RESEARCH INTERESTS âžž DETAILS
Disclaimer:
When you read the following pages, you may find yourself thinking: “Wait a minute—he did all of this?” The answer is both yes and no. The very first steps were entirely my own: my diploma thesis, my doctoral thesis, and my habilitation thesis, as it should be. After that, however, I gradually moved into a different way of working—one based on collaboration: with my doctoral students; with many colleagues in mathematics, mathematics education, and medicine around the world; with teams of teachers in Florida and Bremen; and with my fantastic research staff at CeVis, MeVis Research eGmbH, and Fraunhofer MEVIS.
Much of what I have tried to do falls under the heading of digital transformation, and it seems characteristic of this age of change that innovations rarely come from single individuals, the “lone wolves” of the past. Instead, they emerge from people who enjoy working with others whose talents and abilities complement their own. That is what happened in my case, what I wanted, what I helped to create, and what I was passionate about. I like to believe that the future lies in wiki-like collaboration—small wikis, large wikis, worldwide wikis. I am not referring primarily to the wiki software itself, but to the attitude and spirit of collaboration that underpins wiki culture. The true power of the digital age lies in unleashing the power of such wikis. I like to think of myself as one small example of a local wiki enabler.
Mathematics:
The Beginnings in Bonn
My initial work in mathematics around 1970–71 was in algebraic topology. I succeeded in computing the cohomology ring for simplicial pairs on a computer—something that had never been done before. This involved a twofold challenge. First, I had to go back to the historical beginnings of algebraic topology (for example, the legendary book by Seifert and Threlfall from 1934) in order to develop algorithms that could actually be implemented on a computer. The second challenge was the implementation itself.
The computer available at the University of Bonn at that time was an IBM 7090, considered one of the most powerful machines in Germany. Yet, looking back from the standpoint of 2015, it had a tiny memory of only 128 KByte and a performance of some 100 KFLOPS. It occupied an area as large as a single-family home. Compared to an iPhone 6—with at least 32 GB of memory (around 250,000 times more) and a power of about 100 GFLOPS (roughly 1,000,000 times faster)—it was hardly a computer at all.
During the computations, very large matrices had to be generated and processed, even though they did not fit into memory. In fact, they did not even fit onto the magnetic tape storage that extended the computer’s memory. Today, computational algebraic topology is flourishing, supported by abundant computing power, and most users have no idea what it meant to compute the cohomology ring of a “decent” manifold in 1970.
I eventually left this area and moved into a combination of algebraic topology and nonlinear functional analysis for my dissertation in 1973. Christian Fenske and Friedrich Hirzebruch were the reviewers. The work focused on topological fixed-point index theory, a generalization of the Brouwer degree, and on Lefschetz fixed-point theory. In particular, unstable fixed points in a purely topological setting—without any applications in mind—were of special interest to me.
From there, influenced by a one-month summer school in Montreal, I became interested in what is known as fixed-point theory with applications to functional differential equations. This became the topic of my Habilitation thesis in 1976. My work in this area was greatly influenced by Andrzej Granas and Roger Nussbaum. Much later, Roger and I discovered spurious solutions in the numerical approximation of functional differential equations (see below). Through my friendship with Roger, I soon learned enough about functional differential equations to begin translating my abstract results on unstable fixed points into a framework that could at least be applied to functional differential equations. This led to a series of papers with the late Gilles Fournier on fixed-point theory in cones.
Eugene L. Allgower, a mathematician at Colorado State University, drew my attention to Emanuel Sperner and his famous 1929 result, which, when interpreted appropriately, provides a constructive proof of the celebrated Brouwer fixed-point theorem. Brouwer’s theorem has an infinite-dimensional generalization in the Schauder fixed-point theorem, which in turn is one of the key ingredients in many existence proofs for solutions of ordinary and partial differential equations.
The idea was therefore tempting: if solutions of differential equations exist by virtue of Schauder’s theorem, then their numerical approximations live in finite dimensions, where one might use something like Sperner’s lemma to obtain a numerical approach to finding solutions. I was fascinated by this beyond belief, and my first doctoral student, Michael Prüfer, while I was still in Bonn, developed this idea and pushed it through.
We then turned to another famous result on differential equations: Paul Rabinowitz’s global bifurcation theorem, which guarantees the existence of topological continua under certain conditions. The key ingredient in Paul’s proof is the homotopy property of the Leray–Schauder degree. We looked for a constructive, finite-dimensional version of this and succeeded in providing an algorithmic and computational form of Rabinowitz’s celebrated theorem. This led us into the domain of simplicial continuation methods, which Dietmar Saupe later used for the numerical computation of periodic solutions of functional differential equations.
Ultimately, in this area we were able to provide a completely constructive—that is, algorithmic and computational—framework for the Brouwer degree. This was a highly unconventional approach to topological degree theory and, in a sense, brought me back to the spirit of computational algebraic topology that I had pioneered in 1970.
In 1978 I met Emanuel Sperner himself, when I was invited to give the keynote lecture in honor of the 50th anniversary of his PhD at the University of Hamburg. Remarkably, he had obtained his doctorate at the age of 23.
From Bonn to Bremen
In 1977 I accepted a full professorship at the University of Bremen, which at the time was still trying to find its identity as a newly founded “reform university.” The early years in Bremen were a huge challenge compared to the thriving environment in Bonn. Bonn was then one of the world centers of mathematics, with generous funding and a unique atmosphere for ambitious young researchers. To illustrate this: in 1974, just one year after my dissertation, I had access to funding for several visiting professors for several months and was able to organize an international conference during the summer with a substantial number of invited speakers. Among others, I was able to bring my new mathematical friends from Italy—Massimo Furi, Mario Martelli, and Alfonso Vignoli—and from the US, Roger Nussbaum, to Bonn. I had met all of them in Montreal a year earlier.
Up until my departure to Bremen, I had access to funds that seemed never to dry up, enabling me to invite many colleagues to Bonn as visiting professors and to organize another international conference on Functional Differential Equations and Approximation of Fixed Points, which took place in 1978, while I was already in Bremen. The funding situation in Bonn was so excellent because of two multi-million DFG Sonderforschungsbereiche, one in pure mathematics and one in applied mathematics. Even though I was quite young for such a role, I was involved in writing the proposal for the latter and served on its steering committee. Even if one assumes that I may have looked promising to the leading mathematicians in Bonn, this was still remarkable—especially in the 1970s, when typical German universities were still rather petrified in hierarchical thinking and structures. Friedrich Hirzebruch had somehow infused a very different attitude and atmosphere, and in my immediate mathematical neighborhood, Eberhard Schock and Heinz Unger gave us every freedom and support one could wish for. This had a deep impact on me, and I later cultivated the same attitude in my own research centers, first at CeVis and then at MeVis.
In sharp contrast, I vividly remember an incident after my Habilitation. At that time, the rule in Germany was that you had to leave the nest, so I applied for professorships elsewhere. An offer came from the University of Würzburg in Bavaria. After the colloquium lecture, which served as my formal introduction, I was invited to dinner with the department, as was customary. There, full professors sat at one table, associate professors at another, and the lower ranks at yet another table.
I was a liberal, and I wanted to be among people who disliked rigid hierarchy—and Bremen was, to put it mildly, a very outspoken liberal place at that time. However, Bremen had practically no funding or endowment for professors and drew a lot of criticism for its immature reform projects. Still, I wanted to be there because I believed that the old German university system needed to be replaced, and I wanted to be part of that transformation. Today, the University of Bremen has an excellent reputation, and I consider myself fortunate to have had the opportunity to actively participate in this transformation more than 30 years.
The early years in Bremen, however, were as hard as it gets: no resources, no reputation, and—so many believed, though not we—no future. It still fills me with joy and satisfaction that we proved the mainstream opinion, according to which the University of Bremen would be a lost cause, to be premature and very wrong. My strategy was simple: look for bright students and work with them in the Bonn style. And I did indeed find excellent students in Bremen. Perhaps Bonn had also endowed me with an inexhaustible supply of self-confidence.
It did not take long—only a few years—before we acquired funding from the Stiftung Volkswagenwerk and then, I believe as the first group in Bremen, we obtained substantial support from the Deutsche Forschungsgemeinschaft. Here my Bonn experience in writing successful proposals proved extremely valuable. Together with the late Peter Richter, Diederich Hinrichsen, and Ludwig Arnold, we founded the Institute for Dynamical Systems (IDS), which became an unusually fertile breeding ground for young mathematicians, many of whom later achieved international recognition, such as Dietmar Salamon and Dieter Praetzel-Wolters.
The IDS then became the launch pad for my own trajectory: first CeVis, and CeVis then became the launch pad for MeVis, which in turn led to Fraunhofer MEVIS and MeVis Medical Solutions AG. Around 2015, these two institutions together employed about 300 researchers and software specialists.
Sabbaticals in Salt Lake City
In 1976, while still in Bonn, I attended a NSF-CBMS Regional Meeting in Boulder Colorado on Isolated Invariant Sets and the Morse Index. It was there that I met Klaus Schmitt from the University of Utah, who inspired my interest more and more for differential equations. The conference featured Charles C. Conley's ground breaking work. Several years later Charly would visit Bremen a few times and we had unforgettable summer workshops on the tiny island of Helgoland.
Two sabbaticals, in 1979–80 and 1982–83, changed my mathematical path dramatically. Both were at the University of Utah in Salt Lake City, hosted by my friend Klaus Schmitt. The University of Utah had—and still has—something that not many universities can claim: a remarkably strong mathematics department. At the time of my visits, it also had something absolutely unique: it was the birthplace and home of computer graphics. As Robert Rivlin writes in his history of the field, “Almost every influential person in the modern computer-graphics community either passed through the University of Utah or came into contact with it in some way.”
Beginning in 1979, Klaus and I developed a very fruitful collaboration that lasted many years, combining his expertise in ordinary and partial differential equations with my background in nonlinear functional analysis. We also began to rely on numerical and computer-graphical experiments on the legendary Evans & Sutherland PS/2 for our work on global bifurcation phenomena in nonlinear elliptic eigenvalue problems.
The legendary Evans & Sutherland PS/2 vector graphics system. Dials (above key board) were used to manipulate parameters of the graphics.
The PS/2 belonged exclusively to Robert Barnhill, a pioneer in computer-aided geometric design, and he generously allowed us to use his jewel of a machine. Remember, in 1979 computer graphics was virtually nonexistent in mathematics and computer science departments around the world. In fact, I am not aware of any other mathematics department that had anything comparable at that time.
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Spurious Solutions and the Transition into Dynamical Systems
Being able to visualize our numerical experiments, which were meant to guide our new theory, turned out to be crucial for a remarkable discovery. We found that the numerical solutions fell into two classes: some genuinely approximated the solutions of the partial differential equations, while others, quite mysteriously, did not. These additional solutions were artifacts of the numerical discretization. We knew that whenever the partial differential equations had solutions, they had to exhibit certain symmetries, and these strange solutions violated those symmetries. That is how we first spotted them—on the screen of the PS/2.
We called them spurious solutions and continued developing our theory; at that time we were not yet working in numerical analysis. In fact, Klaus and I did not run the experiments ourselves—we only designed and supervised them. The simulations were carried out by Dietmar Saupe, who had joined me from Bremen as a graduate student. We could not find any mention or explanation of such spurious solutions anywhere in the numerical analysis literature. For many years we tried to understand what was happening.
Eventually, drawing on my interest in chaos theory—and in particular the Poincaré–Birkhoff theory of chaos in Hamiltonian systems caused by homoclinic structures—we found a complete answer by embedding the numerical problem into the framework of dynamical systems. The emergence of spurious solutions turned out to be a rather peculiar homoclinic bifurcation in the associated dynamical system (A and B). We later discovered similar phenomena in functional differential equations with Roger Nussbaum.
When I lectured on this topic at various numerical analysis conferences, I expected great interest—but that did not happen. Instead, I heard remarks such as: “A good numerical analyst would recognize spurious solutions and does not need a theory for them.” Many years later, presenting our work on liver surgery planning, I encountered similar reactions from some surgeons: “A good surgeon doesn’t need the kind of risk analysis you provide; he can rely on his experience.” This attitude changed dramatically only after one of the greatest liver surgeons of all time, Koichi Tanaka of Kyoto University, adopted our methods and made them the standard in liver surgery.
Back to spurious solutions: after my talks, there was never a single question about our use of homoclinic structures. In retrospect, I should have known why. Homoclinic structures were like Chinese to an audience of numerical analysts. Even at the University of Bonn, where we grew up as young mathematicians and were confident that we knew—or at least had heard of—the most important areas of mathematics, we never really encountered the great ideas and achievements of Poincaré and Birkhoff, nor dynamical systems as a field. There was neither a basic nor an advanced course on the subject. What Bonn did teach us, however, was to value cross-disciplinary insights and to build such connections; this was very much what Hirzebruch stood for. Numerical analysis as a discipline, by contrast, often seems surrounded by fortress-like fences, with little tradition of looking beyond its boundaries or connecting to other areas.
When we returned to Bremen from Salt Lake City in the spring of 1980, we managed to secure access to a Tektronix 4014 storage-tube graphics terminal at the university’s computer center and continued our experiments on homoclinic structures, using clever algorithms developed by Hartmut Jürgens. One day I observed a homoclinic bifurcation on the screen, and I will never forget the sense of satisfaction I felt. The full understanding of the mechanism behind the spurious solutions was still to come (A and B), but our path had changed. From then on, mathematical experiments became increasingly important to us—even though we were still far from having our own equipment.
Some of the images we generated on the Tektronix 4014 I found aesthetically very appealing. During a vacation in Hawaii, I began to paint some of them in watercolor—homoclinic structures and chaos in an area-preserving diffeomorphism (December 1982).
Water colored phase portrait of area perserving diffeomorphism with homoclinic structures
Encounter with Fractals and Experiments
The second sabbatical in Salt Lake City, in 1982–83, influenced my path even more profoundly. Through Klaus Schmitt I was introduced to Richard Riesenfeld, head of the Computer Graphics Lab in the Department of Computer Science. Richard kindly gave us permission to use his equipment whenever we were not in the way of his group—typically in the evenings and on weekends. There, Dietmar Saupe and I became acquainted with early pixel-based computer graphics, which was completely different from the vector-based graphics of the PS/2 and would, of course, later become the standard.
Then came December 6, 1982. Leo Kadanoff, the famous physicist from the University of Chicago, came to give a lecture. We expected him to speak about phase transitions, since his student Ken Wilson had just received the Nobel Prize in Physics. But he didn’t. Instead, he spoke about chaos in a rather popular, accessible way. We arranged to meet the next morning in my office and at the blackboard we exchanged ideas. I told him about my work related to Poincaré–Birkhoff, and he described something that I found extremely puzzling.
He talked about the dynamics of
in the complex plane, where c is a small constant, and he drew weird chalk pictures on the blackboard and explained quasi-circles. I must confess that I did not understand a thing. But I was intrigued, and so I twisted Dietmar Saupe’s arm and suggested that we run some experiments in Richard’s lab that afternoon. That evening, we saw a stunning colored image of a Julia set—probably one of the very first of its kind. The beauty of that picture was breathtaking at the time, and we simply could not stop. Very soon we had a whole collection of images.
When we showed them to Frank Hoppenstedt, a mathematician in the department, he said, “Go and look at that book by a guy named Mandelbrot in the library—I just saw it there.” Indeed, Benoit Mandelbrot’s "The Fractal Geometry of Nature" had just appeared, and in it we found everything about Julia sets and, of course, the Mandelbrot set. But compared to his graphics, we had one crucial advantage: we had access to the best computer graphics equipment of the time.
Before long, we were spending most evenings and weekends—over Christmas and New Year—running experiments until we returned to Germany in the spring of 1983. Dietmar did not get nearly as much use out of his newly purchased skiing equipment as he had hoped. When we flew back to Bremen, we carried almost a whole suitcase filled with 35mm slides. In those days, that was how you captured images from a computer screen.
Back in Bremen we had only one goal: how could we get similar equipment? Peter Richter, with whom I was leading a research group, was immediately fascinated as well. With funding from the Stiftung Volkswagenwerk and the Deutsche Forschungsgemeinschaft, we eventually acquired very competitive equipment: an AED 767 raster graphics terminal and the Rolls-Royce of vector graphics, the legendary Evans & Sutherland PS 300. Initially, the DFG was not convinced that mathematicians and physicists needed such top-of-the-line hardware. But we persisted—and prevailed. That is how our own computer graphics lab came into existence, ushering in some golden years of experimentation. We all became more and more deeply involved in fractal geometry and computer graphics, and we produced scientific films with the Göttingen Institut für den Wissenschaftlichen Film (IWF), which went on to receive several international awards: Fly Lorenz, 1985 (https://av.tib.eu/media/59647), Interview with Benoit B. Mandelbrot, 1990 (https://av.tib.eu/media/11636).
Towards the Goethe Exhibits
Quite unexpectedly, this was followed by a public exhibition of framed prints selected from our Salt Lake City slides, accompanied by a catalogue, in the Bremer Sparkasse — a bank in Bremen with a beautiful Art Nouveau main hall.
This exhibition, held in early 1984, attracted an unexpectedly strong response from the public and the media—and, in particular, from Manfred Eigen, Nobel Laureate in Chemistry (1967) in Göttingen, and Friedrich Hirzebruch in Bonn, both of whom headed Max Planck Institutes they had founded. Soon we were invited to mount follow-up exhibitions in Bonn and Göttingen.
Then Arnold Mandell,, a professor of psychiatry from UCSD, gave a crucial impulse. I knew him from interdisciplinary conferences on dynamical systems, and when he saw our images, he insisted that we should aim for an exhibition in the United States. He knew the director of the Museum of Modern Art in San Francisco and introduced us. I visited, and the director in turn connected me with the late Frank Oppenheimer, then director of the San Francisco Exploratorium.
When Frank saw the pictures, he was immediately enthusiastic: “We must show them here. We must have them—but I don’t have funding.” He introduced me to the director of the San Francisco Goethe-Institut, whom I visited with Frank that same evening. The Goethe-Institut director also had no money, but he did have connections. He put me in touch with the head of the Goethe-Institut organisation in Munich, who was interested, but unsure whether he could get the idea approved by his committee of art historians and others responsible for selecting exhibitions for worldwide tours to more than 150 Goethe-Institutes.
Manfred Eigen and HOP in Göttingen
Naturally, the question arose: “Is this art?” Our answer was: “No, not at all. It is pure mathematics.” The art historians had never seen anything like it. We needed high-level support. Manfred Eigen wrote a letter of recommendation that the committee simply could not ignore—and we were in.
The opening of the first exhibition was scheduled for 1985, and we began spending late evenings and weekends redoing many of our Salt Lake City experiments in higher image quality, and conducting many new experiments, especially around the Mandelbrot set. The title of the exhibition was chosen as “Schönheit der Fraktale – Bilder aus der Theorie Komplexer Dynamischer Systeme” (“The Beauty of Fractals – Images from the Theory of Complex Dynamical Systems”).
Catalogue for Goethe-Institute Exhibits
GEO Magazin June 1984
While we were frantically working on our experiments, the University of Bremen’s press officer, Klaus Sondergeld, had the idea of presenting our images to GEO, a very popular German monthly magazine, then renowned for its outstanding photography on geological and natural themes. We were invited to show our work to the editor-in-chief, Rolf Winter,. He did not even consult anyone else before deciding, on the spot, not only to publish our mathematical images but to feature them in a major cover story. On top of that, he offered 15,000 Deutsche Mark, which gave our small company, MAPART GbR—founded to market our pictures and copyrights—a considerable boost. From then on, we used the proceeds from MAPART to further improve our lab equipment.
The Goethe-Institut exhibition began its world tour in mid-1985. We had prepared two complete sets of around one hundred beautifully framed Cibachrome prints. The first showings took place at the Museum of Modern Art in Oxford, sponsored by Oxford University and, in particular, Sir Michael Atiyah, a close friend of Hirzebruch. Almost simultaneously, the second showing opened at the San Francisco Exploratorium, sponsored by Stanford University and especially by the late Robert Ossermann. Sadly, Frank Oppenheimer—who had been instrumental in winning over the Goethe-Institut—had passed away just a few months before the opening in San Francisco. Later, the two copies of the exhibition toured all continents and far more than one hundred cities.
Before the exhibition left our lab, we wanted to show it in Bremen. Since the Goethe-Institut is mostly active outside Germany, the only way to present our work at home was to organize a show ourselves. And so we rented exhibition space in Bremen’s Böttcherstrasse, dedicated primarily to the German painter Paula Modersohn-Becker. Our exhibition ran there from May 11 to June 18, 1985.
The Books and Mandelbrot
Two major paths emerged from the Goethe-Institut exhibitions and the farewell show in Bremen. One day, while we were preparing for the vernissage, Peter Richter said, “Wouldn’t it be wonderful if Mandelbrot came for the opening?” We dug out his phone number, and I simply dared to call him and propose the idea. To our surprise, he agreed. That is how we first met him in person—and soon a lifelong friendship evolved, leading to many new adventures. Mandelbrot opened my eyes to worlds that had previously been unknown to me, and my rekindled love of art and music was, in many ways, an indirect consequence of my encounters with him and his work, and of what his work set in motion within me.
Mandelbrot himself had begun his own journey by studying the rather obscure Zipf’s law, which describes the frequency distribution of words in a language. This curiosity helped galvanize his later work on fractals. In a similar way, crossing his path galvanized my later work in medicine—or, put differently, prepared me for it. Incidentally, there is a wonderful video about Zipf’s law by Michael Stevens, The The Zipf Mystery which I consider a must-see. Zipf’s law is foundational for LLMs because it describes the extremely skewed frequency distribution of words and tokens in natural language, which shapes how models are tokenized, trained, optimized, and ultimately why they are very strong on common patterns but less reliable in the long tail of rare ones.
While we were preparing the images for the exhibition, the irresistible idea arose to produce an exhibition catalogue. So we began working on that in parallel and eventually completed it, including a generous number of color plates. Color printing was still quite expensive at the time, but we were fortunate to have the financial backing of MAPART.
One of the guests at the Bremen vernissage was Joachim Heinze,, then the mathematics editor at Springer-Verlag in Heidelberg. He saw the catalogue and immediately had the idea of turning the material into a mathematics book. During a several-hour walk along the river Wümme, we hammered out an outline for the mathematical text that would be written around the images from the catalogue.
Joachim had to present the project to his advisory board, composed of leading mathematicians. They were skeptical; in the 1980s, lavish imagery was still frowned upon by the Bourbaki-style establishment that largely dominated the mathematical community. But with the backing of his chief, the late Heinz Götze, he managed to overrule the board—and so the book The Beauty of Fractals was conceived. It appeared one year later, in August 1986—a record time for writing and producing a book.
When Springer’s production planners were confronted with the sheer volume of color images, they were at a loss. The printing costs would have forced an extremely high retail price, and the book would likely have been unsellable. The solution was that Peter Richter and I agreed to subsidize the color printing, so that the bookstore price could be set at around 70 dollars (with a contract ensuring that we would be reimbursed if the book turned out to be successful). And successful it was: it became Springer-Verlag’s all-time bestselling mathematics book and was later translated into Russian, Italian, Chinese, and Japanese.
My co-author for the unexpectedly successful The Beauty of Fractals was Peter Richter, Professor of Physics at the University of Bremen, with whom I had delved into chaos theory in the early 1980s. The main theme of the book revolved around fractal geometry, but we also explored key concepts in chaos theory. Around 1985, Peter and I managed to establish our own laboratory for computer graphics, which we used extensively to conduct experiments in mathematics and physics.
Our equipment was inspired by what I had seen and worked with during my visits to the University of Utah — the birthplace of modern computer graphics. At that time, computer graphics were undergoing a major transition from vector graphics to raster (pixel-based) graphics. The Rolls Royce of vector systems was the Evans & Sutherland PS300, a machine costing nearly half a million Deutsche Mark, which we managed to acquire through a grant from the Deutsche Forschungsgemeinschaft (DFG). This was no small achievement, as reviewers initially argued that mathematicians would have little use for such an extravagant piece of technology.
Evans & Sutherland, founded by David Evans and Ivan Sutherland, played a foundational role in the birth of computer graphics — their work laid the technical and conceptual groundwork for everything from flight simulators to 3D rendering systems, and many of their students went on to establish what became the core of today’s graphics industry.
Today, vector graphics are long obsolete, and the computing and graphics power of an ordinary smartphone surpasses that of these once-revolutionary systems by an almost unimaginable factor. One of the experiments was about key features of the famous Lorenz attractor, which we carried out on our magnificent PS300. In fact the experiments culminated in the film Fly Lorenz, which we produced together with the Institut für den Wissenschaftlichen Film in Göttingen. At that time, the only way to capture images from the graphics screen was by literally pointing a 16 mm film camera at it. Here is the film, later transcribed into video.
In 1992 we published Chaos and Fractals: New Frontiers of Science with Springer-Verlag New York, co-authored by Hartmut Jürgens and Dietmar Saupe with a beautiful Foreword by the late Mitchell Feigenbaum. Paul Manning at Springer was largely responsible for turning it into a major bestseller in the United States. A second edition appeared in 2004 with one significant change: the first edition had included two invited appendices—one on fractal image compression by Yuval Fisher, and one on multifractals by Carl Evertsz and Benoit Mandelbrot—which were not retained. A Chinese translation followed in 2008.
The idea of the book was to present chaos and fractals as comprehensively and as accurately as possible, without relying on the usual high-level mathematics. One way to achieve this was through extensive use of illustrations, experiments, and tables; the second edition still contained more than 600 illustrations. Another was to effectively write two books in one binding: a running text in a popular-science style set in a serif font, and more technical passages inserted into the main text in a sans-serif font.
As with The Beauty of Fractals and its large number of color plates, the publisher was concerned about the high cost of typesetting and layout for such an illustration-rich volume. The solution was that we—more precisely, Dietmar Saupe—took on the typesetting and complete layout of the book ourselves. It was quite an undertaking, which Dietmar mastered with great skill, pushing LaTeX to its limits.
There is an amusing anecdote connected with Chaos and Fractals. Barnes & Noble planned a special Christmas sales campaign in 1993 and wanted 1,500 copies of the book hand-signed by me. Paul Hrdy, then head of marketing and sales at Springer, flew me in from Germany, and I spent an entire day in Springer’s warehouse in New Jersey signing books. Every so often I had to switch from ballpoint to a softer pen and take a break because I could hardly write my own signature anymore. Over the course of the day my signature deteriorated from clearly legible to much less readable and significantly shorter—and it has essentially remained that way ever since.
The following spring semester I taught a course on chaos and fractals at Florida Atlantic University based on the book. Students were not required to buy it, but it certainly helped. One student came to me and said: “I bought your book at the Barnes & Noble store in the University Commons, and I got a great price. Guess why? They had several copies, and I found one with your personal signature in it. I went to the cashier and asked for a big discount because someone had scribbled something in the book.”
Chaos and Fractals even made two cameo appearances in popular sitcoms: Murphy Brown and Seinfeld. In Season 8, Episode 9 of Seinfeld, “smart George” is seen carrying the textbook Chaos and Fractals: New Frontiers of Science in his scenes at the diner. In Murphy Brown, Candice Bergen’s character tests a date using her newly acquired knowledge about fractals.
A few years later, Paul Hrdy gave me a wonderful gift. Springer had decided to give up its own warehouse and outsource book order fulfillment, following the advice of one of the big multinational consulting firms for process optimization (a step they later regretted and eventually reversed). As a result, they had to reduce their inventory drastically, and the books sorted out were destined for the trash. I was allowed to spend a day in the warehouse and choose what I wanted. I ended up selecting two huge piles of several hundred books: one for the mathematics department at FAU and one for my institute, CeVis, in Bremen. The latter collection later became a gift to Jacobs University.
Mathematics in the Later Years
Mathematically, my students and I had moved from partial and functional differential equations and their numerical analysis to chaos and fractals. We studied Newton’s method in the complex plane and in two real dimensions from a fractal perspective, explored connections between the theory of phase transitions in a renormalization setting and Julia sets, and much more.
The second half of the 1980s was the period in which we began to accept mathematical experiments as a way of life. The mathematical establishment was not quite sure what to make of this. A small anecdote may illustrate the point. Toward the end of the 1980s—around the time I received (and declined) an offer for a chair at the University of Bonn—Hirzebruch invited me to organize a semester on dynamical systems at his Max Planck Institute for Mathematics. In the middle of that semester, he asked me to prepare a proposal for modernizing the Institute’s computer equipment. At the time, the Institute had almost nothing beyond a few PCs; the most powerful machine was an HP desktop computer that was barely stronger than the PCs themselves.
I proposed a lab with up-to-date SUN or SGI workstations. Hirzebruch asked me to draw up a detailed plan with cost estimates suitable for submission to the Max Planck Society, which I did. The proposal, however, was never submitted. One of the members, Don Zagier—the Institute’s young star at the time and a heavy user of the painfully slow HP for his work in number theory—objected. His argument was: “A mathematician should use a rather powerless computer. Otherwise there is the temptation to compute too much and too fast without thinking.” I am sure he meant a good mathematician. In other words, powerful computers were, in his view, for the less gifted.
Today, of course, computers and computational experiments have become entirely natural and are integrated into mathematics almost everywhere, and pictures are accepted—although some mathematicians still dismiss them as Bildchen, as if they feel a need to apologize for them.
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Later, in the early 1990s, my mathematical interests expanded to include number-theoretic problems involving automatics sequences, finite and cellular automata—some of which have applications in theoretical computer science—as well as Iterated Function Systems (IFS). Here is a nice popular site introducing IFS.
Guentcho Skordev and my student Fritz von Haeseler at CeVis were instrumental in these investigations, which began with a seemingly innocent question from fractal geometry and elementary number theory: if one looks at binomial coefficients modulo 2—replacing even coefficients by 0 and odd coefficients by 1—a pattern emerges that resembles the SierpiÅ„ski triangle. What happens modulo p, where p is a prime, a prime power, or a composite number, turned out to be increasingly challenging problems. We eventually found that a combination of renormalization techniques with finite automata and hierarchical iterated function systems provided the answers. A frequent guest from France, Jean-Paul Allouche, further galvanized our interest.
I had first met Guentcho when both of us were still working on fixed point theory in algebraic topology; when he later came to Bremen, we resumed these investigations after I had not looked at such problems for more than a decade.
When Carl Evertsz arrived at CeVis around 1991, after his postdoctoral years with Mandelbrot, our focus expanded into the mathematics of finance, following Mandelbrot’s critique of the traditional efficient market theory. Peter Singer, an expert on martingales, joined us from the University of Erlangen, and after I had secured funding from the Deutsche Investment Trust (then a subsidiary of Dresdner Bank), we engaged intensively in freeing the Black–Scholes formalism from the hypothesis of efficient markets, using extensive computer experiments. The key idea was to build a new framework in which Einstein’s square-root law would be replaced by Hurst exponents. It seemed this would shape our mathematical direction for quite some time.
But it did not. An unexpected dinner-table conversation with Klaus J. Klose, then Head of Radiology at the University Hospital in Marburg, catapulted me into an entirely new world—like a ball hitting a bumper in a pinball machine: the world of radiology and surgery. I would never have predicted that.
Computer Science:
Enough of Computers
I had my full dose of using computers in mathematics around 1970, when I worked with them in algebraic topology. I remember that time as both wonderful and stressful. I had nobody to talk to about what I was doing; the algebraic topology community was not yet interested in computers. There were enormous technical obstacles and severe computer limitations to overcome. It was far from clear whether what I was attempting—and eventually succeeded in doing—was even possible: computing the cohomology ring of simplicial pairs.
On the IBM 7090, the language of choice was FORTRAN, with some ASSEMBLER and very little in the way of system tools. Everything lived on punched cards. When the work was finally done and published, I felt a strong desire to turn back to pure mathematics and leave computers behind. I had had enough.
It is difficult today to convey what using computers meant at that time: standing in line for a punch-card machine (a sort of special typewriter that punched holes into cards), sorting the cards into a box, and delivering that box to a shelf in the computer center, where operators would pick it up and feed it into the computer. Later, you would collect the paper printout from your personal shelf, stare at the typically unexpected results (something was wrong—but what?), and start all over again.
We never knew when our job would run. It depended on the overall workload and the resources our program required. The computer operated 24/7. As a result, we often lingered near the machine in cramped spaces, smoking or eating cheap food, waiting and checking, waiting and checking, long hours into the evening or through the night. There was no email, no text messages, no notification system—nothing of the sort. What took us a week then could probably be done in an hour today. There were no modern parsers, and tracking down errors was a painful process.
Still, we did not know any better and thought of ourselves as belonging to a very exclusive club. Only two years later, the IBM 7090 was replaced by an IBM 360 with terminals for input and communication—a major revolution that, unfortunately, came too late for my project. Had I continued my work on making algebraic topology computational, my life would have become much easier. But, as I said, I had had enough of computers for a while, and I missed out on UNIX, C, and all the great developments that followed.
Salt Lake City - Salt Lake City
Before the two sabbaticals in Salt Lake City, my group had been working intensively on simplicial continuation methods, but my own contact with computers was, at best, rather loose. This changed during the first sabbatical in 1979–80, when Robert Barnhill allowed my students and me to use his legendary Evans & Sutherland PS/2. For the first time, our interest in computer graphics hardware and software was truly awakened.
During the second sabbatical in 1982–83, Richard Riesenfeld gave us access to all the equipment in his absolutely fantastic, world-class laboratory in the Computer Science Department. As a by-product, we learned a great deal about his groundbreaking work in computer graphics and became thoroughly inflamed with enthusiasm. Richard connected us with SIGGRAPH (the Special Interest Group on Computer Graphics) and presented our work on fractals (Julia sets) at the 1983 SIGGRAPH annual meeting in Detroit. His lab—and the way he ran it—was a major inspiration for me when I later set out to build a similar lab in Bremen.
Back in Bremen the development of our own lab, led by my student Hartmut Jürgens, went through several stages, all inspired by the computer graphics role models we had seen in Salt Lake City:
1981: NORD 100 Computer
Tektronix 4014 (loan from the University Computing Center)
1983: Evans & Sutherland PS300
Matrix Camerasystem for slides and film
AED 767 Rastergraphics
1986: Bull SPS9/60 Computer
Raster Technology One/360
1988: Management Graphics Solitaire
Sony U-Matic editing recorder
1989: Silicon Graphics Iris
1993: Silicon Graphics Onyx
1996: Silicon Graphics O2
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During the years from 1981 to 1988, our lab was probably the only computer graphics laboratory in the hands of mathematicians and physicists. Peter Richter and his group, as well as my own, produced several films with the Göttingen Institut für den Wissenschaftlichen Film (IWF). We also produced two films for Spectrum-Verlag in Heidelberg, which was responsible for the German edition of Scientific American.
The second film, titled Fractals: An Animated Discussion, was distributed in 1990 by W.H. Freeman as a VHS video. It featured two interviews I conducted with Ed Lorenz and Benoit Mandelbrot, along with computer-graphic experiments involving the Mandelbrot set, Julia sets, and the Lorenz attractor. In one of the scenes, we continuously zoom into a particular region of the Mandelbrot set that we named “Seahorse Valley.” This was inspired by the legendary film Powers of Ten by the late Charles Eames. I had met Charles Eames through my friend, the late Raymond Redheffer from UCLA, who was the mathematical advisor for Powers of Ten, and that film was always in the back of my mind while we were creating our own.
