Table Of ContentMathematical Events of the Twentieth Century
Mathematical Events
of the Twentieth Century
Edited by A. A. Bolibruch, Yu. S. Osipov,
and Ya. G. Sinai
123 PHASIS
† A. A. Bolibruch Ya. G. Sinai
University of Princeton
Yu. S. Osipov Department of Mathematics
Academy of Sciences Washington Road
Leninsky Pr. 14 08544-1000 Princeton, USA
117901 Moscow, Russia e-mail:
Preface
Russian mathematics (later Soviet mathematics, and Russian mathematics once
again) occupies a special place in twentieth-century mathematics. In addition
to its well-known achievements, Russian mathematics established a unique style
of research based on the existence of prominent mathematical schools. These
schools were headed by recognized leaders, who became famous due to their
talents and outstanding contributions to science.
The present collection is intended primarily to gather in one book the tes-
timonies of the participants in the development of mathematics over the past
century. In their articles the authors have expressed their own points of view on
the events that took place. The editors have not felt that they had a right to make
any changes, other than stylistic ones, or to add any of their own commentary
to the text. Naturally, the points of view of the authors should not be construed
as those of the editors.
The list of mathematicians invited to participate in the present edition was
quite long. Unfortunately, some of the authors for various reasons did not
accept our invitation, and regretfully a number of areas of research are not fully
represented here. Nevertheless, the material that has been assembled is of great
value not only in the scientific sense, but also in its historical context. We wish
to express our gratitude to all the authors who contributed.
We hope that this collection will induce other authors to write their memoirs
of mathematical events of the twentieth century which they witnessed and par-
ticipated in. This will open a possibility for a continuation of the present edition.
We are very glad to thank Owen de Lange for his help in the preparation of
the English edition of this book.
The Editors
P. S. A. A. Bolibruch played a very important role in preparing this book. His
untimely death was a great shock and unrepairable loss for all people who were
fortunate to work with him.
Heidelberg, 2005 Moscow, 2005
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V
D.V. Anosov
Dynamical Systems in the 1960s: The Hyperbolic Revolution . . . . . . 1
V. I. Arnold
From Hilbert’s Superposition Problem to Dynamical Systems . . . . . . . 19
A.A. Bolibruch
Inverse Monodromy Problems of the Analytic Theory of Differential
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
L.D. Faddeev
What Modern Mathematical Physics Is Supposed to Be . . . . . . . . . . 75
R.V. Gamkrelidze
Discovery of the Maximum Principle . . . . . . . . . . . . . . . . . . . . 85
Yu. S. Il’yashenko
The Qualitative Theory of Differential Equations in the Plane . . . . . . 101
P. S. Krasnoshchekov
Computerization . . . Let’s Be Careful . . . . . . . . . . . . . . . . . . . . 133
V. A.Marchenko
The Generalized Shift, Transformation Operators, and Inverse Problems . 145
V. P.Maslov
Mathematics and the Trajectories of Typhoons . . . . . . . . . . . . . . . 163
Yu.V.Matiyasevich
Hilbert’s Tenth Problem: Diophantine Equations in the Twentieth Century 185
V.D.Milman
Observations on the Movement of People and Ideas in Twentieth-Century
Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
VIII Contents
E.F.Mishchenko
About Aleksandrov, Pontryagin and Their Scientific Schools . . . . . . . 243
Yu.V. Nesterenko
Hilbert’s Seventh Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 269
S.M.Nikol’skii
The Great Kolmogorov . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
A.N. Parshin
Numbers as Functions: The Development of an Idea in the Moscow
School of Algebraic Geometry . . . . . . . . . . . . . . . . . . . . . . . . 297
A.A. Razborov
?
The P=NP-Problem: A View from the 1990s . . . . . . . . . . . . . . . 331
L. P. Shil’nikov
Homoclinic Trajectories: From Poincare´ to the Present . . . . . . . . . . 347
A.N. Shiryaev
From “Disorder” to Nonlinear Filtering and Martingale Theory . . . . . . 371
Ya.G. Sinai
How Mathematicians and Physicists Found Each Other in the Theory
of Dynamical Systems and in Statistical Mechanics . . . . . . . . . . . . 399
V.M.Tikhomirov
Approximation Theory in the Twentieth Century . . . . . . . . . . . . . . 409
A.M. Vershik
The Life and Fate of Functional Analysis in the Twentieth Century . . . 437
A.G. Vitushkin
Half a Century As One Day . . . . . . . . . . . . . . . . . . . . . . . . . 449
V. S. Vladimirov
Nikolai Nikolaevich Bogolyubov — Mathematician by the Grace of God 475
V. I. Yudovich
Global Solvability Versus Collapse in the Dynamics of an Incompressible
Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
Name Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529
D.V.Anosov
Dynamical Systems in the 1960s:
The Hyperbolic Revolution
Translated by R. Cooke
Probably everyone would agree that the theory of dynamical systems underwent
a profound change during the 1960s, that its general features as then formed
have been retained to the present time despite significant changes in content,
and that this theory turned into a separate discipline at that time, having been
up to that point a branch of the theory of ordinary differential equations. Here
I intend to discuss certain events of this period, after which I shall say a few
words about what came after.
The theory of dynamical systems consists of three branches, corresponding
to the general character of the objects and questions it considers: differential
dynamics — the theory of smooth dynamical systems, the ergodic theory — the
theory of metric dynamical systems (in the sense of measure theory), and topo-
logical dynamics — the theory of topological (continuous but not differentiable)
1
dynamical systems.
