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Украина чтит И.Гельфанда и Д.Мильмана

Биография Д.Мильмана

Публикации Д.Мильмана

Диссертация Давида Мильмана

David Milman's Thesis

Виталий Мильман

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to facsimile of D.Milman's Ph.D. thesis

In order to start browsing the photocopy of the original David Milman's second Ph.D. thesis, click the link "to facsimile of D.Milman's Ph.D. thesis" found in this text right below the photo of the general view of David Milman's Ph.D. Thesis that in this photo is shown open on its title page. You will then find yourself below this my INTRODUCTION note looking at the sequenced complete list of relatively small photos, each being the photocopy of one page of the original Thesis, all pages of it listed this way. Clicking on the first photo of this list, or on the text-link by it that says "Title Page" you will see the readable photocopy of the original title page of the Thesis with shown on it year 1950, and, further, clicking on this photocopy of the original title page of this Thesis, at any spot of it that you may want to see better, you will get significant zoom into this spot. Similarly, you may read and/or closely examine a photocopy of any other original page of your choice of this Thesis: Table of Content, Introduction, pages of the main body of Thesis.

Although this INTRODUCTION note addresses wide variety of readers not at all limited to mathematicians, but, rather, it addresses general population of those capable of thinking and interested in it people, since the Thesis is in the field of Mathematics, it will be fair to devote this just one paragraph to in precise accurate Math terms listing the key elements of this Thesis. I recommend that the readers, who are not mathematicians, skip it and continue reading from the next paragraph. The major quality of this Thesis is that, on one hand, it covers wide variety of topics that had just started flourishing at the time, so, the related to them research within this Thesis did very meaningfully enrich those topics, e.g., ergodic theories and dynamic systems, while, on the other hand, the author offers strictly and only his own completely authentic approaches and takes. Some of them are totally unique - such are the offered by him methods of the established by him science of geometry of Banach spaces, of which the powerful initial point is given by his own extreme points theorem. Others are offered in parallel with contemporary to him vanguard mathematicians, for instance, he firmly stands on the grounds of defining qualities of the sets being formulated in much richer, than the sets themselves, functions over these sets: e.g., this way he defines both the topologies of the sets, and the dynamic systems. In all of this can be clearly traced the constructed by him characteristic method of the science he establishes - the geometry of Banach spaces. Enlisting the key points of this Thesis, it consists of four chapters:

- in the first one, David Milman, based on his own extreme points theorem, develops into separate meaningful science the geometry of Banach spaces, this substantial and integral branch of the functional analysis of Banach spaces, his extreme points theorem being the central element of it, and introduced by him T-boundary (theoretical boundary) with all the geometry that is attached to this notion becomes the peak of this his research,
- the second chapter researches the multi-metric and multi-normed spaces, develops geometry of their convex compact sets, of their isometric transformations, centre instances and fixed points, researches ergodic qualities, and draws the picture of the invariant decay (sub-division) of such spaces into sub-spaces,
- the third chapter is devoted to research of defined by him through the spaces of functions compact dynamic systems, filtering their central parts, their invariant averages and their extreme invariant averages, and to the completeness of the linear isometric representations,
- and, finally, in the fourth chapter are collected three separate applications of his developed general method consistently carried over through this whole thesis - the isometry of the properly normed spaces, the integral representations of the maximal ideals through the ring border, and the application to the theory of rings with involution.

Further, this Introduction addresses general public, as well as mathematicians, and, consequently, I replace each Math term with my attempt to make it relatively acceptable for a reader who is not a mathematician.

This presented thesis is what in the Soviet Union times used to be called "doctoral thesis", but in the Western countries it would be right to consider it the second doctoral degree thesis. The first one used to be called in the Soviet Union "the candidate thesis" - it too corresponds to Ph.D. in the Western countries, but of somewhat weaker level of the expected science. I don't know what is the approach in Russia now. As to the presented here doctoral thesis, it had been defended and approved in 1950 by the oral Ph.D. exam committee with the made comments of this committee on its highly outstanding scientific level.

