The meeting of the seminar at the
Institute of Mathematics,
dedicated to the memory
of A.V. Skorokhod

Anatolii Volodymyrovych Skorokhod was a brilliant scientist, an excellent teacher, a marvelous person. He was a creator of new methods and theories in the field of probability theory, in particular, the theory of stochastic differential equations and the theory of limit theorems for random processes. Skorokhod was (jointly with I.I. Gikhman) an author of the excellent monographs, providing a high level of training for young specialists in probability theory. Being a profoundly decent person, he realized the social problems objectively, through an independent mind. He treated his friends, colleagues and students gently, with humor but with exactingness. I'm thankful to fortune that I was granted to live, to work and to be friends with Anatolii in the period of his outstanding creative work.

In 1957, Anatolii Volodymyrovych Skorokhod began to lecture at Kiev University after his return from Moscow, where he had his postgraduate studies. I was lucky to be among the first students at his lectures and, later on, to work under his leadership at the Institute of Mathematics in Kyiv and, for more than 30 years (from the end of 1950s), to observe the process of birth and formation of the Kyiv probabilistic school and the exceptional role played in this process by Skorokhod’s scientific activity, his original ideas, and the exclusiveness of his personality.

At the third year of my studies, a new specialization ("probability theory") was organized at the faculty, and 10 students of our group were enrolled to it. Every semester, we had some special courses. Anatolii Volodymyrovych permanently delivered special courses. We were quite diligent and attended his lectures at another, younger, course too. That lectures were not similar to those which he delivered to our group. They contained a lot of new facts, and even those facts that he gave at our lectures, were proved differently.

In September 1960, six of us, the fourth-year students, had a pleasant surprise: we were recommended by the faculty student scientific society to go to the scientific conference in Vilnius. Those were the unforgettable days for me, real holiday which is always with me. First of all, we had the opportunity to become acquainted with Lithuania, such a distinctive country, and to feel the patriotic spirit of the Lithuanians. On the other hand, these days in Vilnius gathered all the best creative forces of the Soviet Union, a large cohort of young talented scientists, who worked in the field of probability theory. They were so passionate, as if they shone by creative ideas.

Among the other wonderful events of those days, I was very impressed by Skorokhods lecture. He talked about the estimation of probabilities that the sequence of normalized sums of independent identically distributed random variables were between certain curved boundaries. To solve this problem, he suggested to replace the sequences of the sums by the values of the Brownian motion process at certain random points of time such that the distributions of both the sequences coincided. But we already knew these results from his lectures at the special course! It turned out that these beautiful, elegant theorems were his own and were just born! But we believed that they were already well-known... So we opened that Anatolii Volodymyrovych had also acquainted us with his own results at our special courses! [It is worth noting that the idea of this Skorokhod’s result later on became the basis for a new research direction in the theory of random processes and this direction is now called as Skorokhod embedding problem].

At Kiev University, there was a tradition of "The Last Bell". Before the last winter session, one of the lecturers appealed to the students with a farewell speech. Moreover, the students elected the lecturer for the Last Bell speech by themselves and invited him. In December 1961, the students of our course invited Skorokhod to read such a lecture, because he was our first lecturer.

All the participants of this action (students and lecturers) gathered in the large audience. Skorokhod’s speech was very bright and thoughtful. He spoke about the nature of mathematics, its difference from other sciences. Its main points are the following:

Mathematics has no authority. Mathematician never says: "This thought is right, because some respectable person considered so. In mathematics, each assertion must be logically proven. Mathematics develops the logical thinking. So, regardless of the need of application, all the persons who need the ability to think logically have to know mathematics. No wonder, even Abraham Lincoln being a President of the United States had studied the "Geometry" by Euclid. Mathematics contributes to the development of science in general, to the development of the technology, and to the human development . In our progressive movement, we are going to build a better society, and the role of mathematics will be increased with the time in this track. Mathematics thrills, it has its own laws of development and its uniqueness.

- "While you are young, - Skorokhod appealed to the audience, - and are not burdened by the need to take care of the earnings for keeping the family, try to penetrate deeper into the essence of mathematics and to appreciate its beauty."

The official boss, the Dean of the faculty, did not like this speech. He had a speech after Skorokhod and subjected the first speaker to a sharp criticism. His was resented by the indifference of the previous speaker in politics, who did not even mention the benefits of our social system relative to a capitalism and, speaking of a better future, did not even pronounced the word "communism"! "This speech, - said the Dean, – could pronounce a teacher from capitalist countries, such as, the U.S.A."

We all were sitting as electrified. Finally, I could not stand and wrote a note to the Dean that we, the students, did not agree with his assessment of Skorokhod’s speech, that he spoiled all the holiday of the Last Bell for us. After reading my note, the Dean immediately went out. The meeting was closed immediately.

