The interweaving of the trajectories of life

Vladimir Igorevich Arnold left us on June 3, 2010, but all the time it seems that this is not true, that he is about to come, that we will hear his energetic and cheerful voice. I want the real Arnold not to be forgotten, the more important the memories of his closest people. We publish an article in memory of V.I. Arnold of his friend and colleagues Viktor Pavlovich Maslov, doct. physical. sciences, professor, academician of the Russian Academy of Sciences.

If death is destined for me
in the arena,
The death of the tamer,
I know now:
This invisible beast for the public
The first to have my knees.

Nikolai Gumilyov

I can’t even imagine that Arnold is no more. He is "more lively for me to all living." And death crept up to him from the wrong side from which his wife-doctor El put a reliable defense. At the most tragic moment of my life, when my wife was dying, I rushed to Arnold, and his wife El said that the doctors had to make her to save her.

My life as mathematics and personal life in a slightly bizarre way intertwined with Arnold's life. In our career, he, then I pulled forward, and we helped each other. At some point, for some reason, the trajectories came together at one point, and we both (rare-two Russian mathematicians at once) received an invitation to make plenary reports on the same Stockholm mathematical congress (1962).

When we were young, V. Arnold, Yu. Manin and I, we felt the need to make a “coup d'etat” in mathematics, how the “coup d'etat” was committed in physics, when the quantum mechanics, which even the venerable professors of the physical department of the Moscow State University did not understand, began to teach 4th year students.

Once at the beginning of the century V.A. Pyast published an article entitled “Council”, in which the fact of the transition of representatives of modernist literature from the position of “outcasts” to the position of “recognized” was noted.

I noticed that the applicants who perfectly surrendered mathematics did not fall into the physical faculty, and thought about gaining a group of students from the “outcast” and conducting such a variety with the help of such a “coup d'etat” group. I agreed with the rector of MIEM (Moscow Institute of Electronic Engineering) E.V. Armen, that I will move from Moscow State University to MIEM and the head of the Department of Applied Mathematics, if I have the opportunity to take the “outcast” group every year without exams. V. Arnold, Yu. Ma-Nin, M. Fedrayuk, A. Kostyuchenko, and from physicists, the student Bogolyubov B. Medvedev undertook to read courses in modern mathematics, according to a completely new program, for this group of “outcasts”. We all experienced the inspiration of futurists in mathematics. Each lecture by Arnold and Maninin was a unique masterpiece. There were only 15 listeners, they perfectly understood how lucky they were.

In Yu. Manin, preparations for a lecture occupied at least 10 hours, and it was like a competition with the world's best lecturer Arnold. At these courses, Manin, Medvedev and I published a new type of textbooks. Unfortunately, the second such group from the “outcasts”, selected based on the results of a written exam in mathematics, could not be recruited. The rector of Armensky was called to the Central Committee, he received a bit, and on this group the experiment ended. Unfortunately, no matter how I fought and no matter how scientists of various institutions helped me, the laws of the so -called “distribution” scattered these guys in different directions. There was no Landau minimum, who collected the talented young scientists about one leader. It was easier at MHMAT MHMU, and Arnold overgrown with a wonderful school; In Miem'e, after catching up, it became more complicated: “outcasts” became even more than at the physical faculty of Moscow State University.

Arnold used to protect my doctor. Before the defense of my doctoral dissertation, one of the opponents fell ill, I asked Arnold to urgently be my opponent. For 5 days I told him the geometric part, and from my words he wrote a brilliant text, setting all the points over the “I” in the geometry of the intersection index with the design of Lagranzhev’s diversity design to coordinate hyperpravianness. I had a more general algebraic concept, and its geometric interpretation was given mainly with examples. Algebraic evidence occupied a whole chapter of the book “Theory of Resentment and Asymptotic Methods” (Publishing House of Moscow State University, Moscow, 1965).

For a long time I said Arnold to publish the text of the review, and finally it went under the heading "On the characteristic class included in the conditions of quantization." The number of references to it was huge. As far as I remember, in this article he basically did not change anything in comparison with the review. Mathematics historians, who, of course, will explore the legacy of such a large scientist, can compare these texts on the dissertation archive.

