Handling editor: Hinke Osinga
In Memoriam Floris Takens
by Henk Broer,
Groningen University, The Netherlands
Els, Leida, Vincent, other family members,
friends, colleagues and acquaintances!
The scientist
Floris Takens became a full professor of
mathematics at Groningen University in 1972, when he was 31 years old.
He had gained his doctorate at the University of Amsterdam in 1969
under the supervison of Nico Kuiper. The title of his thesis was
The minimal number of critical points of a function on a compact
manifold and the Lusternik-Schnirelman category. Following his
doctoral studies he spent a year as a guest researcher at the Institut des Hautes
Études Scientifiques (IHES) in Bures-sur-Yvette near Paris
(1969-1970). There he was influenced both by René Thom and
David Ruelle (known in Groningen from the 1999
Johann Bernoulli Lecture). With Ruelle he wrote the paper On
the Nature of Turbulence, published in the journal Communications
in Mathematical Physics (1971) [1].
This was a groundbreaking paper, in which a new idea was introduced
that contradicted the established theory about the onset of turbulence in
fluid motion as developed by the leading physicists Landau and
Lifschitz and the eminent mathematician Hopf. The new concept proposed by
Ruelle and Takens was baptized strange attractor, and it
later became part of the theory of chaos.
Context
Takens's chair was in the field of Differential
Topology, in particular Dynamical Systems. Geometrical ideas had
been introduced into Dynamical Systems research by the pioneering work
of Poincaré, in celestial mechanics among other things. In the
1960s and 1970s the field of Dynamical Systems received an enormous
impulse in this direction by the contributions of the Fields medalists
Stephen Smale (University of California at Berkley) and René
Thom (IHES) and their schools. Both had become famous in Geometry
(Topology). Thom is renowned for Catastrophe Theory, which
has later become somewhat controversial. The former topologist
Christopher Zeeman (University of Warwick) also participated in the
development and application of this theory (Zeeman held the
1993 Johann Bernoulli Lecture).
This geometric culture was perfectly suited for Floris Takens. The
Brazilian Jacob Palis, who obtained his Berkeley doctorate from Smale,
is of the same generation. Floris and Jacob maintained an extensive
and extremely fruitful collaboration from 1971 onwards. For a long
period Floris was a guest researcher for several months each year at
the beautifully situated Instituto de Matemática Pura e
Applicada in Rio de Janeiro.
Research Themes
Floris Takens wrote dozens of important papers,
many of which are influential to this day for researchers all over the
world. Roughly speaking, two research directions can be distinguished
in his work, and taken together, he supervised about 20 PhD students in these areas.
Stability, Hyperbolicity, Bifurcations.
With his contributions to structural stability and moduli in the
setting of (almost) hyperbolicity and the bifurcations from simple to
complex behavior, Takens is surely one of the founding fathers of the
modern discipline of Dynamical Systems. His major scientific standing
and his numerous international contacts have greatly benefited both
the University of Groningen and the Dutch mathematics community as a
whole. His PhD students in this scientific direction at
Groningen University were Albert Hummel, myself, Gert Vegter, Fopke
Klok, Jan Barkmeijer, Cars Hommes, Ale Jan Homburg, Bernd Krauskopf,
Florian Wagener, Evgeny Verbitsky and Renato Vitolo. Outside
Groningen we can add Freddy Dumortier, Bert Jongen and Sebastian van
Strien to this list.
Nonlinear Time Series.
Around 1980 Floris Takens initiated a new direction of research in
which information regarding characteristics of the dynamics, such as
dimensions of attractors, entropy, Lyapounov exponents, etc., can be
obtained from time series generated by deterministic systems for which
the equations of motion need not be known [2].
Many non-mathematicians have applied and adapted this theory, whose
main result is now known as the Takens Embedding Theorem.
