that young Polish mathematicians should rather concentrate their thinking and research efforts on new arising areas of mathematical research than to compete on the international “research market” with established, classical so to say, chapters of mathematics. As the new promising directions, the Janiszewski program named three: the first was set theory and topology, the second Real Analysis and the emerging functional analysis, the third was foundations of mathematics and logic. It was considered that in these areas Polish mathematicians had good chance to develop new ideas in their creative work, obtain important new scientific results at the front line of contemporary mathematics.

FCL: Was Sierpiński also a logician?

BB: No, Sierpiński worked in set theory, general topology and real analysis; Also, in arithmetic and number theory.

FCL: Ah-ha

BB: Important mathematical names appeared in the foundations of mathematics and mathematical logic. Let us recall J. Łukasiewicz as the father of many ideas, but later appeared the great figure of Alfred Tarski, before him S. Leśniewski, K. Ajdukiewicz. Later, from the time which I still remember, A. Mostowski was a mathematician of international importance. In the area of real analysis great names appeared also. Let’s recall S. Saks, A. Zygmund and J. Marcinkiewicz in complex and real analysis. At the parallel time J. Schauder started his famous seminal activity in the area of newly arising functional analysis, infinite dimensional topology and in strong connections with breakthrough new concepts in linear and nonlinear problems of new emerging theory of partial differential equations. Before J. Schauder and somehow without direct contacts with the ideas of the new Polish school of mathematics was the personality of S. Zaremba who independently developed a relatively small but active group, mostly in Cracow, in the classical theory of partial differential equations and applications in theoretical mechanics. That group was developed on the basis of contacts with French school, mostly H. Poincaré school, Picard, in classical potential theory and partial differential equations. S. Zaremba from Poincaré’s school. But real analysis group was developing under the influence of H. Lebesgue from Paris, and N. Luzin and his group, A. Khintchin, A. Kolmogorov etc. of Russia. Later Stefan Banach appeared as the founder of general functional analysis. Both these directions were developed on the basis of the scientific activity of the groups of W. Sierpiński and H. Steinhaus.

TPL: Isn’t Banach also Polish?

BB: Yes, sure. S. Banach was undoubtedly the most influential Polish mathematician in the world of mathematics. But before him Z. Janiszewski and his closest mathematical friends elaborated and formulated the general new concepts and a strategy of research program. Unfortunately, Z. Janiszewski unexpectedly died at the age of 24 or so, very early age, and he was active only at the start of the XX century Polish school of mathematics. After him came Mazurkiewicz and the famous Banach, later S. Saks, Zygmund, Marcinkiewicz, Schauder in analysis, Knaster, Kuratowski, Borsuk, Eilenberg in topology. Many Chinese mathematicians knew Kuratowski. But of course, again predominant were his contacts with French-Russian school. P. S. Alexandrov was influencing Kuratowski and so was Urysohn, also from Moscow. Unfortunately, Urysohn also died at an early age. As a matter of fact, quite a number of mathematicians active at the beginning of Polish school and Russian school died young.

TPL: But this is a list of fields. You say that the Polish decided to go into these fields instead of more classical fields. What are the more classical fields?

BB: For instance, complex analysis, classical analysis, even differential equations have not been much developed in Poland in the early decades of the XX century. Later in the thirties it was mainly J. Schauder who studied everything and was able to achieve remarkable progress in partial differential equations, functions spaces, boundary value problems for linear and non-linear equations. It is enough to recall the concepts of Schauder à priori estimates or Leray-Schauder index theory in infinite dimensional topology. Also T. Ważewski from Cracow should be recalled for his seminal works on ordinary differential equations. There also was another remarkable, very deep mathematician, with a broad range of research activity who started his career in Poland. This was Leon Lichtenstein.

FCL: That’s a French, right?