1 These explanations of the subject matter of each of these three parts, being rather general and
brief, are not entirely accurate. Thus, in reality the first part also encompasses one-dimensional
dynamics, including real dynamics, while the mappings considered in the last part are nonsmooth
and even have discontinuities. Such failure of the subject matter of real one-dimensional dynamics
to comply with the norms of the theory of smooth dynamical systems is compensated by their
conceptual proximity. The picture is different for symplectic systems. They have not lost their
2 D. V. Anosov
The rise of dynamical systems as a separate discipline was due mostly to
two events in differential dynamics: the development of KAM theory (its seeds
were planted earlier, but did not immediately receive wide development) and the
“hyperbolic revolution.” The chronology of all this is framed by the penetration
of ideas of probabilistic origin into ergodic theory: the entropic theory arose (and
2
immediately began to develop vigorously) in the mid-1950s. The Ornstein
theory (which within the limits of its applicability sometimes establishes an
even closer connection of dynamical systems with the most ordinary random
processes such as coin tossing than does the entropic theory, while at other
times it points up subtle differences between them) arose in the 1970s.
It should be mentioned that two topics vary with particular vigor in the
3
theory of dynamical systems. In their “purest” form they occur in differential
dynamics as quasi-periodicity, for which a certain regularity is characteristic,
and as hyperbolicity, which is connected with those phenomena that are descrip-
tively named “quasi-randomness,” “stochasticity,” or “chaos” (see the article of
Yoccoz [1]). The use of spectral concepts and methods in ergodic theory, which
began as early as the 1930s, can be regarded as a sort of implementation (in a
4
suitable context) of the first topic. The second topic found an adequate expres-
sion in ergodic theory just in the 1950s and 1970s, when the entropic theory and
the Ornstein theory arose. Kolmogorov, whose name provided the first letter in
the acronym KAM, was the only one at that time who made an equally large
contribution to the study of both regular and chaotic motions (but in different
parts of the theory of dynamical systems — the differential and the ergodic).
His colleagues in the acronym exhibited a lively interest in chaos and of course
interest on the part of scholars of such stature was bound to have consequences;
but these consequences pale noticeably in comparison with their contributions to
the study of regular motions. The same people who succeeded in distinguishing
former smoothness at all, but specifically “symplectic” considerations, questions, and so on, have
increased to such an extent and have acquired such great importance both within the theory of
dynamical systems and (mainly) outside it that symplectic dynamics, which previously belonged
to the differential theory, now appears to have attained the status of a separate, fourth branch of
the theory of dynamical systems.
2
More precisely, the first papers by Ornstein on this topic appeared at the end of the 1960s.
3
And not only in that theory. Kolmogorov was guided by similar considerations in a broader
context when he allocated a significant portion of his papers into the first two volumes of his
collected works.
4
This by no means goes all the way back to the original source. A continuous spectrum, and
even more, a Lebesgue spectrum is more characteristic of a dynamical system of “chaotic” type.
But, because of its very origin in the analysis of sufficiently regular motions, the spectral approach
does not suppress certain essential specific features of “chaos” (which, of course, does not make
it superfluous or even unimportant).
Dynamical Systems in the 1960s: The Hyperbolic Revolution 3
themselves in “chaos” left no marks at all at that time in matters connected with
regular motions. Only later was this “tradition” violated by Ornstein [2], [3] and
Sinai [4], [5], but again, despite the importance of these papers, the authors do
not owe their fame to them. All this goes to show that the differences between
regularity and chaos do not simply lie on the surface, but are buried deep within,
so that it is difficult for even the best specialists to switch from one to the other.
Only in recent times have the contributions of a few brilliant representatives of
French mathematics (Yoccoz, and Perez-Marco) in the two areas become more
equally balanced.
The question arises: Is there some other “sufficiently substantial” class of
motions that could occupy an intermediate position between the quasi-periodic
and the hyperbolic motions (or, perhaps, lie somewhere to the side of both)?
Could horocyclic flows and (or) nilflows (or, perhaps something of the sort that
we do not yet know about) play this role? In any case, one would like to
investigate what takes place in perturbations of these flows. Passing to discrete
time, one could pose the same question in relation to pursuit mappings for
nilflows. In the simplest case this involves perturbations of a mapping of the
two-dimensional torus
(x,y) →↦ (x+α,x+y) (x,y mod 1; α irrational).
For a start it would be good to develop at least a formal theory of pertur-
bations (like the formal series in celestial mechanics, which may be divergent,
but nevertheless contain significant information, being, in particular, asymptotic
expansions of real solutions). As far as I remember, such questions were posed
in the early 1960s by Arnold.
I was a witness to all these events from the late 1950s on and a direct
participant in the “hyperbolic revolution.” The memories of a participant, of
course, are more complete than those of a witness; and I hope that their value
is higher.
But first I must relate something of my life before the “hyperbolic revo-
lution” began. I was a student, and later a graduate student, of Pontryagin,
who had switched entirely from topology to ordinary differential equations and
associated questions (variational problems of automatic control theory, and later
to the theory of differential games, which was created by him, to a significant
degree). It seems to me that Pontryagin’s greatest achievements are nevertheless
in topology, and that he was past his prime when I met him. But the beginning
of the decline was very gradual, and the level was still very high. The only
critical remark about this period of his activity, in particular, is not connected
with his activity itself, but with a certain frame of mind that came over him:
