In Memoriam for Floris Takens, 12 November 1940 – 20 June 2010

By Henk Broer
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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.
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