Russian version

Denis Artemievich Vladimirov

(1929 -- 1994)

      D.A.Vladimirov was born on February, 7, 1929 at Leningrad. He spent all the blockade in the besieged city and fully experienced all the burdens of the war (Vladimirov has left his notes about the blockade which contain a number of bright observations and personal memories as well as interesting information about the life at that time).
      At 1950 Vladimirov entered the Department of Mathematics and Mechanics of the Leningrad University. Having graduated from it at 1955, he became an assistant at the Chair of Mathematical Analysis. His research interests were formed under the influence of L.V.Kantorovich, G.P.Akilov, A.G.Pinsker. He was an outstanding representative of the branch of functional analysis created by L.V.Kantorovich and based on the concept of a semi-ordered space.
      Besides the general theory of semi-ordered spaces, the research interests of Vladimirov included spaces of measurable functions, invariants of measurable functions with respect to metric isomorphisms of their domains, properties of integral operators, the theory of Boolean algebras and the measure theory as well as their applications to general topology and the probability theory.
      The first publication of D.A.Vladimirov was a little masterpiece. It contained solutions of three difficult problems from a well known survey by Kantorovich, Vulikh, Pinsker (1951). The first part of the work presented a negative solution of the (o)-completeness and the (*)-completeness of an arbitrary K-space. The second part of the work dealt with Pinsker's theorem, "if the Diagonal Sequence Theorem holds for some K-space, then this space is regular". In the proof of this theorem Pinsker had used the Continuum Hypothesis (CH), and it was not clear whether one could dispense with it. Vladimirov showed that one could not. Moreover, he found a statement of the set theory (now it is known that this statement is independent in ZFC) equivalent to Pinsker's theorem. It allowed Vladimirov to make the following principal conclusion: "usage of the set theory hypotheses in the theory ... is not just a convenient technique but deeply relates to the essence of this theory". For that time, the idea was new. Among mathematicians whose works were not closely connected with logics, the opposite opinion dominated (in spite of Godel theorem it seemed to many mathematicians that CH was about to be proved and, say, Suslin hypothesis was about to be refuted). Two cardinals introduced by Vladimirov in this work are now widely used in logics and general topology.
      The first research work of Vladimirov was made while his being a student, long before the first paper appeared. It was devoted to the concept (introduced by him) of strongly compact linear operator in the space of  measurable functions. The results obtained in this field were not published till 1965--1967.
      For a number of years, Vladimirov returned to the problem of the metric type of measurable functions, i.e. of complete system of invariants characterizing a measurable function up to a mod 0 isomorphism of the measurable space. For functions on a Lebesgue space, this problem was solved by V.A.Rokhlin in 1957. In a joint paper with A.A.Samorodnitsky Vladimirov pointed out a class of measurable spaces in which the distribution function was not only one of invariants of the metric type, but completely defined it.
      The central subject of Vladimirov's works was the theory of Boolean algebras. One can regard the Boolean algebra as an object of logics, algebra, topology, the probability theory, analysis and the measure theory. It attracted Vladimirov mainly as an object of the three latter disciplines. He devoted a number of profound research works to the theory of Boolean algebras.
      In a series of works Vladimirov dealt with the problem of normability of a complete Boolean algebra (a Boolean algebra is said to be normable if there is a strictly positive countably additive measure on it) as well as the problem of existence of such a measure satisfying an additional condition of invariance under a given group of automorphisms of the Boolean algebra. These principal problems were treated by many authors (D.Magaram, A.G.Pinsker, G.Kelly, E.Hopf, A.Haian, S.Kakutani etc.). Vladimirov obtained a number of beautiful theorems in this field which implied the results of the above authors.
      In 1965 Vladimirov defended brilliantly his Ph.D. thesis which was based on these works but included some other results, in particular, the ones concerning the class of decomposable Boolean algebras introduced by him.
      In 1969 Vladimirov published a monograph "Boolean algebras". This excellent book may be an introduction into the subject for a mathematician of any specialization. At the same time, being devoted to the general theory of Boolean algebras, this monograph is the only work where the theory is treated from the standpoint of analysis (and partially of the probability theory). Though it exposes in details the classic matters, the central part of the book is entirely original. The book was translated into German and had two editions in Germany.