I had considerable troubles because of my quick temper. But the history of the Last Bell gave rise to my civic enlightenment an understanding that there is an extensive system of pressure on people in our society. One could imagine how many troubles had Skorokhod after his speech. But the next year, students again invited Skorokhod to deliver a speech at the Last Bell, and during several years, Skorokhod was invited for the farewell lecture. And the watchful universities officials were forced to retreat and did not expose the apoliticity of his thoughts.

One of the universities teachers encouraged students to be bold and self-assured. Since, on his opinion, only such a person can create something new. Skorokhod has always been an extremely modest person, but he actually did not leave a sense of the inner freedom. He tried to develop such inner freedom and independence in his students and colleagues. He repeatedly stressed that it is important to be not only a good specialist, but a good person, first of all.

Anatolii Volodymyrovych was always distinguished by his organization and ability to concentrate the thoughts. He wrote reviews at once, in one sitting, without corrections. He was always ready to talk with students, never refused to consult them, never complained for a lack of time. According to the evidence of his relatives, he fulfilled the main daily research from 6 to 8 a.m. and did not allow himself to refuse such a work in the morning time. It is worth mentioning here Montaignes words that greatness is not determined by the ability to a high rise but the ability to be always ordered in all things.

In the spring of 1968, there happened an outstanding event in social life of Ukraine: 139 researchers, writers, artists, workers, and students expressed their protest against the illegal political repressions, against the closed court sittings in Ukraine during 1965-1966. They signed up a collective letter addressed to the state authorities. Among those who participated in this action were mathematicians, three of them were employees of the Institute of Mathematics: Yu.M. Berezansky, A.V. Skorokhod, and Yu.D. Sokolov. It should be noted that all three of them were members of the Academy of Sciences of Ukraine. At that time, to decide to such a step meant to be condemned by a public censure, and even more, to cross their own future. All the persons who signed this letter were exposed to repressions. Many of them lost the job. In the Institute of Mathematics, nobody talk aloud about this letter and the fact that some of our colleagues had signed it, only in whispers, between good friends. At last, a special meeting of the Scientific Council of the Institute was announced. It was closed session, but, at the last moment, four young employees, including me, were allowed to be present at this meeting. As it was often done in such cases, the essence of events has been moved to another plane. The main reporter at the meeting was academician of NAS of Ukraine J. Z. Shtokalo. On his point of view, those who signed the letter were simply provoked by enemies of our country. So, all of them showed their political short-sightedness and have to be punished. It was a provocation because the letter was prepared intentionally and transferred to foreign correspondents before the session of the United Nations to put our representative at the UN in the awkward position and do not give him an opportunity to speak about the principal propositions concerning the destiny of oppressed peoples around the world.

The rest of the speakers tried to protect "those who blame the staff". They said about them as outstanding scientists, who, except for this ill-considered act, did not stigmatize themselves and always behaved like the honest soviet citizens. There were many such speakers, but, after all these numerous positive speeches, the resolution of the meeting sounded as a sharp dissonance to all that occurred. It was adopted almost unanimously without debate. In it, the colleagues step was convicted. Taking into account their political short-sightedness, the resolution suggested:

1. to limit their opportunity to contact to young people, that is, to forbid them to lecture at Kiev University, to have postgraduates, etc.
2. to withdraw them, as politically short-sighted persons, from the editorial boards of scientific publications.

Anatolii Volodymyrovych always was far from the vain worries and was indifferent to the various honors. Was he pained that his creative work was not properly evaluated in the Soviet country? The procedure for awarding prizes was too bureaucratic in our society, the awards were given routinely for signs often far from the values of scientific work. Several times, his works (made mainly jointly with I.I. Gikhman) were nominated for Prizes of different ranks. But, as a rule, they were rejected. They were appreciated considerably later on. Time puts everything on its place. Now A.V. Skorokhod became a well-known scientist by his outstanding numerous scientific works. Moreover, the Kiev school of probability theory which was created before our eyes (that is, in the second half of the XX century) is, to great extent, a result of the existence of the Skorokhod phenomenon. The long-term development in this branch of science underwent the strong influence of his creative thoughts, constantly productive ideas, his generous advices, his demanding remarks. He was as inexhaustible source of ideas for all of us, which renewed our forces and led to the new achievements.

To appreciate a great person, one must be at a certain distance from him. Of course, over time all of the estimates and the emphasis in the work of Anatolii Volodymyrovych will be put in the rank by the future generations of mathematicians. But now we can certainly say that the Theory of Probability with Skorokhod’s contribution into it is a fundamentally different thing as compared to what it was before.

It should be noted that even prominent mathematicians of our days are impressed by Skorokhod’s achievements, not only by the volume of his contribution, but also by his exquisite designs in the construction of new theories and elegant swing thoughts in grounding of certain assertions. I heard many times the comments of the visiting mathematicians about the impossibility that all Skorokhod’s works could belong to one person. In general, Anatolii Volodymyrovych avoided talking about how he managed to do it, but he told to some of his close Moscow friends that he had taught himself from the young years to the daily intense thinking on the problem. For him the day during which he had not learned something new was lost. I think that there have been not many such days in the life of Anatolii Volodymirovych. Owing to his intense work day after day, the creative spark given to him from God became a bright shining star of the first magnitude on the mathematical frontier.

Talking about the method of Anatolii Volodymyrovych, I remember his words that one should go directly to the problem, without any tricks. This implies the direct probabilistic methods about which there are so many conversations. Of course, it is not so easy to explain what it means "to go directly to the problem". Perhaps, from the place where Anatolii Volodymyrovych stood he could go straightly to the problem. But if you stood in some other place, perhaps, there could be the "direct approach to the problem" too. These are the subtle things. They affect the very essence of the incomprehensible creative process. In this regard, I remember the following episode:

It was about 30 years ago. Once, I turned on the TV and saw on the screen Volodymyr Vyshensky who summarized the results of the math competition held before. Among others, he analyzed the proof of such assertion: a finite number of parabolic shapes (parabolic shape is a part of the plane bounded by the branches of a parabola) cannot cover the entire plane. The author’s proof impressed me by its beauty. It was based on the fact that a straight line on the plane which is not parallel to the axis of symmetry of a parabola, has as a common with the corresponding parabolic shape no more than a finite interval. Therefore, if there is a finite number of parabolic shapes on a plane, then each line on it which is not parallel to the axis of symmetry of any of those parabolas will have in the intersection of those figures at most a finite number of bounded intervals. But a straight line cannot be covered by a finite number of bounded intervals. Once after I met Anatolii Volodymtrovych and told him that I was impressed by the beauty of the solution of a problem, which I learned from Volodymyr Vyshensky. "And what is a problem?", – he asked. I stated the problem. He thought a few seconds and said the following: "Every parabola can be enclosed into an arbitrarily small corner. Therefore, if there are a finite number of parabolas, they can be enclosed into corners such that their total value is strictly less than the full angle. But the whole plane cannot be covered by such angles. Then he asked about the solution, which I learned from Vyshensky. I told him, ending with the words: "You should agree that it is a beautiful solution”. He said being in the deep thought: "Yes, beautiful". But in his words I did not feel any enthusiasm.

In fact, his concept of a beauty in mathematics was special. In his lectures, he had never said any monologue about the beauty of a consideration or a formula. I think that the train of thoughts in the proof of a fact seemed to him so that it was beautiful itself and don’t need decoration. His lectures were capacious. In my teaching practice I tried many times to fit in one of my lecture the material of Skorokhod’s one lecture, but all such attempts appeared futile. Anatolii Volodymyrovych had a principle, which he always kept to in his scientific work, namely, he never used the assertions of other mathematicians which he was unable to prove or did not see how to do it. In other words, he had never constructed his proofs on the facts that seemed to be unclear for him. I knew only one exception to this rule in his work. I mean some inequalities proved by the Moscow mathematician Nicolai Krylov (who is now a professor at the University of Minneapolis, USA) that now are well-known to the experts on the theory of stochastic differential equations as Krylov’s estimates. I think that Skorokhod made a lot of efforts to find a "direct" proof of those estimates, and so he believed in their correctness. Therefore, he also used them. But once I was a witness of his discussion with Krylov, during which Anatoly Volodymyrovych expressed his displeasure at how his friend argued to prove his estimates. Of course, the author had his own view on that proof, and this conversation was very interesting. Now let me say a few words about Anatolii Volodymyrovych’s attitude to his students. I think in this respect he is close to the Grigorii Skovoroda: there should be no authority to a student, no mentoring tone in discussion, only the personal interest to the problem which both of them were trying to solve, joint search for the ways to solve it. In conclusion to my speech, I want to mention about another passion of Anatolii Volodymyrovych. I mean his passion to poetry. During our walks in Lansing in the United States or here, in Kiev, he could read by heart poems by Osip Mandelstam, Anna Akhmatova, Boris Pasternak, Lina Kostenko, Joseph Brodsky et al. by hours.

I am deeply grateful to the Fate for being in my life such a Teacher and a Friend.

For me, Anatolii Volodymyrovych was a teacher first of all. When I was a graduate student and joined the department of the theory of random processes, Skorokhod’s system of training researches had already existed there. Here, I would like to say a few words about his attitude to the education of young researchers. Apparently, the main his aim was to form a responsibility in their work. Skorokhod always watched my notes and notes of other graduate students to the first significant error. Incorrect mathematical statements were not even discussed. This forces us, quite young people, to try to represent him a small, but the new mathematical result, clearly written. In the same way Skorokhod treated the performances at the seminar. He often interrupted the speaker saying that he told the obvious things and then showed it. This behavior seemed first to be impolite. But then I realized that in this way speaker was given a very clear understanding of the importance of speaker's own results at first, and secondly, time was saved for everyone who was present. Thus Skorokhod managed to create in us, his students and colleagues, the feeling of some absolute level of mathematical research, a kind of "Hamburg score", which each of us had to seek and satisfy. Of course, this was possible only thanks to the greatest talent of Skorokhod as a mathematician and teacher.

His ideas and teachings will remain forever in our memory.