There is another scene in this video that grew out of our desire to show how chaos and fractals are intertwined. The experiment we designed turned out to be one of the nicest ways to explain what chaos actually is and how it relates to fractals. We used a simple pendulum—an iron ball swinging freely—in the presence of three magnets. We then placed the iron ball at various starting positions and observed over which magnet it would eventually come to rest. The entire computer experiment is based on a sophisticated mathematical model.
1985
​In 1985, two major events occurred that boosted our visibility far beyond anything we had expected. In August, Scientific American decided to publish a simplified version of our computer code for generating images of the Mandelbrot set and chose one of our images for the front cover. This triggered a worldwide “fractal mania.”
The Mandelbrot set is probably the one mathematical object on which the cumulative global computer time devoted to visualization exceeds that of anything else. The internet is still saturated with images and videos of it. Yet only a few have truly captured a sense of discovery or beauty. Two more recent examples stand out for me:
– The almost meditative video Hardest Mandelbrot Set Zoom: 1 Quadrillion Iterations.
– The iPad app Frax HD by Kai Krause and collaborators.
The second event was the SIGGRAPH annual meeting in July 1985, held in San Francisco. Benoit Mandelbrot had been invited to give a course on fractals, and Dietmar Saupe and I were invited by him to present our work as part of it. This was also where we first met Richard Voss. SIGGRAPH itself was an experience beyond words. Around 40,000 participants (not in our course alone, of course) celebrated the emerging power of computer graphics in scientific visualization and Disney-style animation. We all felt as if we were witnessing history.
The following year, we taught a course at the SIGGRAPH annual meeting in Dallas, and in 1987 I was invited to organize the course, for which I asked, among others, D. Saupe, Robert Devaney, Richard Voss, and Michael Barnsley to join. From the outset, I intended to publish the course notes, and we did. The book The Science of Fractal Images, co-edited with Dietmar Saupe, appeared in 1988 with Springer-Verlag New York and became a remarkable success, thanks in part to the efforts of Rüdiger Gebauer, Springer’s mathematics editor. In 1989, it received an award from the Society for Technical Communication. This book brought fractals into the computer science community, and especially into computer graphics.
In summary, our core team—Dietmar Saupe, Richard Voss, and I, often joined by Benoit Mandelbrot—gave a total of ten invited SIGGRAPH courses between 1985 and 1994, at annual meetings in San Francisco, Dallas, Anaheim, Atlanta, Boston, Dallas again, Las Vegas, Chicago, Anaheim, and Orlando.
The cover shows an image that combines a rendering of the true electrostatic potential of the Mandelbrot set near its boundary—shown in “fantasy” colors that resemble a snowy landscape—generated by Hartmut Jürgens, together with fractal clouds based on fractional Brownian motion, generated by Dietmar Saupe. It is one of my all-time favorite images, because it brings together the worlds of mathematical fractals and random fractals.
Incidentally, the region of the Mandelbrot set shown on the book’s cover is the same as on the Scientific American cover: in our book we see the electrostatic potential rendered as a mountain ridge with steep cliffs (indeed, the potential falls off exponentially), while Scientific American used a rendering of the same potential with selected equipotential curves highlighted. Note that, to compare the two images properly, one must perform a mirror flip. The header image on my website is yet another beautiful rendering of the Mandelbrot set by Hartmut Jürgens, again showing a 3D representation of its electrostatic potential. A Japanese translation of the book appeared in 1989.
That is how we became involved in computer graphics. From 1991 onward, this involvement took a new and much more serious turn when we began our journey into medical image computing, a field rich in innovative visualization and rendering techniques. Before long, however, our focus shifted from purely scientific questions within computer graphics and computer science to a much broader—and in many ways quite different—set of goals.
Medical Image Computing:
For more reading go to MeVis - The First Ten Years.
Founding MeVis
​The non-profit research center MeVis Research gGmbH was founded in August 1995. From 1991 to 1995, our work in medical image computing was carried out within CeVis, the institute embedded in the Department of Mathematics at the University of Bremen. It became increasingly clear to me that we had reached a bifurcation point: should the primary focus of our research be scientific merit in mathematics and computer science, or impact in medicine? I knew what I wanted—I wanted medicine, without compromise.
Consequently, embedding our research permanently within a traditional mathematics or computer science department would have been problematic. With the support of the City State of Bremen, I founded MeVis and became its CEO. At the beginning, our research staff consisted of just a handful of visionary people who dared, together with me, to chart unknown territory. There was nothing like MeVis in Germany at that time. There were somewhat comparable institutions like Max Viergever's ISI at Utrecht University, Hans Reiber's LKEB at Leiden University Medical Center, or Kunio Doi's Kurt Rossmann Lab at the University of Chicago. I visited all of them before founding MeVis, to learn and gather ideas for my own center.
In two essential respects, however, MeVis was designed differently. First, it was—and remained—independent of any university. I very much wanted independence and freedom. I did not want to work under a department chair, dean, or president, because I knew they must balance many competing interests. I wanted to follow our own trajectory, and nothing but our own trajectory, fully aware that this meant taking care of ourselves without the umbrella and safety net of a university. The City State of Bremen promised a base funding of about 20–30%, and we had to secure the remaining budget through research contracts from the DFG (Deutsche Forschungsgemeinschaft), the BMBF (Federal Ministry of Research), European agencies, and industry. Second, unlike ISI, LKEB, or the Rossmann Lab, we were not embedded in the strong research environment of a university hospital—which, according to prevailing opinion, is almost a prerequisite if one wants sufficient medical input and impact.
The University of Bremen did not—and still does not—have a university hospital, nor is it formally associated with one. When we designed and built MeVis, this strategic disadvantage had to be turned into a strength. The idea was to develop a regional, national, European, and eventually worldwide network of clinical research partners. This sounds appealing in theory, but in practice it was extremely difficult to achieve, because this was not the usual way research was organized at university hospitals, and we were largely unknown during our early years.
Funding was also a tremendous challenge at the beginning for two reasons. First, we had no track record yet. Second, the digital transformation of radiology—our main focus at the time—was still in its infancy, and most radiologists, especially in Germany, doubted that research would be crucial for their future workplace. As a result, virtually no national funding programs existed in this area.
Mammography and Breast MRI
In 1995 I met the late Jan Hendriks from Radboud University in Nijmegen. Jan, a radiologist, was at that time responsible for building the Dutch national breast cancer screening program together with his friend Roland Holland, a Hungarian-born pathologist. Jan and Roland were European pioneers in breast cancer screening and were remarkably open-minded about technological and methodological innovation. They soon became lifelong friends and mentors. Through their vision and tremendous experience, I was put on a steep learning curve regarding the intricacies, pitfalls, enormous challenges, and responsibilities of screening. Mammography at that time was film-based, and digital detectors were still far from providing the resolution required for mammography.
Jan, however, was looking ahead. He and I knew it would only be a matter of time—three, five, perhaps ten years—before digital detectors with sufficient detail would become available. We began discussing what kind of software could support breast radiologists in reading mammograms. At MeVis, we started developing an initial prototype, which had to overcome huge technical obstacles. One example is worth mentioning: we could work only with digitized film mammograms, and to preserve their diagnostic quality they had to be scanned at very high resolution, resulting in enormous data volumes per image. Naive loading times from memory to display were prohibitive, and innovative software strategies were needed.
This led to a very fruitful collaboration, which ended abruptly when Jan died in 2004. In the meantime, however, we had expanded our joint activities year by year, participated in two Europe-wide research programs that brought together all major European breast cancer experts, and gone through numerous cycles of development, field testing, and redesign. By 2001, we had solutions ready for routine use in European screening centers. That was when MeVis BreastCare GmbH was founded, with Carl Evertsz as CEO—a joint venture between Siemens Healthcare and MeVis Technology GmbH, which was our first commercial spin-off founded in 1997.
Ironically, a few years earlier Siemens had dismantled its own digital mammography efforts because it did not believe the technology would be viable. Another company, Hologic in the United States, then introduced detectors that promised the necessary high image resolution—but they had no software for reading digital mammograms. Siemens chose to license Hologic’s detectors and, as a partner in our company, to offer our software in return. MeVis BreastCare grew with the success of Hologic and Siemens and became the worldwide market leader in digital mammography reading software.
If MeVis had been under the umbrella of a university medical center, it is very likely that this success would never have happened. Unlike in my original scientific home, mathematics—where international collaboration is actively encouraged—medical research is often quite peculiar: cross-institutional collaboration is not cultivated and, in many cases, explicitly discouraged or even prohibited.
Henny Rijken, Roland Holland, HOP, Jan Hendriks, Carl Evertsz
Another early impulse in the development of MeVis came around 1996. Through Burckhard Terwey, who ran a very successful private MRI practice (ZEMODI) in Bremen, we learned about the benefits and challenges of breast MRI, which had been introduced by the German radiologists Werner Kaiser and Sylvia Heywang-Köbrunner. (There is a wonderful anecdote about Sylvia that I will not forget to tell.)