BB: No, he was a Polish mathematician. He started his scientific activity in Cracow, most active in the years 1908 – 1916. Later after the first world war he moved to Germany, and he settled in Leipzig. He was working in classical analysis and under the influence of the German school in partial differential equations, potential theory and hydrodynamics. In the thirties he published in German the monograph Hydrodynamik, in the famous Grundlehren series of Springer Verlag. This series of mathematical monographs contributed very much to the progress in mathematics research in the XX century. That‘s the Yellow Series. So I think that Hydrodynamik of L. Lichtenstein is a suitable book to start the study of the problems of hydrodynamics. When I was a student, I liked this book because it describes the hydrodynamical problems in realisticaly clear terms and at the same time in a mathematically rigorous way. So it is suitable to all levels of rigour expectations. It’s unlike most of the books on hydrodynamics and mechanics, which are very difficult to follow for a mathematician and to be fully convincing about the formulated scientific statements. When I was a young student it was my feeling that unless you can prove yourself the theorem, you don’t feel the theorem. For a theorem to be a creative tool in getting new results you have to hold it strongly in hand, to feel it, i.e. to prove or reprove if needed. And this is one of the basic things which decide about the attractiveness of mathematical thinking. In mathematical world there exists (a tendency to) some kind of selfsecurity attitude: either you know a fact, then you fully understand it, or you don‘t know then you admit that a mystery is before you. Either you engage in clarifying the mystery or you live with it. But you cannot claim about understanding unless you are able to clarify all doubts. This peculiarity of mathematical thinking somehow happened to be decisive for me in the process that I have finally chosen to be a mathematician. I actually started my university studies in physics, even experimental physics. In the exciting atmosphere of the first years after the World War II, when we all, boys and girls, have been so hungry for knowledge, education and reading, I got in my hands, while still in the secondary school, several prewar editions of books on topics in Sciences: biology, chemistry, physics, astronomy, astrophysics. Among them also The Expanding Universe by British astrophysicist Sir A. S. Eddington. It influenced my thinking and inspired my imagination so much that I decided to participate in the efforts to understand the mysteries of the interstellar universe as well as atomic universe. Though I was from rather poor teacher’s family, nobody bothered about the costs of education and future practical life. The enthusiasm for work and learning will solve everything!! No doubt the State will supply the most necessary means!! That was our attitude. So after the secondary school I went to study physics, astronomy. The shock came with the laboratory work and introductory courses in chemistry and physics. I started to argue with teaching assistants, wanted to understand the discussed processes and experiments better and deeper, asked more and more questions. The obtained answers didn’t satisfy me, the arguments didn’t match, the introduced concepts were not clear enough, not precise etc. On the whole all that created a feeling of confusion and disappointment in my mind. And then came a course in mathematical analysis I, as it was called in our universities, given by Prof. Zahorski, a devoted high level specialist on real function theory and trigonometric series. His course was presented in absolutely precise and rigorous way. It took him several hours to give the full proof of commutativity of basic operations on the reals! I felt some kind of enlightment!

FCL: Zahorski, right?

BB: Zahorski is my teacher in real analysis. Zygmunt Zahorski.

FCL: When was that? When you were studying in Poland?

BB: Yes, Yes, in the 50’s, after 1948, when I started to study at the university, in Łódź, Poland. So Zahorski was presenting in the basic mathematical analysis I (calculus) the real numbers theory following E. Landau.

FCL: Yes, Yes, foundation of analysis.

BB: Yes. In his analysis course it took Z. Zahorski about half a year to discuss the real numbers theory. His course started from the properties of the semi-ring of positive integers and introduced the real number field as sections of the ordered field of rational numbers.

FCL: Dedekind cut.

BB: Yes, Dedekind cuts and all the detailed proofs of their arithmetic, continuity, and completeness properties à la Landau. And I was shocked again by the full clearness, integrity and transparency of the presented theory. Suddenly, I was completely convinced and sure, you know, when I really understand the discussed theorem and if anybody wants to argue with me he has to present fully controlled argumentation. He cannot just say, “you are stupid, you don’t understand, the statement is obvious etc.” He has to argue, he has to convince you. And teaching assistants (and some professors even) in physics, for instance, did not have this quality. Teachers of physics (or chemistry), you know, they are appealing to imagination, fantasy or experimental “obviousness”, but these words have not been so convincing as Landau or Dedekind or Zahorski. And so that's why I switched back to mathematics.