The algebraic abstract approach was not lost. S.P. Novikov applied it in a cycle of brilliant work on K-Theoria, which later developed by his wonderful school. It is also used by physicists for the specific calculation of the so -called Maslov index.

Arnold was not a corresponding member when I, bypassing the title of member of the corrar, despite the opposition of the CPSU Central Committee, was immediately elected to academics. Not still approved at the general meeting, I nevertheless came (not quite legal) to the elections of a member of the corps. I came not in vain. The influence on the member of the corps who still does not know how the newly-minted academician will vote cannot be discounted, but, frankly, the main role in Arnold’s elections to the corresponding members was played by L.D. Faddeev.

Elections to academics are another matter. Here we needed painstaking work, analysis, who and how will vote in the future, who has a chance to get away from scientists who will vote for Arnold. This principle (the minimum) was for me as a litmus paper in the election of the next members of the academy.

When, finally, Arnold became a real candidate, he doubted the voice of the largest scientist (we both considered him ingenious), before which the cheism experienced a certain complex of inferiority. He, by and largely, told me about his doubts, and I vowed to fully control this moment in the elections.

Arnold was never a diplomat, he was always sincere to naivety. For example, when he spoke in support of one of his brilliant students nominated a member of the corrary, he said that the latter is stronger than all candidates, including among a member of the corps nominated to academics. I later told him that he thereby offended the voters, and calculated how many votes he lost as a result of this. It was necessary to see how he repented and scolded himself.

Arnold was helpless when his superiors - deputy deputy or the head of the department - were eaten on him. I described our vicissitudes at Congress in the book "Unarmed Love." I will add that his head. Department of O.A. Oleinik (to put it mildly) asked to mention her in the report. I laughed and said: "And please look, do not mention me." He fulfilled my request, but began his report with the words: "All these tasks came from Miem." In our reports, we helped each other change transparenes, and when it was my turn to help Arnold, I circled and emphasized the name Oleinik on Transpensers.

Arnold told his famous disaster theory. Indeed, I was an initiator of this work to a certain extent. But I have an irresistible aversion to all sorts of praises addressed to me (especially in person and especially if it is connected with unambiguous hints of my now rather advanced age). For example, to the unexpected praise of my scientific merits in Novaya Gazeta, and even in comparison with other scientists (No. 36, dated May 22, 2008), I was able to immediately respond, since in the next issue there was my note “On Risks and Acquisitions” (“New Gazeta”, the “Centaur” application, No. 11, dated May 29, 2008), in which (due to the reduction of the important text of the economy about economics about the economy about the economy about the economy about the economy about the economy about the economy. The upcoming crisis) I protested: “I categorically cannot agree with the excessively high assessment of my contribution to science, this edition of the“ Centaurus ”to No. 36)” , etc. for 1/6 of the volume of the article.

And the problem solved in the famous theory of Arnold's disasters has tormented me since 1965, when I described in detail the kaustics and focal points using the canonical operator (now it is called the Fouri-Integral Operator). In the work “Theory of Indignation and Asymptotic Methods”, I managed to reduce the task to the minimum number of integrals, but I would like to express these integrals even easier, through the specifications that D. Sheffer and V. Guillaimin made in a number of important cases.

L. Hurmander, having learned about my theory from the words of Yu. Egorov, gave another representation of the canonical operator, which was worse than the canonical operator with the above point of view. His plenary report on Congress in Nice in 1970 was published in the form of a preprint and translated into Russian in the magazine "Mathematics" (16: 1, 1972, p. 17-61).

