Century Mathematics

PREFACE Let me enlist the reasons, why a professional Mathematician, like I am, who is expected to be thinking and writing only on the topics of Mathematics, decided to write the articles of this collection. Around 20 years ago three events happened almost simultaneously. It all started with Russian Academy of Sciences at some point deciding to collect and publish articles on major events In Mathematics during the XXth century. Among others, I too was invited to contribute to that collection, and one of the editors of that publication, Yakov Sinai, suggested that I write for it also about the phenomenon of migration of significant part of Russian (the Soviet) Mathematics to Israel. I had written such an article, and it was published in that collection (Phasis, Moscow, 2003, and English translation by Springer-Phasis, 2005). Now, in present collection this piece of that article makes the first one. The outcome of that publication was not anything that I had expected: many both wrote to me and told me personally that the article was very well written, and that it had made a very interesting read. (Such was the reaction to Russian version, which I include here; I think that English translation with its formal tone was not much of a success.) The second trigger was that by a strange coincidence at the same time I came across three essays by William Somerset Maugham ("Yesterday I turned 70...", " Looking Back on Eighty Years..." and then "On My Ninetieth Birthday…”). It was quite an observation, the transformation of human approach to life, as the age of this human advances. So, I decided to record my own state of mind, upon reaching 70. I did that, and the result of the exercise makes the second article of this collection. It was too first written in Russian. Much later I translated it into English (with assistance of “Dimersky Translation Services” – I appreciate their help!) The third factor was something I considered funny at the time, but only recently, a few years after my retirement, it came to me, how serious it was. Michel Talagrand and Wansoo with Luda, Paris 2018 1 I have a very good friend, Michel Talagrand. He is an outstanding mathematician, the member of French Academy of Science, and, e.g., he recently received a one million dollar Shaw prize for his achievements in Mathematics. My wife, Luda, and I know both him and his wife Wansoo for around 40 years. He is 13 years younger, than me. Around 20 years ago his wife asked me, what I had been doing to prepare myself for retirement. I considered myself too young to think about it, but she explained that such things should be taken care of much in advance, and that she had already been preparing Michel for this (he was, still, below 50 at the time !). She, Wansoo, was coming from an aristocratic South Korean family, and was herself a scientist of a very high caliber. Apparently, already in School (I think it was a girl-school) they had been learning how to be taking care of their families, and planning much in advance the retirement of their husbands. I am ashamed to recall how much not seriously I reacted to this back then, because now I do understand her point. Adding the first and second reasons to this third, I have decided that it does make sense to me putting down in writing a few episodes of my life; this may make a “correct” occupation for me during my retirement. I live meaningful filled with diverse events life. There are many stories rotating in my mind. So, hopefully, I will have enough to do for some number of years. I would like to comment on one of the article presented only in Russian in this collection. It is about my participation in 1982 War in Lebanon. The interview, upon which this article is based, was taken from me in New York, a month and a half after my return from the front line. My father, David Milman, passed away, while I was in Beirut, and I arrived home to his funeral. This was a very emotional period of my life. Of course, later I would have talked very differently on the topic. But the facts inside the story remain the facts, and for the unknown to me reasons they so far remain not uncovered. I will add to this that for many months after returning home I could not sleep at night. With its silence falling upon me, my brain would had been reviving before me the endless rotation of the front lines life images. The body adrenaline of that period must had been too high in me, letting me go ever so gradually. Yes, this continued months, perhaps, 8 or 9 of them, time and time again, remaining the same. And, yes, I would have told this story very differently now, about the life on the front line of a war, and about how one felt “after”, upon returning home, and, also, why so. But, probably, this won't be right to do, so, I don't change anything in that article, it stays, as it was told by me at the time I had been interviewed. On a couple of occasions back then the interviewer had changed my words, so, that it became even wrong. But the matter on those occasions was so insignificant that it does not worth correcting. And, finally, let me also comment on the Chapter “Life and Consciousness” from “I am 80 to-day”. This is the only chapter with a lot of fantasy. Among some funny and, perhaps, not well known facts I add my own thinking (fantasy) on creation 2 of life and us. Sorry for this. Acknowledgements. I am infinitely grateful to Ludmila Milman for taking care of me, to Vladimir Milman for taking care of my bad English in a number of places of these texts, to Tatyana Petrova and Alex Segal for the computer editing, and to Jan Rauchwerger for his help in all aesthetic matters. Vitali Milman – January 2020." 3 V.D.Milman Observations on the Movement of People and Ideas in Twentieth- Century Mathematics Translated by R. Cooke The title of this article reflects my interpretation of the purpose of this collection. As I understand it, the articles are to be partly historical and partly mathematical. As a result, this article consists of three parts. The first part contains the words mathematics and mathematicians, but not mathematics itself; it describes the relocation of Russian (Soviet) mathematics to the West, more precisely, to Israel. I use the word relocation rather than emigration because of the size and scope of the process. As it happened, I found myself at the center of that event. The second part is a short historical remark on the ancestry of Banach and the origin of the term Banach spaces, so that this part is closer to mathematics. In the third part I present a little mathematics, including some of my recent observations on functional analysis, a field that played a large role in the mathematics of the mid-twentieth century and then blossomed into a large number of new areas. Here I shall sketch a picture of an area that is just now in the process of separating from functional analysis. Provisionally, we call it asymptotic geometric analysis, although I am not sure that this name will stick. [ NOTE: The original article published by Springer-Phasis in 2005 in the Collection “Mathematical Events of the Twentieth Century” Edited by Bolibruch-Osipov-Sinai. I copy here only the first section of this article adding some photos. - V.M., 2020. ] 4 1. From Russian to Israeli Mathematics The emigration of mathematicians from the Soviet Union to Israel began in the early 1970s, and as a result the so-called "Russian" mathematics began to relocate to the West. This emigration of mathematicians had a significant effect on both the West and Russia as early as the 1970s, and it became an avalanche in the 1990s. Every mathematical center in the West was touched and enriched by this movement. But only a few people understood that, while beneficial for these individual centers, it bore elements of tragedy for mathematics as a whole. At one stage, the Russian mathematical school looked as if it might disappear altogether, but today it can be said that the reality turned out to be less dramatic. Despite a hundredfold difference in salary, many first-class mathematicians remained in Russia. It is particularly gratifying to observe a very young generation of outstanding students graduating from Russian universities (as well as Ukrainian and other universities). The best universities of the West (and Israel) are striving to get them as graduate students. The concept of the "Russian mathematical school" is distinct from the concept of "the Luzin school," "the Kolmogorov school," or "the Gel'fand school," although it includes these schools and many others. This concept, which is extremely difficult to explain to a Westerner, encompasses traditions that prescribe ways of studying mathematics and a code of behavior for mathematicians. It is more an intellectual necessity (and a game) than it is work. Scholars raised in the traditions of the Russian mathematical school do not study mathematics for the sake of a salary. That is why the "chats" in the corridors of mathematics departments go on for hours at a time; and that is the most effective forum for studying and exchanging the latest mathematical news. That is why seminars have a beginning but no definite end, and a seminar lasting less than two hours is inconceivable.

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