FCL: So you were turning to mathematics while you were still in the university, before you went to Russia?

BB: Yes, to Russia I went after obtaining master‘s degree at the Polish university. It was in late Fall, 1951. The political and economical situation in Poland and in the region around was very tense at that time. My family – especially my mother – was very worried and opposing my plans. About life in Moscow, Russia, in general very little was known and controversial opinions were abundant. Despite all that, together with some university friends from the Łódź University we decided to go as Ph.D. students sponsored by the Polish government. In my group of 4, 2 mathematicians, I was the youngest, 20, the others have been older by at least 4, 5 years. So we arrived to Moscow in November 1951, greedy for science, mathematics, philosophical and political discussions, literature, art, theatre, music, inspiring or just sincere emotional international and Russian country music and country songs. And we found all that in the Moscow academic and student community despite, sometimes, poor accommodation and life conditions and various shortcomings. For the first year and a half or even a little longer we were living in the Moscow University students‘ dormitory with 6 persons in one room. Nevertheless our life was very satisfactory and effective in the sense of academic activity and achievements, social and cultural life. Philharmonic concert halls, theatres, Opera and social life were complementary to our academic activities. Situation drastically improved when we were transferred in Summer 1953 to the new MGU university building on Leninskiye Gory where each of Ph.D. students got a separate room with all necessary modern supplements. This considerably increased the intensiveness of our academic activity, participation in lectures and mathematical seminars, personal scientific contacts and international and interdisciplinary interaction in mathematics.

TPL: At Moscow that time, you have audited the courses given by Kolmogorov and others, right? Can you talk about the people in the lecture, and so on?"

BB: When I got to know Kolmogorov, there was rather peculiar atmosphere in Moscow in the relations between students and professors. So, for instance, Kolmogorov was world famous, leading contemporary mathematician but he also was directly involved in teaching activity with the youngest students, just freshmen or even secondary school students. Teaching these freshman courses gave him the possibility to get in direct contact with the intelectually active and promising students with the purpose of attracting them at the very early stage of their intellectual development to genuine mathematical research. I was already a graduate student when I came to Moscow and my strict mathematical speciality was different from other students close to A. Kolmogorov. So I was somehow outside of the circle of students around Kolmogorov at that time. However Kolmogorov knew me, also from my reports which I was supposed to present to him each year (Kolomogorov was also a supervisor of all Ph.D. in mathematics). I was also attending many seminars and lectures at MECHMAT, practically all the time present at the department. So he started to invite me to various activities of the strict circle of his students, also of social character, recreational and touristic activities on weekends, which he liked very much: swimming, walking in beautiful pri-Moscow forests, country skiing in Winter. Later, when I was already married and got my Ph.D. degree, he invited us, me and my wife, to the famous vacation house in Komarovka, a village some 40 km outside Moscow, which he shared with P. Aleksandrov. Besides having dinner, we were skiing (he was a very good country skier), or swimming in the nearby river, listening to classical music (Bach, French baroque composers), talking about friends, his students, art (he liked very much the painter Petkov-Vodkin), mathematics. When you open many publications of Kolmogorov (or Aleksandrov), you can find there often at the end of the book or article a reference to Komarovka. These all were unforgettable meetings...

TPL: What’s your impression when you talked to Kolmogorov? A great mind, something else?