Breast MRI exploits a peculiar property shared by many cancers: as they grow, they stimulate the formation of nearby blood vessels to feed them (angiogenesis). These newly formed vessels, however, are structurally imperfect—they are more “leaky” than normal vessels. If a contrast agent is administered and its passage through the breast is monitored over time, this leakiness can be detected by measuring and comparing the dynamics of contrast enhancement in the tissue, thus providing a potential indicator of cancer.
At that time, breast MRI faced many obstacles: it was not reimbursed by insurers, examinations were slow, there were no well-established criteria for its use, and most radiologists were overwhelmed by the sheer volume of data and information. A single case could easily take an hour to read. The need for software support was obvious. Breast MRI also promised to fill some gaps left by mammography, which made it particularly attractive to us.
Together with Burckhard Terwey, we concluded that dedicated software for analyzing and interpreting these massive data sets could dramatically speed up diagnosis and improve its reliability—and, hopefully, encourage wider adoption of breast MRI among breast cancer specialists. His advice provided the lift-off for our breast MRI project, which began in 1996 and continues to this day. Another radiologist who played a key role in guiding our research was Joachim Teubner from Heidelberg.
When the first prototype was ready in 1997—running on an SGI O2 workstation (PCs and Macs were still far too weak at the time)—it was chosen as the first product for our new commercial spin-off, which Carl Evertsz, Hartmut Jürgens, and I founded in 1997 using our own funds (including a substantial bank loan) and venture capital. We named the company MeVis Technology GmbH. We initially hired an external CEO, which did not work out; as a result, Hartmut took over the role of CEO.
Multimodal breast view fusing breast MRI with digital mammograpgy: the way to go in the future
Our product achieved moderate success in European breast cancer centers, particularly in the Netherlands, Sweden, and the UK. However, the time was not yet fully ripe for this innovation. Through my professorship at Florida Atlantic University, I met Kathy Schilling and Jon Wiener. Jon is a first-rate MRI radiologist, very open to innovation, and Kathy was head of the Boca Women’s Center, one of the largest breast cancer centers in the United States. Like Jon, Kathy was highly receptive to new ideas and collaboration.
As a result, one of our SGI O2 workstations was installed in Boca Raton, and an extremely productive and rewarding collaboration began—one that continues to this day. We carried out many joint projects, including work on mammotome breast biopsy and computer-assisted teaching tools for Johnson & Johnson. Most importantly, our breast MRI software matured through this collaboration, and Kathy and Jon conducted a significant patient study, which they published in the American Journal of Roentgenology in 2005.
Without going into too much medical detail, their results showed that in a group of women for whom mammography had detected a cancer suggesting local surgery, breast MRI revealed additional malignancies to such an extent that local surgery would have been the wrong treatment. This finding greatly supported the broader adoption of breast MRI in the United States. Today, breast MRI is standard practice there, established for several indications and of proven benefit for certain subgroups of women.
Left to right: Richard Voss, Kathy Schilling, Karin Peitgen, HOP, President Frank Brogan, Nanette Peitgen, Benoit Mandelbrot, Carl Evertsz in the FAU's President residential mansion 2006, after Benoit and I had been inducted into the Hall of Fame of the Charles E. Schmidt College of Science
Eventually, our product became the leading software for breast MRI analysis and reading in the United States through a partnership with MRI Devices, a company that was later merged into Invivo and is now part of Philips Healthcare. This partnership was ideal for us and led to several innovations—the most important, from a clinical perspective, being the first software system for MR-guided breast biopsy. The matchmakers, unsurprisingly, were Jon Wiener and Kathy Schilling.
Tom Schubert, the head of MRI Devices at the time, wanted to expand his business from purely hardware—such as the market-leading head and breast MRI coils—into complete application solutions in which software was seamlessly integrated with, and finely tuned to, their products. Together we developed a whole palette of innovative software solutions: first for breast cancer, then for neurological applications, prostate cancer, and eventually lung cancer, all under the brand name DynaCAD. The name captured two essential features common to each application: CAD for Computer-Aided Diagnosis and DYNA for the dynamic analysis of MRI contrast-agent flow in the respective organ.
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But I promised an anecdote about Sylvia Heywang-Köbrunner. Around 2000, we wanted to intensify our work on breast MRI at MeVis and sought her advice. She was, as always, extremely busy—at the time she held a position at the University of Halle in eastern Germany—so I persuaded her to squeeze in a visit to Bremen between two other commitments. The deal was simple: I would pick her up at Leipzig–Halle airport in my small single-engine Piper Arrow and fly her to Bremen.
Remarkably, she showed no sign of fear, even though we flew the entire route in instrument conditions. On the ILS 27 approach into Bremen, we were descending at 500 feet per minute and still fully in the clouds. Suddenly the cloud layer broke, and we found ourselves between two decks of clouds. Within seconds we saw a Cessna coming straight toward us. I deviated, the other pilot deviated—we passed each other by a frighteningly small margin—and almost immediately were back in the clouds, still descending at 500 feet per minute.
After we landed safely, Sylvia remained in good spirits and surprisingly calm. We went up to the air traffic control office to report the incident and to learn what they thought had happened. Bremen Approach Radar was—and is—computer-supported. Among other things, the computer filters out radar returns that it classifies as erroneous. How does it do that? In controlled airspace around airports, all aircraft must have clearance. IFR traffic like us has it automatically and transmits an assigned squawk code. VFR traffic must squawk 0021 at all times.
The explanation for our near miss was chilling: the pilot of the other aircraft had turned off his transponder. No VFR squawk code was transmitted, so the computer concluded that this was a “ghost” target with no valid squawk and filtered it out from the controller’s display. As a result, the controllers could neither see him nor warn us, as they are required to do for IFR traffic, and they could not talk to him because, from their perspective, he simply did not exist. Worse still, he should never have been flying between cloud layers as a VFR pilot—that is another serious violation.
The pilot who almost killed us was never identified and never caught.
Sylvia and I learned a profound lesson that day about the dangerous mishandling of so-called “false” signals by computer systems—a challenge that remains central to computer-aided decision support in medicine in general, and in breast cancer detection in particular.
Liver Surgery Planning
Our work on liver surgery planning began in 1992—long before MeVis—in the institutional framework of CeVis, which I had founded that same year as an institute within the Department of Mathematics. The driving force behind this effort was Klaus J. Klose, then Head of Radiology at the University Hospital in Marburg. He and his team had been struggling for some time with a profound problem in liver surgery, and when he learned about fractals and my research, he managed to approach me in a rather unusual way.
In November 1991, I gave a keynote lecture for the company Schering Pharma at Schloss Reinhartshausen, a well-known German winery and hotel. The audience consisted of several hundred private-practice radiologists from the region. As a university hospital chief, Klaus did not really “belong” there—but his wife did, and so he slipped in. He managed to be seated right next to me at the festive dinner and, although I was exhausted after my two-hour lecture, he spoke with such contagious enthusiasm about a particular challenge in liver surgery that he won me over to the idea of collaborating. Shortly afterward, a small team from Bremen visited Marburg, where his group gave us a comprehensive introduction.
Back home, I found myself genuinely intrigued by the problem—especially because it connected with two areas of our existing expertise: fractals and numerical continuation methods. To a good approximation, the vascular trees in the liver—the portal vein, hepatic veins, hepatic artery, and bile ducts—each form an almost perfect fractal. One of the major complications arises from the fact that these trees vary enormously from patient to patient and are intricately intertwined.
​The picture shows a photograph of the portal venous tree obtained by a delicate casting technique. Jean H. D. Fasel, a liver anatomist from the University of Geneva, provided the image. He is one of the few anatomists who still masters and practices the art of anatomical casting. Jean contacted me, I believe in 1994, after our first publication on computer-assisted liver anatomy and offered his help. His expertise—and especially his extremely rare and therefore highly valuable liver casts—became crucial for our work.
So what was the problem that Klaus J. Klose presented to us?
In oncological liver surgery, the surgeon must remove a tumor that is embedded within these wildly intertwined vascular trees. The tumor has to be resected with an appropriate safety margin to ensure that all tumor cells are removed. This inevitably means operating in healthy liver tissue, which is densely permeated by vascular structures. The central challenge, therefore, is to remove the tumor safely while minimizing damage to the liver and preserving as much functional tissue as possible.
In 1957, the great liver surgeon and anatomist Claude Couinaud proposed a schematic solution after nearly a century of anatomical research. Having studied numerous livers from deceased patients, he introduced a unifying anatomical concept for the organization of the hepatic vascular trees. This can be seen in the following illustration:
In essence, Couinaud claimed that the anatomy of the liver is organized by the portal venous tree and can be partitioned into eight segments—the famous Couinaud segments. His schematic partition became the guiding principle of liver surgery. Every surgeon used it, and most probably believed that Couinaud had uncovered a law of nature, that the vascular architecture of the liver really was like this.
