Ann. I. H. Poincaré – PR 38, 6 (2002) 807–810

2002 Éditions scientifiques et médicales Elsevier SAS. All rights reserved

S0246-0203(02)01143-3/PRP

ILDAR IBRAGIMOV’S BIOGRAPHY

Formal space-time frames of my life are the following. I was born on July 15 1932,in Leningrad (now St. Petersburg), USSR (does not exist anymore). I graduated in 1956from Leningrad (now St. Petersburg) University and after the graduation I worked and Iam still working at St. Petersburg (formerly Leningrad) University and the St. Petersburg(Leningrad) branch of the Steklov Mathematical Institute of the Russian (Soviet) Acad.Sci. (formerly LOMI and now POMI).

The nonformal part of this biographical sketch is on books and men, some books andsome men of course, several books and men who greatly influenced my life, forming anddeveloping my mathematical (and not only mathematical) personality.

The first such book was “Arithmetic” by A. Kiselev. Kiselev was a famous Russianeducator, the author of Algebra and Geometry books which were basic mathematicalschool books for a few generations of Russian people. The book “Arithmetic” waswritten for teachers and had compliments written by A. Khinchin. I encountred the bookat the age 10 or 11 and it was a great discovery for me. From this book I learned thatmathematical facts can be proved. For example I knew at this time that a number isdivisible by 3 iff the sum of its digits is. But here was the proof! I was greatly impressed.Usually I believe children learn that there are such things as proofs from Geometrybooks. For me it was from Arithmetic.

The next and may be the most important book of my life was “The course of differen-tial and integral calculus” by G.M. Fichtengolz. The author of the book, G.M. Fichten-golz, was the Professor of Leningrad University where he taught Mathematical Analysisfor many years.

In fact, my first meeting with Analysis was the Grenville and Lusin book ondifferential and integral calculus. Initially the book was written by the Englishmathematician Grenville for engineering schools and it had been translated into Russianby Lusin. Lusin continued to work on the next editions, they became Grenville–Lusinbooks and then only Lusin. The book contained theorems about continuous functions:Cauchy’s theorem on zeros and Weierstrass theorems. There were no proofs, only nicepictures and reasonable explanations as to why these facts should be true. But there wasa sentence that one could give strong proofs. For a half of a year I tried to find suchstrong proofs. No way.

At this time (I was about 15) a very great event occured. My father brought me fromMoscow (I lived then in the city of Tavda in Ural region) the above mentioned booksof Fichtengolz, two volumes. The third volume (it appeared later) I bought two yearsafter on my first salary, which I got for summer work (during the school vacation) in aplywood factory.

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Fichtengolz’s book became my main occupation for the end of school years. Very soonI had studied the proofs of the theorems on continous functions and was very desolatethat I was so stupid and had not be able to find such “simple” proofs (I was really naiveand stupid at that time).

Nevertheless till my last year in the school I was not sure whether I would apply to theDepartment of Mathematics of a University (I dreamed of Leningrad University) or toa Medical Institute. Medicine was my second inclination and I had read all the medicalbooks belonging to my mother.

Anyway, in 1951 I applied to the Department of Mathematics and Mechanics ofLeningrad University and was not admitted. It was the severe Stalin epoch and somefacts of my biography (or rather the biography of my parents) did not fit. To avoid the2-year military service I tried other Institutes and was at last admitted to the Departmentof Mechanical Processing of Wood of the Forest Academy.

Here I met Professor Nikolai Vyacheslavovich Lipin, whom I consider to be my firstTeacher. He gave us a course of Higher Mathematics. Once, in lecture, he suggesteda few relatively difficult problems. I was able to solve them rather quickly. It was thebeginning of our acquaintance. He invited me regularly to his home where we spokeabout mathematics. N.V. Lipin was rather old at that time and half-blind and he hadstopped doing new things, but he was devoted to mathematics and these conversationsbrought me a lot. In fact, I realized only much later how deep and benificial were hisinfluences on me. N.V. Lipin tried very persistently to help me move to the Dept. ofMath. With a letter of recommendation written by him I visited even the minister ofhigher education of the USSR. Nothing helped.

In September of 1952 at a party, N.V. Lipin met the new Rector of the University,a famous geometer A.D. Aleksandrov and the happy end followed. It was only one ofmany good undertakings made by Aleksandrov. This one proved decisive for my fate.Thus in September of 1952 I became a very happy 2nd year student of the Departmentof Mathematics and Mechanics of Leningrad University.