BB: hmm…. when you were listening to his concise expositions of his ideas, his lectures or his handing-up talks – all that was leaving unforgettable impressions, though often it was not easy to follow the course of his mind. I was not a direct student of A. Kolmogorov, but I often was listening to his public lectures or seminar talks.
Special role was played by the weekly meetings of the Moscow Mathematical Society. These were on Tuesdays, 8 p.m. Kolmogorov was often visitor there and gave lectures. The lecture hall was then always overcrowded.
The meetings of the Moscow Mathematical Society were attended by very many active mathematicians of various generations, getting together to discuss current research topics, not only those presented by the main speakers. These were some kind of mathematical research market. Usually the lecture hall was filled up. The meetings usually extended over the time assigned to the invited speakers with people gathering in small groups in lively discussions during breaks, and walking around in the neighbouring corridors of the University building, and exchanging ideas on lectures and other actual mathematical events. The meeting could extend through the late hours, making it on the whole an evening of hard, though inspiring work and really strong interaction.
From the time at the Moscow University I remember also very well S. Sobolev on his seminars, for instance his enthusiasm in whatever he was involved in. He was a very lively man, very lively, and by this direct behavior, very attractive to young generations of mathematicians. All were impressed by this vitality. His cute remarks during seminar lectures, his intellectual energy was sparkling, just radiating to his companions in any scientific discussions, thus wonderfully influencing their creative abilities.

A. Kolmogorov liked very much all types of outdoor activities. He was a devoted participant in various touristic excursions and activities. He was very proud that he could stand long, difficult mountain climbing, country trips, swimming, skiing and all that. But these activities have been always intermixed with intellectual alertness...
Some years later, in 1957 or 1958 he visited Poland, in wintertime. A. Kolmogorov, myself and four other young Polish mathematicians, my friends, we spent a week in the Tatra, high in the mountains, in Zakopane and around. The conditions were very hard: winter, frost, intensive snow falls, isolation. For few days we were staying in a lonely hut high in a mountain valley, with nobody around, except us and an old woman, helping us with some food. He was sharing with us all hardships of life in these austere conditions. But there was always also some “scientific” component: in the evenings, after skiing or walking, or during meals or tea hours (not much, but some wine also) we were talking also mathematics. He was telling us about the theorems and concepts which he conceived or understood in his thoughts in the time between. And he was very proud of it, with the joy of a young man!
Our impressions and what we all learned from Kolmogorov during those Tatra days are unforgettable for all our life. In his everyday behaviour, being a mathematician, an intellectual of highest world standards, in contact with his friends, students and colleagues, A. Kolmogorov was like everybody else.
And similarly S. Sobolev or M. I. Lavrentiev in their everyday behaviour have been just people who were very energetic, curious, moving around with alertness and curiosity and lively expressing their reaction to news and observations of any kind. It was also a big chance for me personally to get to know Lavrentiev. Lavrentiev was the referee of my dissertation for the degree of doctor of Science, habilitation.

FCL: This is Vavrentiev?

BB: Lavrentiev

FCL: Lavrentiev, oh, he wrote a book on calculus of variations?

BB: Yes, Yes, he wrote a good book. Lavrentiev, Mikhail Alekseevich. In the first years of my Ph.D. studies I attended the seminar of Dmitrii Menshov, and Nina Karlovna Bari, in real analysis, Stechkin, Efimov in geometry, and Vekua…

FCL: Vekua is your advisor, right?

BB: Say later, after I moved to the Chair of Differential Equations. I started my Ph.D. studies at MGU with D. E. Menshov as my advisor. Menshov was a remarkable man and outstanding mathematician in real analysis and trigonometric series, as Zahorski, my teacher in Poland. After one year and a half of study with Menshov I had a conversation with Menshov and Kolmogorov. Kolmogorov’s status was then to be also a supervisor of all Ph.D. students at Mechmat. So he was meeting with us systematically during the year, asking about progress of our work. And when you were considering making special decision about your program of future study you had to talk to Kolmogorov. So during the second year of my Ph.D. studies I talked to Menshov and then to Kolmogorov about my thoughts of changing my research directions. Actually, the idea was initiated in my discussions with P. L. Ulyanov, who was also a Ph.D. student in Menshov Chair, with N. Bari, at his last year of Ph.D. studies. We concluded that our common research area, the trigonometric series theory was very narrow piece of mathematics, somehow off the broad avenues and burgeoning far reaching perspectives of other seminars, active at Mechmat at that time: topology seminar, partial differential equations seminar of Sobolev, geometry seminar, strong functional analysis seminars. So I decided to ask the opinion of D. Menshov. He told me that he understands me and approved the intention to go to broader areas of mathematical research. So I presented my plan to Kolmogorov. He approved it also. “In partial differential equations there are many problems to be solved, you can try”, he said. And I was transferred to the chair of Sobolev-Petrovsky. Besides S. Sobolev and I. Petrovsky, who was the Rector of Moscow University, but still was involved in the scientific activity, at the chair was a group of remarkable younger professors, O. Oleinik, E. Landis, M. Vishik and many others. I. Vekua, from Tbilisi, also recently joined the chair, and was not overloaded with Ph.D. students. So it was natural to propose that I. Vekua acted as my Ph.D. advisor. After a qualifying conversation he agreed and we started to work. I was very glad, because it turned out soon that the decision was very good for me. With my background in real and complex analysis I started to learn quickly singular integral equations and related boundary value problems. I learned also quasi-conformal mappings theory and related Beltrami equation that I spoke about on the conference near Taipei two weeks ago. It turned out that my studies under I. Vekua were very fruitful and continued for over two decades.

