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ogy under Heinz Hopf (“Hopf algebras”, “Hopf ?berings”). Of course not

personal contact, but Hopf told me about the famous 1935 topology con-

ference in Moscow. At that conference Kolmogorov had introduced a very

important new concept (independently of J. W. Alexander): namely cohomol-

ogy. Homology is due to Poincare (around 1900 already), and cohomology

is its dual; what topologists had not expected was the product structure for

arbitrary complexes. Well, cohomology in very different appearances has

been taken over by so many people, more and more in close connection with

algebra, functional analysis, measure theory and so on — that it is not well

known that the concept goes back to Kolmogorov! Although my thesis was

about homotopy, cohomology became one of my favorite research topics

until today.

Could this be the same person as the probabilist? Believe it or not, Hopf

said, it is! Not only that; it is in every respect a remarkable young man, of

unusual physical strength, climbing mountains of over 4000 meter altitude,

skiing enormous distances, swimming in ice-cold water-almost like his very

close friend Paul Alexandroff. Alexandroff in turn was a very good friend

of Heinz Hopf; their joint book on topology is well-known.

During these early years Kolmogorov made many other important con-

tributions to algebraic topology, for various types of spaces. At that time

not many could anticipate the importance topological ideas and all their

rami?cations would gain during the century — but Kolmogorov did! Thus

we must admire the really independent mind of this young man — inde-

pendent of fashion. Fashion was in that part of the century to work on the

Hilbert problems. Surely he did that too, in the axiomatization of the applied

?eld probability.

Let me remark that in the famous Hilbert 1900 list algebraic topology

(homology, analysis situs) is not mentioned with one single word! It was not

considered to be a respectable ?eld for a long time. I heard this much later

from Hermann Weyl; he even told me that around 1925 he had published

two papers on algebraic topology in Spanish in a South-Amedican Journal,

because he did not want his colleagues to read them!

Clearly I very much wanted to meet that man Kolmogorov. But there

was world war II and then the cold war. Communication by letter was slow

if not impossible, contacts were limited. Finally in 1954 I could meet Kol-

mogorov, he gave the famous plenary lecture at the International Congress

in Amsterdam — about Dynamical Systems, the beginning of what would

later become the KAM theory. As for myself, surely enough I was lecturing

about cohomology of groups.

83

Now things go on in the same way: At the next International Con-

gress 1958 in Edinbourgh I heard Kolmogorov lecturing, this time, about

Functional Analysis (just a short communication), at the 1962 Congress in

Stockholm about Optimal Approximation of Functions and so on and so forth.

No need to continue, you all know what I want to say: That I was lucky to

know Kolmogorov and to see him develop into one of the truly universal

mathematicians of our time, covering all ?elds (with the exception of number

theory, as far as I know), original, creative, deep and broad — whatever you

want and whatever will be said in this conference.

Contacts with Russia became easier during the years 1952 to 60 when

I was Secretary of the International Mathematical Union. It was the time

when we succeded in making the Union truly international, in spite of great

political dif?culties. All the countries of Eastern Europe, and China became

members. As a new member of the Executive Committee of IMU Paul Al-

exandroff was appointed — for me it should have been Kolmogorov, but I

understood that he was unpolitical in every respect. Of course we liked Paul

Alexandroff. He told us many stories about his friend Kolmogorov. For ex-

ample about their long voyage of several months on the Wolga and beyond in

1929 — what impressed me tremendously was the fact that Andrei had with

him on that long trip, apart from mathematics books, the Odyssey! And the

better I knew Kolmogorov the more I realised that his cultural universality

went much beyond mathematics, into logic and foundations, into arts and

poetry and history and education. His human and humanistic universality

enabled him to be an extraordinary teacher.

His students are here, they can tell about that better than I. He inspired

them to do mathematics according to its true nature and unity: abstract, valid

within its strict context, universal and precisely for that reason eminently

practical.

With the passing away of each human being a mystery disappears from

the world, a mystery that nobody else will be able to rediscover (Friedrich

Hebbel). Words, and certainly my words, are inadequate to describe the

mystery of Andrei Kolmogorov. But with regard to our common profes-

sion we can learn from him that mathematics is a manifestation of the free

creative power of the human mind and the organ for world understanding

through theoretical construction. And that it is part of the cultural tradition

we have to transmit to the next generation.

To achieve more we dare not hope, to achieve less we must not try.

84

Ïðèëîæåíèå Xa

Russian Academy of Sciences and Moscow State University

International Conference

Kolmogorov and Contemporary Mathematics

(Moscow State University, June 16 –21, 2003)

SCIENTIFIC PROGRAM

General Plenary Sessions

June 16, Monday

One-Hour Talks (Assembly Hall)

15:15–16:15 V. I. Arnold Kolmogorov and natural science

16:30–17:30 L. Carleson Kolmogorov and the convergence of

Fourier series

June 17, Tuesday

One-Hour Talks (Culture Hall)

10:00–11:00 S. Smale The mathematics of intelligence:

dealing with data

11:20–12:20 S. P. Novikov Fermi surfaces and dynamics systems

June 18, Wednesday

One-Hour Talks (Culture Hall)

10:00–11:00 Yu. V. Prokhorov Randomness and uniform distributions

11:20–12:20 U. Frisch Back to the primordial universe by

a Monge–Ampere–Kantorovich mass

transportation method

June 19, Thursday

One-Hour Talks (Culture Hall)

14:00–15:00 J. Palis A global view of non-conservative

dynamics

15:15–16:15 L. D. Faddeev Algebraic structures generated by

quantum integrable models

June 20, Friday

One-Hour Talks (Culture Hall)

10:00–11:00 A. G. Vitushkin The Thirteenth Hilbert problem and

some related problems

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11:20–12:20 F. Hirzebruch Topology and geometry in the work of

Kolmogorov

June 21, Saturday

One-Hour Talks (Culture Hall)

10:00–11:00 S. R. S. Varadhan Large deviations for random walks in

a random environment

11:20–12:20 Ya. G. Sinai Number-theoretic dynamical systems

Section 1

DYNAMICAL SYSTEMS AND ERGODIC THEORY

June 17, Tuesday

45-minute Talks (Auditorium 02)

14:00–14:45 A. B. Katok Entropy in dynamics: 1958–2003

14:50–15:35 V. V. Kozlov The statistical theory of Hamiltonian

systems

15:35–15:50 Break

15:50–16:35 A. Douady Present state of the question of Julia

sets of positive measure

20-minute Talks

Dynamical Systems and Ergodic Theory. I (Auditorium 14-14)

17:00–17:20 N. N. Nekhoroshev Strong stability of the approximate

fundamental mode of the nonlinear

string equation

17:20–17:40 M. B. Sevryuk “Atropic” invariant tori in Hamiltonian

systems

17:40–18:00 V. V. Basov Bifurcations of invariant tori

18:00–18:10 Break

18:10–18:30 Yu. N. Bibikov On the stability of the state of

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