When we began our work on the problem, Couinaud’s scheme was so deeply ingrained in surgeons’ minds that, in retrospect, it seems almost naive to think we could challenge it, let alone replace it. One of his colleagues and one of the greatest liver surgeons of the century, Henri Bismuth, defended Couinaud’s concept up until very recently. Yet in one of our last anatomical papers with Jean Fasel (2014):
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Majno, P., Mentha, G., Toso, C., Morel, P., Peitgen, H.O., and Fasel, J.H.D. Anatomy of the liver: an outline with three levels of complexity. A further step towards tailored territorial liver resections. J Hepatol. 2014; 60: 654–662
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we demonstrated how far the schematic partition actually is from real anatomy. Bismuth wrote a remarkable editorial about our paper, in which he showed his full intellectual generosity. We quote:
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“Why is this leap in complexity needed? Because specialized hepatobiliary surgeons now operate on more advanced disease, with more frequent reoperations on more fragile patients, and are under the constant pressure of minimizing complications and of sparing as much liver tissue as possible. Surgery based on the real vasculo-biliary anatomy is the only way to achieve these goals, and I am not surprised that it is becoming more customised. The call to free the operation from Couinaud’s 8 segment scheme is in fact not subversive, it is the natural evolution of the concept we introduced in 1982 to the new imaging and surgical techniques, and to the identification of patients who can benefit from it. Of note, the articles I wrote in 1982 were before CT imaging, and the surgeon had to rely on preoperative and intraoperative ultrasound that by definition was done on the individual patient. There was no discrepancy to worry about.”
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Two of Bismuth’s insights are crucial: first, that more sophisticated and daring liver surgery must be based on the real anatomy and will benefit from it; second, that at a time when our kind of computer-based methods did not yet exist, schematic views such as Couinaud’s were better than nothing and therefore genuinely helpful.
Getting to this point was anything but a smooth journey for us. I still remember eminent liver surgeons looking at our results and saying: “A good surgeon doesn’t need this. We have learned to work with Couinaud—that is good enough.”
In the images below we show a study developed with Jean Fasel, where the competing schematic concepts from Couinaud’s time and before are compared with the true vascular anatomy. Simply by observing the vast differences, one can see how problematic the schematic approach was and still is: none of the schemes is truly right and none is entirely wrong. All of them force nature’s rich variability into a rigid cage. We compare partitions corresponding to the schemes of Hjortsjö, Couinaud (with subsectors), the “original” tree views, Goldsmith, and Platzer.
So how did the liver surgery community come to accept that there might be something beyond Couinaud?
Our first software solutions to support risk assessment, risk minimization, and surgical planning were ready around 1995/96. The software was colossal—so complex that it took more than one expert in our lab just to operate it. The response from the German surgical community, however, was anything but enthusiastic.
This changed in 1998, when I met Karl Oldhafer at a conference in Hanover. Karl immediately recognized the potential of our work and became both a loyal research partner and a close friend. He introduced me to Christoph Broelsch, Jürgen Fuchs, and Peter Neuhaus. All four were students of the legendary liver surgeon Rudolf Pichlmayr from Hanover.
Broelsch became famous for pioneering living-donor-liver-transplantation in 1990 at the University of Chicago and later adopted our methods in transplantation surgery. Fuchs is a world-renowned pediatric abdominal surgeon with special expertise in neuroblastoma surgery, and Neuhaus is one of the leading liver surgeons worldwide. Eventually, we would collaborate with all of them—but Karl was the first to truly catch fire.
Broelsch later became the preferred physician of German President Johannes Rau.
Koichi Tanaka
In 2002, our efforts to convince surgeons of the value of our software support received an unexpected boost. In June of that year, Japanese and German surgeons held a joint scientific meeting in Berlin to celebrate their long-standing tradition of friendship and professional exchange. Peter Neuhaus, who chaired the conference, had invited me to give a keynote lecture. This was my window of opportunity to present our work to leading surgeons from Japan. Strictly speaking, it was not quite the topic Peter had originally asked me to speak on—but I did not want to miss what felt like a chance handed to me on a silver platter.
After the lecture, I was immediately surrounded by several Japanese surgeons. They wanted more details and asked: “How can we use this? Can you make this available to us?”
The evening before the conference, there had been a festive dinner with many speeches, as is customary at such events. And then something happened that I could hardly believe and will never forget: the Japanese surgeons stood up, gathered at the front of the room, and began to sing German folk songs—in German—and invited us to join them.
One of the surgeons who cornered me after my talk was Hiroshi Shimada, head of abdominal surgery at Yokohama City University. We soon entered into an extremely fruitful collaboration, which was later continued by his successor, Itaru Endo, after Hiroshi’s retirement. Our joint focus with them was the refinement of our methods for surgery on cholangiocarcinoma (bile duct cancer), which poses an enormous challenge for liver surgeons. In a later study, Itaru showed that using our software support increased the one-year survival rate by 20%.
In the fall of 2004, Hiroshi invited me to give a keynote lecture at the 7th Meeting of the Japanese Society of Clinical Anatomy in Yokohama. Mine was the only lecture in English. In the middle of my talk, I suddenly noticed that people in the audience were disappearing under the tables in front of them, and I wondered what was going on. Seconds later I understood: we were experiencing an earthquake.
At a lecture in Yokohama minutes before an earthquake hit
The digital transformation of medicine is being driven by the still-growing power of computers – especially microprocessors and storage. Over the past few decades, computing performance has increased by orders of magnitude, while the cost per unit of performance has collapsed.
Today’s iPad Pro models (2024/25) ship with Apple’s M-series chips, offer up to 2 TB of storage and 8–16 GB of unified memory, and deliver desktop-class performance with GPUs capable of several trillion operations per second – all on a system-on-a-chip roughly the size of a fingernail.
By contrast, our SGI Onyx graphics supercomputer in the 1990s, which was the powerhouse for the entire institute, filled the space of two or three dishwashers, cost well into six figures, and required its own climatized machine room – yet it provided only a tiny fraction of the compute power and memory bandwidth that now fit into a thin tablet at well under one hundredth of the price.
This miniaturization and price collapse have enormous consequences for surgical planning. The video shows our collaboration with Itaru Endo at Yokohama City University, where a surgical plan prepared by MeVis Distant Services is brought directly into the operating room and interactively used on an iPad – a device that, not so long ago, would have required a full-blown graphics supercomputer.
The other surgeon who approached me after my lecture in Berlin was Yasuhiro Fujimoto, the right hand of the legendary living-donor liver transplantation surgeon Koichi Tanaka from Kyoto. I suggested to Yasuhiro that he visit us in Bremen before flying back to Japan, and with his chief’s permission he agreed.
In Bremen, we learned about Tanaka’s remarkable achievements, and Yasuhiro, in turn, absorbed as much as he could in a single day about our liver surgery planning software. When he left, we were hopeful that the next step would be a direct connection with Tanaka. But for months, there was silence. We kept asking Yasuhiro whether Professor Tanaka might be willing to receive us and let us demonstrate our work. Eventually, he agreed, and in late October 2002 we traveled to Kyoto.
I was accompanied by Guido Prause and Holger Bourquain from MeVis. Holger, a neuroradiologist by training, had greatly improved the usability of our software and was a master of live demonstrations.
When we arrived in Kyoto, we were told that Professor Tanaka would be very busy over the next few days but could spare some time on Monday evening at 8 p.m. in his office—perhaps for an hour. What happened during that hour would give our work an unimaginable boost for years to come. I briefly introduced the focus of our research, and then Holger began a live demonstration of the software using a case study.
At first, Tanaka sat at his desk, keeping a noticeable distance, clearly in a cautious “let’s see what they have” mode. But soon he moved closer. Within minutes, his attitude shifted visibly—from polite curiosity to intense interest and active engagement.
First meeting with Koichi Tanaka in his office October 28, 2002
(left to right: Holger Bourquain, HOP, Koichi Tanaka, Yasuhiro Fujimoto)
The meeting that was supposed to last one hour went on until after midnight. At the end, Tanaka said:
“On Wednesday I will perform a living-donor liver transplantation. Do you think you can use your software to analyze the risk for the donor and generate proposals to minimize that risk?”
I looked over at Holger (silently begging he wouldn’t say no) and replied: “Yes, I think we can give it a try.”
Tanaka continued: “Please prepare your results so that we can present them at the Wednesday morning conference with the international surgeons, before the operation. And one more thing: I am just finishing a book on living-donor liver transplantation. I would like to invite you to write a chapter for it. Will you accept?”
“With the greatest pleasure,” I answered.
These were two unbelievable, spontaneous recognitions and endorsements of our work from the leading pioneer of living-donor liver transplantation. When we returned to our hotel, we were walking on air and needed several drinks to calm down.
At that time, Tanaka had already performed more than 1,300 operations. He had introduced this highly specialized form of surgery in Japan and enjoyed a reputation in his community that can hardly be captured in words. Japan was then the world leader in the number of living-donor liver transplants, because transplants from cadaveric donors—common in the US and Europe—were essentially not an option there for cultural and religious reasons.
On Tuesday, Holger worked a small series of miracles. First, he managed to get the donor’s CT scans onto his laptop. That was no trivial feat, given that we were in a Japanese hospital, spoke no Japanese, and were navigating an unfamiliar IT environment. Fortunately, we had Yasuhiro at our side. He had spent professional time in the United States and was fluent on his beloved Mac—a devotion I happen to share since my first Macintosh in 1985 at the University of California in Santa Cruz.