Regarding the canonical operator, the following was said there. “The work of Egorov, in fact, is the application of Maslov’s ideas set forth in the book“ Theory of Indignation and Asymptotic Methods ”. The author regrets that he was not directly familiar with this book, which, according to Maslov’s report at the International Congress in Nice, contains ideas attributed to Egorov and Arnold, and more general and accurate operator calculation than that will be described in this work. Nevertheless, due to the fact that this book is extremely inaccessible and, possibly, the evidence is not quite strict, we hope that this work will still be useful. ”

I was asked (including O.A. Oleinik), whether I agree with this formulation of Hurmander. I said that I categorically disagree. Firstly, a comparison with other mathematicians is incorrect: the work of Yu. Egorov generally refers to the transformation, long open to V.A. Foc. The contribution of Arnold and his brilliant interpretation, as I have already said, from my point of view, is invaluable, and the work of Hurmander illuminates a number of the most important aspects of this problem.

Secondly, it is incorrect to say "perhaps the evidence is not quite strict." The work was checked by such subtle and wonderful mathematicians as G.I. Eskin and O.A. Ladyzhenskaya, and I exhaustively answered all questions that arose in them in the presence of such specialists as V.P. Palamodov and S.P. Novikov. Their questions helped me in my lectures -I realized what moments should be covered in more detail.

Then, at the Congress in Nice, I asked the question of representing a canonical operator in the form of simple formulas to the great mathematician Sir Michael Atier. I presented a trivial case of the amount of degrees in the exponent (action) of the integral and showed that the degree of features according to the parameter H (the Planck Constant) does not change after replacing variables, which means that the features of the design of the Lagranzhev introduced a variety of the coordinate plane introduced by me. But this did not interest him. I paid attention to this fact of other topologists, as well as A. Mishchenko and V. Arnold, but no one became interested in them.

Finally, I still managed to arouse interest in this problem of engineers of the Ministry of Radio Industry. I suggested that they solve the problem of reflecting the radio reeds from the Hevisaid layer. This is a task in which the occurrence of very complex tricks is extremely rare, and it is necessary to consider focal points and caustic in the overall position.

Having concluded in a row, I hired Arnold's team to solve an extremely laborious problem on the classification of Kaustik in the overall position. Arnold and his team brilliantly solved this problem (I played a role only as tumor and overseer). Arnold, however, thanked me in his articles with such enthusiasm that in the book of Poston and Stewart “Theory of Disaster” I almost attributed 50% of Arnold's merit, although I had no more relation to this work than that general who put his signature in the work of reflecting radio waves in closed journals.

In the end, Minister Pervushin himself wrote an idea of ​​the recommendation of my candidacy for member of the USSR Academy of Sciences, which in no extent compensated for the negative opinion of the Central Committee. I want to emphasize that, according to the Hamburg account, it was Arnold who earned such attention of the highest authorities of the radio industry, and not even me and not even the aforementioned general.

When I calculated the inevitability of default in Russia in 1998, I decided to sell the cottage that was next to the Arnold dacha, and came to him to tell about my new concept of chaos. He subtly felt the shades of my story and said: “You tell as if you were going to call me to opponents again.”

He guessed. I explained that I want to leave the country to the UK, at least for a while, to wait out the consequences of the default if they lead to the collapse of the Russian Federation. In UK, my secondary daughter can continue to study at school (there is a 13-year-old secondary education)-by that time she graduated from school in Troitsk, but had not yet chosen specializations. Arnold told me that he was just asked at Hong Kong University to persuade me to accept their invitation to work, for at least one year, on very good conditions. In Hong Kong the same education system as in England.

In addition, Arnold turned to our common friend Sir Michael Ber-ri, and he also sent me an invitation to come to Bristol for a year.

The invitation from Hong Kong had many favorable parties that Arnold and other scientists described me. But at the last moment I preferred the UK-after all, this country is closer, only 3 hours to summer to Moscow. Defolt, fortunately, was quickly overcome, and my family and I returned to Russia, leaving the eldest daughter in England to study at the university.

The last mystical coincidence of our trajectories - his funeral coincided with my anniversary. After the funeral, our common friend said sadly: "Arnold" ordered to live long " - it turns out that he ordered you." For us and Arnold, to live - this means creating. When he was taken to the hospital on the day of death, sheets with unfinished formulas remained on his pillow.