With this INTRODUCTION I aim to relatively briefly explain, why is it, actually, that it was decided to publish this thesis this way. I will start with at least briefly explaining the general meaning and sense in Math of David Milman's research, and, wider - in understanding of the whole we live in: already his early research carries over his deep view on what the whole is. For instance, his theorem on reflexivity of Banach spaces firmly connects the inherently local (that is, those concentrated in "small" proximities) qualities of infinite-dimensional spaces measured by norms (these making the analogue of distances from some "centre") with their inherently global qualities. That is, it links together the "direct" and "indirect" their qualities, gives conditions that sustain such connections. Interestingly, one of the early works of his oldest son, Vitali Milman, is the joint with David Milman research on how from reflexive "inside" it is possible to "get to" not reflexive "outside" of a space measured by norms (which space, consequently, is made of those "inside" and "outside", and is not reflexive, that is, it does not satisfy the conditions given by David Milman in regard to it being reflexive). Of course, the apotheosis of such kind of research by David Milman makes his extreme points theorem. The weight of this theorem in science and in global understanding is tremendous. By the way, it is carved on David Milman's grave stone, which is not a trivial fact, because Jewish religion and tradition strictly forbids on Jewish grave stones other symbols, than the letters of Jewish alphabet. Vitali Milman, though, managed to convince rabbis (Jewish teachers and priests), explaining to them that this carved theorem carries the weight and meaning of Talmudic statements. And, indeed, if we can agree that Talmud is the collection of Jewish high wisdom (it is compiled by the Medieval Jewish philosophers and scientists), then Vitali Milman is right, this theorem carries the weight of Talmudic wisdom. The connection of the regularity of the surrounding us Whole with its extreme points carries in itself principal for understanding meaning. In terms of the science of Math, such research of David Milman made him the founder of the branch of Math called "Geometry of Banach Spaces", that is geometry of infinite-dimensional spaces measured by norms (again, norms are so to say distances to the "centre" of space). Since in this INTRODUCTION I have consciously chosen the stylistics of stressing the role of events in the "history of what is inherently human", including the "inherently human Math", I should, probably, mention the peculiar from this standpoint fact that the proof of the extreme points theorem came to David Milman in his sleep, that is, through the work of purely his sub-consciousness.

Back to the extreme point theorem, when David Milman delivered his proof of this theorem to his thesis supervisor, Mark Krein, the recommendation was to immediately send this research signed by them both for publication into an internationally acclaimed scientific journal. The argumentation for this was that the results of such grandiose calibre come simultaneously to many scientists, so, the publication should be rushed, which would be insured by his, Krein's, participation, the participation of the scientist with established reputation. He, also, claimed that he could already see the important applications of this theorem, which they would too publish together. David Milman agreed, and the theorem acquired name of Krein-Milman theorem (letter "k" is ahead of the letter "m" in the alphabet). It soon became clear that Krein did not, actually, envision any application at that time, while David pretty quickly started pouring out the chain of very important applications. When Krein, finally, came up with some application, he published it together with his brother Selim, not involving David. Sure, I have no proof of any of this, but this story reached me coming out of the mouth of the very participant/witness of these events, David Milman, who was his whole life notoriously known for his honesty and decency, who only told it to a few closest to him individuals, and, at least by his will on this Earth, he did not want it to become a public knowledge. Yet, I think, that the will on this Earth differs from the will that comes from "out there", and that fairness "out there" is deemed even more important, than here on Earth. For the sake of this same fairness, I will stress that much later Mark Krein helped David Milman out of the situation characteristic for his naive nature, situation that under the Soviet Union reality could had costed David Milman his life.

The mentioned applications of the David Milman's extreme point theorem made the content of his being published on this site doctoral thesis (second doctoral thesis, per above explained), and, according to the references in it, his research resumed as soon as his post war circumstances started normalizing in the beginning of 1947. The delay was first due to the second world war itself, then the devastation of the country in ruins, and all along the burden and complications of the Stalin's rule. Nevertheless, by 1950 his thesis was ready for defence, this date can be seen on the title page that is too presented here, and, meanwhile, he continued publishing his research, including the thesis related one, in 1947, 1948, 1949 and 1950. It remains unknown, whether he treated his research the way I do, but I am happy to be seeing that four chapters of this his thesis offer important Math elements of the understanding of the surrounding us Whole. Approximately one quarter of the thesis context is made of direct consequences of his above-mentioned extreme points theorem. To put it briefly, they explain how the dots of an infinite dimensional space with norm can be represented by the means of probability measures upon some set that David Milman called "theoretical boundary" (T-boundary). Slightly simplifying, probability measure is nothing else, but breaking number 1 into a set of positive numbers that sum into 1, for instance, 1 can be broken into two positive numbers p and 1 - p.

In accordance with the rules of the time and place, after being with great success defended orally, thesis was sent to special state committee that sent it to two independent reviewers. These were Israel Gelfand and Andrei Kolmogorov. Gelfand gave great review, but it wasn't that simple with Kolmogorov. Upon the end of the war Stalin raised glasses for the great Russian people. This was understood in peculiar way. According to David Milman, in Odessa Jews were thrown off the street cars. Never directly announced antisemitism invaded the country; it never fully stopped, definitely not in the Soviet Union, but years 1950 to 1952 were the worst. It is believed that through these years not a single Jew had her/his second doctoral thesis approved.

Kolmogorov very well knew this. Such circumstances were bad in themselves, but it was worse in case of David Milman. While the Communist Party announced cybernetics, genetics and the theory of probability being the ideological diversions invented by the rich capitalist countries, David Milman openly and enthusiastically had been insisting that those were highly important new sciences that are to be under intense development. In the Soviet Union of that time people like him had been labeled "bastard cosmopolites", and they had been confined into Stalin's prison camps, but fate did have different plans in stock for Milman. Rumours were that Kolmogorov had been instructed not to allow bastard cosmopolite to have his second doctoral thesis approved. Was it so or not, but quite politically experienced Kolmogorov knew that David Milman's thesis should had better be "sitting low" through the bad times. So, he wrote to David that one of the chapters needed to be better edited. In this regard Gelfand commented that each chapter of this Milman's thesis was in itself a strong second doctoral thesis.

But David Milman was the man with principles. Upon receiving such review from Kolmogorov, he dropped his thesis onto the shelf of his cottage, and made his decision that he will never again return to the topic of its approval. And, indeed, when two years later Kolmororov let him know that his thesis could again be sent for review, David Milman ignored the offer. Yet, in addition to the already published research papers, he continued publishing out the key achievements of his thesis in form of laconic articles into (Soviet) Annals of the Academy of Science (Dokladi Akademii Nauk, DAN) and, partially, in the (Soviet) Achievements in Math Science (Uspehi Matematicheskih Nauk, UMN). The last such publication happened in 1952, but, unfortunately, these science journals back then probably had not been translated from Russian; consequently, this David Milman's research back then remained beyond attention of the mathematicians around the world. This must had been the reason that three years after the last publication by David Milman of his results associated in his second doctoral thesis with representations of the dots of space by the special kind of its dots, French mathematician Gustav Choqet, probably not knowing about further research in this direction of the author of the extreme points theorem, in December 1955 reported to Burbaki seminar his repetition of the most of these results - only this was taken, as his original research. This is how the whole such theory was called Choqet theory, and the mentioned special dots that David Milman called "T-boundary" (Theoretical boundary) acquired name Choqet boundary. These representations (in Math terms they are integral representations by some measures located on the mentioned special dots) may be treated as representations of the arbitrary dots through the centres of "probability" mass concentrated on the mentioned special dots. Although Choqet's research in this area was, as I have just explained, not original - it repeating David Milman's reseach, one his result was new and important: he formulated and proved the criteria for T-boundary possessing the measures needed for the above representations. On the other hand, David Milman introducing topologies (in Math topologies are the means of "structural measuring" of spaces) through them being constructed by the functions that are defined on these spaces (he calls them "functionally defined topological spaces", "f.d.t.s."), offers not only deep research of geometrical qualities of the infinite-dimensional spaces measured by norms (again, norm is sort of distance to the "centre" of space), but, also, "dynamic" applications. The whole of the third chapter in his (second) doctoral thesis is devoted to this. Similarly to topologies, he defines this "dynamic" side of things through functions. This way defined dynamic systems he calls "functionally defined dynamic systems" (abbreviated, as "f.d.d.s"). (Already in his paper published in the Soviet journal Annals of the Academy of Science in 1948 he had been researching dynamic systems this very way .) As to his thesis' chapter 2, it is devoted to geometric definition of ergodic systems, the systems that in a way explain what should be meant by talking about deterministic everywhere dense transformation of space. All of this, on one hand, solidly prepares mind to accepting what I call here "global UNDERSTANDING", and, consciously or not, is inspired by it, while, on the other hand, one day, I am sure, it will serve, as basis for the quantitative research in support of the constructive-qualitative global UNDERSTANDING. Of course, it will be fair to start calling "Choqet's theory" by David Milman's name, "Milman's theory" (I definitely from now on will be referring to it this way), with its Milman-Choqet boundary, thus, breaking the principle stated by V.I.Arnold that no science discovery is ever named after its discoverer. I strongly hope that such breaking of the Arnold postulate will, actually, happen. (Searching the Internet, I found very old and just one attempt in this direction, the research article by Uri Alekseevich Shashkin called "Milman-Choqet Boundary and the Approximation Theory", Functional Analysis and Its Applications, 1:2 (1967), 170-171; just one, but this is the start, after all.)

David Milman, though, while in the Soviet Union, remained with his first step doctoral science title of the Soviet Union tradition, which beyond Russia too corresponds to the Ph.D. title, and he, consequently, remained there with corresponding to it title of the education instructor position; the latter beyond Russia corresponds to University senior lecturer or assistant professor, or college professor, or something like this, although beyond Russia he was always referred to, as "professor". On one of David Milman's jubilee parties the director of the educational university level state Institute Of Communications, I.P.Pishkin, in his greeting speech said something like this (in my own words from my memory): "I have all the reasons to be proud of my achievements. After all, such world acknowledged outstanding scientist, as David Milman is, carries the same scientific title and the same educational position title, as I do". This reminds me of legendary statement made once about I.M.Gelfand that Soviet Academy Of Science must be the most worthy Academy Of Science in the world, after all, even I.M.Gelfand carries only the position of the correspondent member in it. (Unfortunately, many Soviet mathematicians that were full members of the Soviet Academy Of Science were anti-Semites, so, for a very long time they had been blocking Gelfand's full Academy Of Science member election; it is curious in terms of how much not selectively spreads the disease of anti-Semitism - these anti-Semite scientists themselves were solid good mathematician.)

*Vladimir Milman*

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