      In a series of papers (1979, 1983) Vladimirov found a criterium of metric independence and presented two examples of its application.
      In the last (by the time of publication) work Vladimirov solved a very difficult problem of the isomorphic classification of all pairs {a normed Boolean algebra, its proper subalgebra}. He introduced two invariants completely characterizing the pair (up to an isomorphism). The obtained result is a far-reaching generalization of Magaram--Kolmogorov classification theorem.
      Not long before his death Vladimirov completed preparing the second Russian edition of his book. It considerably exceeds the first one. The second edition includes (in considerably extended form) many new results of Vladimirov on ordered topologies, on homomorphisms of Boolean algebras, as well as the results of his pupils I.I.Bazhenov, A.V.Potepun, A.A.Samorodnitsky).
      Vladimirov liked to work with the youth. About fifteen postgraduate students began their research activities under his guidance.
      Several generations of students of the Department of Mathematics and Mechanics remember Vladimirov mainly as a talented teacher. Teaching was the work of his life, the subject of constant reflections and anxieties. He chanced to teach at different levels and at very different audiences -- from special courses and special seminars for future professional mathematicians up to general courses and not only for mathematicians; from lectures on the history of sciences and methods of teaching up to lectures for schoolchildren. Vladimirov devoted many forces and time to his TV lectures for university enrollees which he delivered for more than 15 years. Several years Vladimirov was the dean of the Department of refresher courses for teachers of mathematics and gave lectures to its students. His lectures at each level, for each audience, were always marked with particular clearness and methodical ingenuity. He generously shared his ideas with his collegues. Many of them remember gratefully his advices. Vladimirov liked direct contact with students and used to say that practical courses gave him as much satisfaction as lectures. His views on teaching mathematical analysis was mainly the result of comprehension of the intrinsic logics of its development. That is why he paid more close attention than usually to logic foundations of the course and early introduction of general notions. When considering each topic, he tried to choose a small number of main ideas and concetrate the exposition of the rest material around these ideas.
     Vladimirov was an outstanding personality. He struck those who knew him by breadth of interests, rare erudition and wit. Books (of quite various contents) were for him not only a pastime and a source of information, but the mode of life. Vladimirov was a passionate patriot of Petersburg--Leningrad. His library contained a rich collection of old and new maps, guides, reference books on the city. He possessed a deep knowledge of the topography, history, toponimy of the city and its environs.
     Images of the city which were imprinted on his soul since the childhood and supplemented with the terrible shock of the blockade had merged with his ego, and there was no place for other landscapes in his soul. He almost never left Leningrad and we had not listened him praise the places he had visited.
      Classical music occupied a great place in his life. His understanding of music struck by insight and profundity suprising for a non-specialist. He understood and knew the art (and he painted well himself; we remember the elegance of his pictures which he drew when delivering lectures).
      One particular trait differed Vladimirov from his collegues who usually shunned philosophy and braved their hostility with respect to it. He knew the classical philosophy (he had begun his higher education at the Department of Philosophy) and was inclined to look in a general, philosophical way, whether at scientific, pedagogical, political or everyday problem. Maybe that is why his opinions on current events were so sharp, shrewd and free of cliches. He often shaped his statements into an aphoristic, hyperbolized and sometimes even shocking form. Usually they were based on the position that was considered more deeply and thoroughly than the position of his opponents. And even those who did not agree with some of his opinions fell under his influence and felt the need to associate with him again and again.

      A.I.Veksler, S.A.Vinogradov, G.A.Leonov, A.A.Lodkin, B.M.Makarov, G.I.Natanson, A.N.Podkorytov, A.V.Potepun, B.A.Samokish, A.A.Florinsky, V.P.Khavin.

  (Translated from the paper in Vestnik of the St.Petersburg University (1), 1994, 4(22).)


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Last updated: 16.07.2017