Using his theory, Floris made a significant contribution to chemical
process technology, and this earned him an honorary doctorate at Delft
University of Technology --- an award in which he rightfully took
considerable pride. In this research direction he acted as an advisor
for a number of external PhD students, namely, Jan-Pieter Pijn, Pieter
Been, Cees Diks and Marcel van der Heijden. In Groningen he was also
coadvisor of Svetlana Borovkova.
|
|
|
University of Groningen's
Academy portrait of Floris Takens by Jacqueline
Kasemier. |
`Totalmathematiker'
For Floris Takens the discipline of mathematics was
one organic entity that includes applications. This point of view
applies to his own career, during which both `pure'
Differential Topology and `applied' Time Series Analysis
coexisted in a brotherly fashion. There is a natural place in his
papers for Analysis, Geometry (in many manifestations) and Measure
Theory, among other things. Furthermore, he wrote programs himself in
computer languages such as Matlab and C++ when the need
arose. It should be noted that the area of Dynamical Systems is
closely connected to Mathematical Physics, as is apparent from
Takens's early work with Ruelle.
Floris always fiercely resisted the constant threat
of fragmentation of the mathematics curriculum. One of his ideals was
that all professors should be able to teach all courses in the first
three years of the undergraduate (Bachelor) degree program. This never
came to pass in Groningen, but I'm sure that he himself would easily
have been able to do it without any problem.
Takens was an editor of the Springer Lecture Notes
in Mathematics, an honorable task that he continued to perform for
another decade after his retirement in 1999. Bernard Theissier, one of
his fellow editors, sent a condolence message that highly commends
Floris's `immense culture.'
After his retirement Floris Takens was still
closely involved with the PhD theses of Renato Vitolo (2003, where he
acted as co-advisor) and of Olga Lukina (2008). One of his interests
in the last decade of his life was the geometry of torus bundles that
arise in integrable and nearly integrable Hamiltonian systems.
This is an interesting direction of research, which is related to
earlier work by Hans Duistermaat and Richard Cushman and has both
classical and quantum-mechanical applications in theoretical chemistry
and other fields. Here, Floris could whole-heartedly give free rein to
his old passions for Differential Geometry and Algebraic Topology.
Floris faithfully kept attending PhD defenses, colloquia and
relevant seminars. Together, we worked on the advanced textbook Dynamical
Systems and Chaos [3] and on Volume three
of the Handbook of Dynamical Systems [4].
|
Giethoorn, spring 2006;
photo by Els Takens.
|
Additional activities
Apart from numerous teaching duties, among which
the famous (and somewhat dreaded) Bachelor course
Differentiaalrekening in \( \mathbb{R}^n \), Floris Takens regularly taught
Master courses in Analysis on Manifolds, Differential Geometry,
Differential and Algebraic Topology, and he held weekly seminars in
his office. These mostly dealt with Dynamical Systems, but
occasionally also with Riemann Surfaces, Sheaf Theory, and many other
subjects.
Floris also performed his share of administrative
duties. From around 1990, he served for several years as Head of the
Mathematics Department. He also served a term as chair of the national
Mathematics
Research Institute (jointly with the Universities of Nijmegen,
Twente and Utrecht). Floris was one of the founders of the Dutch
FOM/SWON program Mathematical Physics and also acted as its chair for
a period of time. In 1991 he became a member of the Royal Netherlands
Academy of Arts and Sciences (KNAW); already much earlier he became a
member of the Brazilian Academy. At the KNAW he was also very active,
including as chair of the Mathematics Section.
Floris Takens never shirked the demanding tasks
that came his way. An example is the University Teaching Assessment
that took place in 2008 in the Netherlands and Flanders. When the
acting chair Jacques van Lint suddenly passed away, Floris took over
this responsibility.
`De mensch'
Floris had a reputation for being a meticulous man
with a strong sense of duty and consistently high standards. This
applied both to his daily practice of the flute as well as his
precision in all matters of the department or elsewhere. He used to
arrive at work with a Spartan punctuality every morning, whether or
not he had partied long and hard the previous night. I remember
numerous joyful open-air sessions in Rio de Janeiro and in Trieste,
with a view of the Atlantico or of the Adriatico and a table filled
with empty bottles: eat, drink and be merry, meanwhile talking about
life itself.
