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).)
Back to the first page
Last updated: 23.08.99