FCL: I remember last time when I was in Warsaw, we had a chat with Ulyanov, and you were talking about Menshov, right? So everything on the wall?

BB: On the table.

FCL: On the wall, right?

BB: Where?

FCL: I mean at Menshov’s room.

BB: Oh, no, Menshov had a rather small room, very tiny, as his whole flat (yes, yes). There was one bed, one narrow small table near the window and two chairs. The room was very narrow too, so the channel between the bed and the side wall, leading from the window to the entrance door, was very narrow too. It didn’t allow to pass the second chair from its place near the entry door to the table at the window, where we have both been working, unless carrying it up over the bed. I knew very deep and difficult results discovered by D. E. Menshov in his work. After my first visit to him at his house, as a young man of 20 years old, I was overwhelmed by the feeling of admiration to D. Menshov for his determinacy and ability to create such deep mathematics in the most modest, even austere life conditions. Leaving the place I will remember that feeling for all my life; I seemed to understand how much of devotion I should be ready to give to discover deep mathematics!!

FCL: I’m saying that if he wrote many mathematical things on the wall or…

BB: No, no, that as I’ve remembered, referred to his university seminars. In the seminar room there was a long blackboard fixed to the front wall. A rather narrow strip separated the blackboard from the long narrow front table with the following bench rows. D. Menshow used to sit in the first row. As he was rather tall and of thin posture with long hands, it was possible, in the case he didn’t agree with a formula written on the blackboard by the lecturer, just to stand up lean over the table and correct the formula with the chalk in his hand, directly from his chair. His informal, though in essence very strict manners and behavior impressed me very much as the youngest participant of his seminars. He was very remarkable in his comments arousing respectful admiration and enthusiasm of the young students for doing mathematics. All that was very, very stimulating.

FCL: So that is actually a very interesting story.

BB: While D. Menshov was living rather solitary and strictly peculiar everyday life, his impact on the mathematical thinking and career of many young mathematicians, including myself, was tremendous. Kolmogorov and his many famous students, e.g., V. Arnold, Y. Sinai and a long list of other big names, testify to that in many of their writings. Very many high rank mathematicians who started their career at the Moscow University, in the years 1950 – 1990, spread around the world and settled in many prestigious universities and academic institutions all over the world. I had the chance to talk to very many of them on various conferences and occasions in America, in Poland, Warsaw or Będlewo or elsewhere in the world. All of them recalled and agreed that the creative atmosphere in mathematics, scientific enthusiasm and direct everyday contacts between professors and students available at the Mechmat MGU – Moscow University – were incomparable to what they could experience in academic mathematical centres elsewhere in the world. The activity of seminars, exchange of mathematical ideas on various scientific meetings and levels was so intensive that everybody was interested to participate in the stimulating mathematical life of Mechmat. They were very, very glad to have the chance to participate in that activity and the gained experience was very important in the places where they had later to work and realize contacts between teachers and students.

FCL: So that kind of environment is actually stimulating and helpful for students and for training students. I think this is a nice talk.

- Fon-Che Liu and Tai-Ping Liu are faculty members at the Institute of Mathematics, Academia Sinica.