As I said, Floris never shirked his duties but,
rather, fought hard for his beliefs --- not always effectively. In
fact, his attitude was a bit soldierly, stoically accepting one's
responsibilities and doing one's duty without complaint. In return, he
could not bear tardiness in others. It must be said that Floris was
not always an easy-going person, neither with himself nor with others.
Over the past twenty years I remember Floris to an
increasing degree as a good colleague, and a warm friendship grew
between us. This also had to do with the change in his personal
circumstances, finding himself in calmer waters after his move to the
village of Bedum.
|
At a gate in the middle of
the woods near Bad Bentheim, Germany, in spring 2009; photo by Els
Takens.
|
Culture
For Floris Takens Mathematics was embedded in a
much larger scientific culture, in which Minnaert's De Natuurkunde
van 't Vrije Veld, the Feynman Lectures on Physics, as
well as Gravitation by Misner, Thorne and Wheeler, were never
far from his desk. Aside from this, he was also very interested in
painting and music. He owned a large collection of paintings and
regularly visited museums and exhibitions. He could be seen cycling
around the province of Groningen through all kinds of weather to
listen to musical performances in Leens, Feerwerd, Thesinge or the
Groningen Oosterpoort.
He was also actively involved in performing
music. I cherish precious memories of numerous evenings in Bedum where
Floris and I practiced and performed flute sonatas by Händel and
Bach, now and then relaxing with Mozart's Andante for Flute in C or
Gluck's Dance of the Blessed Spirits. Floris played the traverso and I
accompanied him on a virginal. An integral part of these evenings were
the conversations afterwards, obviously with a good glass of
wine. Apart from small talk and (local) politics, we also got around
to matters of philosophy and theology. Floris had a clear affinity
with the ideas of Spinoza and he was somewhat inclined to a form of
pantheism.
Unfortunately, Floris was witness to the undeniable
decline of the scientific culture in our society over the past 40
years, as demonstrated by the decline in educational standards. Overly
strict attention to market principles in research and education tends
to replace true scientific quality and interest with a mere quest for
funding in the name of science. In a sense, the university itself has
somewhat degenerated into a PhD-thesis factory, often at the expense
of scientific depth. The type of research that is driven by personal
curiosity has too frequently been forced to abandon field. In my
opinion these developments sadly contributed to the fact that Floris
Takens took early retirement at the age of 59.
A fitting anecdote in this regard is Floris's
farewell lecture in 2001, where he discussed a report in the Dutch
newspaper NRC some years previously on the Monty Hall
problem. A series of articles and letters on the subject had been
summarized by a journalist with the words: "Stop, stop, stop
sending letters. The misunderstanding between common sense and the
mathematicians is clearly unbridgeable." Floris observed this
contempt for mathematics in a much broader sense, also within
science. This trend seems to have grown only stronger with time, and
Floris reacted with a resigned nostalgia.
|
|
To end
on a more joyful note, let me say something about Floris's own
independent research, which gave him much pleasure until the end of
his life. It concerned mainly the geometry of torus bundles, as
mentioned before. In the very last months of his life Floris produced
an interesting sketch of a Morse Theory of monodromy and Chern
classes, which will keep a number of us occupied in the near
future.
Finally, I wish to say that a fascinating human
being with unmistakable grandeur has passed away. And to Floris
himself I say: "Old soldier: it was an honor serving with
you!"
|
Floris working in spring 2010;
photo by George Huitema. |
References
[1] |
|
D. Ruelle and F. Takens, "On the nature of turbulence,"
Comm. Math. Phys. 20 (1971), 167–192; 23
(1971), 343–344. |
[2] |
F. Takens, "Detecting strange attractors in turbulence,"
in D. A. Rand & L.-S. Young (Eds), Dynamical systems and
turbulence, Warwick 1980, LNM 898, pp. 366-381,
Springer-Verlag, 1981. |
[3] |
H.W. Broer and F. Takens, Dynamical Systems and Chaos,
Epsilon-Uitgaven 64, 2009; Appl. Math. Sc. 172,
Springer-Verlag, 2010. |
[4] |
H.W. Broer, B. Hasselblatt and F. Takens (Eds), Handbook of
Dynamical Systems Vol. 3, North-Holland, 2010. |