It happened that I won a competition on problem solving held by the Dept. In thespring of 1953 the chairman of the jury of this competition Yurii Vladimirovich Linnikinvited me to his home to discuss my solutions. I returned from the visit as his studentand with 3 books which he gave me. One of these books was Kolmogorov’s “Foundationof Probability”. Later Yu.V. Linnik gave me Doob’s “Stochastic Processes” which hadjust appeared and told me to study it. I believe that for a few years I was the only personin Leningrad to have read Doob’s book from the first till the last pages. I remember thefollowing interesting story. In 1954 the students of my year passed exams in Probability.After the answer of my collegue Grigorii Tseitin our Professor O.V. Sarmanov asked if Iknew such and such variant of Kolmogorov’s inequality for max of sums of independentvariables. Of course I knew. I knew about martingales, I knew Doob’s inequality. I hadread Doob! Tseitin knew nothing about martingales, he invented them preparing hisanswer. And neither me nor Sarmanov knew that these inequalities had been discoveredby Sarmanov’s teacher S.N. Bernstein in the second half of the thirties.

Later I added to Kolmogorov’s and Doob’s books Cramer’s “Mathematical Methodsof Statistics” and Gnedenko and Kolmogorov’s “Limit theorems for the sums ofindependent random variables”. Just these books became the foundation of all my

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Probability knowledge. I do not want to say that I did not read other books. I did. I think Ieven read too much. For me books are more important than lectures, talks, conversations.But these four books, like the book of Fichtengolz became in a sense my blood andflesh. They made me. I could add to this list the book of Akhiezer on approximationtheory, the book of Akhiezer and Glazman on functional analysis and maybe Van derWaerden’s “Modern Algebra”. Alas, there are no geometry and topology books in thelist and sometimes I feel it very strongly.

In 1956 I finished Leningrad University and became an aspirant (graduate student)of this University. My adviser was Yu.V. Linnik. In fact he never gave me advice andnever interfered with my mathematical independence. We met very often and discussedour work. It was the time when Linnik produced a great number of results in numbertheory, probability, statistics and he spoke about his results with great enthusiasm. Theseconversations with my teacher were so stimulating! And his naive joy of the creator whohad seen that his doings were good was so infectious!

At the end of 1956 A.N. Kolmogorov and P.S. Aleksandrov visited Leningrad.It was planned that they would meet the New Year in the company of Leningradmathematicians at the house of L.V. Kantorovich. Linnik wanted to introduce me toKolmogorov and invited me to this company (of old people from my opinion of thattime). I was not very happy because I had different plans how to meet the New Year butI was to obey. It was also planned that for the next few days Kolmogorov would staywith me in a University rest home and we would ski. In truth, I was afraid to stay afew days alone with one of the greatest mathematicians. It happened that Kolmogorovwas not only the great mathematician, he was also a good companion, in company withwhom I felt no tension (almost). Such was the beginning of my long aquaintance withKolmogorov which Andrei Nikolaevich called the friendship (I felt like this but I neverdared to use the word).

In 1969 (68, 70?), a meeting was held in a small place in the mountains of Armenianamed Dilizhan. One of the participants was A. Kagan who gave a series of veryinteresting talks about estimation theory. In particular, he told about a joint work withLinnik concerning invariant estimators and on a difficult problem they met: how to provethat Pitman’s estimators are asymptotically normal. At that time we (I mean me and myfriends) were relatively young and liked to tease one another. Thus Rafail Khasminskiiand myself began to tease Kagan saying that the problem was difficult only for him, thatLinnik had never really tried etc, and that we would easily solve the problem (of course ifwe wanted to do it). Alas, Kagan was never angry, he only laughed, it was not interestingto tease him and we stopped doing it. But it happened that after the meeting we bothbegan to think on the problem and got some similar results. After having exchanged afew letters (Khasminskii lived in Moscow and I in Leningrad) we decided to write a jointpaper. It happened that we both had been invited to visit the University of Erevan. Wemanaged to do it at the same time and to use the visit to write the paper. The month inErevan gave us the material for many papers and for our book on estimation theory. Itwas the beginning of the long collaboration with Rafail Khasminskii which brought meso much joy.

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Such were the most important meetings in my life: with the book of G.M. Fichtengolz,with my Teachers Nikolai Vyacheslavovich Lipin, Yurii Vladimirovich Linnik, AndreiNikolaevich Kolmogorov, with Rafail Khasminskii.

And last but not least. For Soviet mathematicians it was not always easy to visitforeign countries. I was lucky in this respect. In 1973 I spent 4 months in the USAand in 1979 I spent 3 months in Orsay, France. I found new friends in the USA andFrance and amongst them were Didier Dacunha-Castelle and Jean Bretagnolle.

Ildar IBRAGIMOV