Second, Holger was able to process the data and, with Yasuhiro literally looking over his shoulder, analyze the donor’s individual surgical risks and generate concrete planning proposals.
On Wednesday morning, we gathered in a lecture theater filled with dozens of physicians—many of them from around the world—who had come to learn from the master. Tanaka asked Holger and Yasuhiro to present our results. He questioned, discussed, challenged—and then paused and said:
“Because of the MeVis software results, I have changed my mind. I will now operate this way.”
He then explained his new insight, gained through our risk analysis. I was allowed to observe him performing this historic operation throughout the entire day.
And there was one more thing: after the donor operation, we performed an ad hoc perfusion experiment on the transplanted liver segment, following our planning proposal. This allowed us to verify that the risk assessment we had provided was indeed correct.
For us, this was a monumental breakthrough. It marked the beginning of a collaboration and friendship for which I will remain grateful for the rest of my life.
Koichi Tanaka, left, with HOP on October 30, 2002
after the first living-donor-liver-transplantation based of the MeVis planning results
Back home, we quickly developed software that allowed secure data exchange with Tanaka’s group over the internet. That was anything but trivial at the time: bandwidth was limited, data protection requirements were strict, and the CT datasets involved were huge.
From then on, Tanaka used our methods for all of his operations, even when he was operating in other countries, and he used them to refine and extend his surgical techniques, constantly pushing the envelope. He also propagated our approach worldwide through his lectures and communications and inspired other surgeons to follow his example.
For me, the most remarkable aspect of Tanaka’s attitude is this: here was Mr. Living-Donor Liver Transplantation himself, with more experience than almost anyone on the planet — and he did not say, “A good surgeon doesn’t need this.” These are the people who move medicine forward and tear down barriers. They are deeply confident in their own abilities (otherwise they wouldn’t dare remove half a healthy person’s liver), yet somewhere deep inside they know there is still room for improvement beyond their own skills. They take patient care seriously in an exemplary way.
I have been very fortunate to meet Koichi Tanaka and many other outstanding physicians around the world who share this mindset and became partners in our research at MeVis. They are a prerequisite for true medical progress — and they are relatively rare, not always easy to find. And this is not just a “men” thing: I have been equally blessed to work with exceptional women who embody the same attitude.
Although the world of transplantation surgery is still overwhelmingly male, there are remarkable exceptions. In the United States, Liz Pomfret, Chair of the Department of Transplantation at the Lahey Clinic in Boston, is one of them. She has adopted our methods and has been using them in her own work for several decades, providing powerful stimulus and feedback for our research – together with Christoph Wald, who initiated and built our collaboration in many other clinical domains, such as lung cancer screening.
Our methods for surgical planning are by no means limited to liver surgery, even though that is where we started. Over time, we extended them primarily to oncological surgery of the lung, brain, kidney, and pancreas.
MeVis Distant Services
What we first did with Tanaka – receiving CT data, performing a risk analysis, and generating a surgical proposal in the form of images and movies – became the template for a worldwide service. We initially established it in a research setting funded by the German BMBF and later turned it into an internet-based commercial service, embedded in rigorous quality control procedures to the extent that we were able to obtain FDA clearance. In Bremen, the work was initially supervised by Holger Bourquain and carried out by a team of specially trained radiology technicians.
The company we founded in 2004 was called MeVis Distant Services AG. In 2006 it merged with MeVis Medical Solutions (MMS) AG, which has continued to expand and refine the service over the years. Today, the liver surgery service is available for both transplantation and oncological surgery, and more than 20,000 cases have been supported for surgeons on all continents.
Breast Cancer Screening in Germany: The First Model Project in Bremen
Towards the end of the 1990s, Germany entered into an intense political debate about mammography screening. Countries like Sweden (the first in Europe) already had population-based breast cancer screening programs, built on extremely thorough and laborious epidemiological studies. The Netherlands followed Sweden with a national program, and the UK established its own within the framework of the NHS. The Dutch program, in particular, distinguished itself through very strict quality standards and an exceptionally refined quality control system.
The situation in Germany, by contrast, was abysmal. Studies had shown that, although many women were receiving mammography (so-called “grey screening” – screening without systematic quality control), the outcomes were completely unacceptable: the vast majority of detected cancers were already large (hard to miss on a good mammogram) and the disease was very likely metastatic.
In Sweden, the Netherlands, and the UK, carefully structured programs with dedicated quality measures had demonstrated that screening can detect “baby cancers” – small tumors that are much less likely to have metastasized and can often be treated with minimal aesthetic impact on the breast. The Swedes had even provided evidence that screening reduces breast cancer mortality, which is the ultimate goal and the key criterion of success.
However, screening comes with major challenges. Some arise from the limitations of mammography itself: although numerous studies have shown it to be the most appropriate method for breast cancer screening, it is far from perfect. For instance, some women have dense breast tissue, which appears more or less opaque on a mammogram and can easily obscure a tumor. Other challenges result from the nature of screening programs: they address large populations in which the vast majority of participants do not have the disease the program is designed to detect.
Quite typical for medicine, everything good comes with side effects and those can be so severe that screening is doing more damage than good. This is the case when the quality of screening is low (like in Canada in the 1990s and in the US at that time). Screening is for non-symptomatic women between 50 and 70, when the number of breast cancer cases is at its peak in a general population. Non-symptomatic basically means most likely healthy. There is an excellent website which I would like to refer to by the American Cancer Society. This website contains numerous statistical details that demonstrate how difficult it is to speak about screening in a few words and not oversimplify (unfortunately that is what the antagonists too often do). So let's say 5 out of 1000 non-symptomatic women should be detected with cancer. That means that more than 99% should come out of screening with a negativ result. Now, if the mammogram is technically poor or badly positioned, the radiologist may not have a chnace to find the cancer. Ot, if the radiologist has not enough experience cancers likely will be missed. At the same time the unexperienced radiologist may select many cases with suspicious spots in mammograms, which would lead to unnecessary biopsies and surgical interventions, although they are not cancer. Therefore quality and quality control is mandatory in screening or it can turn into a nightmare. In order to be experienced a radiologist needs very dedicated training for screening and has to see a lot of cases per year (some 5000). In the best of all worlds the radiologist's performance is continuously checked and there is retraining if necessary.
This is basically the route that the Dutch program had chosen and the late Jan Hendriks together with his colleague Roland Holland were the pioneers in setting up a very rigorous system of quality control, where every element in the screening chain is under observation, assessment, control and continuous improvement.
So here we are at the end of the 1990s in Germany. I knew all of this – and I got involved.
Germany has a very peculiar healthcare system. Alongside the legislative, executive and judicial branches, it effectively functions as a kind of “fourth power.” It is largely self-governed by the main stakeholder groups: the statutory health insurers and the providers. They decide what is done and how it is done. Health policy from government can set frameworks and boundaries, but the real power lies with the healthcare players themselves – in largely closed, non-transparent circles.
A simple example illustrates what I mean: pilots must undergo recurrent training, pass regular proficiency checks, and obtain medical certificates. At the very least, their eyesight is tested periodically (an airline pilot every six months). All of this is obviously desirable for passenger safety. Now imagine, just for a moment, that pilots could decide for themselves whether they are proficient enough and whether their eyesight is still good enough.
That, in essence, is the situation in general radiology. Periodic eye exams? No. Bad eyesight – stop working? Also no.
Now try telling such a profession that they need systematic training, recurrent retraining, external monitoring, documented quality control, and minimum case volumes. You can imagine what happened in Germany when the debate started about what kind of breast cancer screening we should have – and what quality standards should be mandatory.
Given the requirement of at least 5,000 cases per year for a participating radiologist, it was immediately clear to many radiologists and gynaecologists that a large number of those who had previously offered “screening” would no longer be needed in a certified program. And it became even more uncomfortable when someone without a medical degree, like me, began to argue for such standards.
Eventually, the situation in Germany led to a nationwide competitive call for model projects, which would be evaluated by a panel of European experts. I tried to organize a proposal from Bremen, initiated out of MeVis. At first, there was broad support within the local community of interested physicians. But once they realized how strongly my thinking was shaped by the Dutch approach – quality here, quality there, quality everywhere – much of that initial enthusiasm turned into resistance.
With the help of a few brave colleagues at MeVis and in the city, we nevertheless pushed ahead and submitted a proposal that was fully honest about the quality standards derived from the Dutch experience. The battles to get to that submission were unbelievable. Still, I felt that compromising on quality at this stage would be a lost opportunity. Put differently: it is relatively easy to relax quality requirements in a program later on if needed – but it is extremely difficult to raise them once the system is already in place.
​​But we prevailed. In the end, we secured the support of the Bremen Cancer Society under its president Heinrich Schmidt, as well as several other institutions, including the Senator for Health of the City State of Bremen. We submitted a proposal without a single compromise on quality. Christian Beck, Antje Bödicker and Markus Lang from MeVis were my co-authors.
The result: out of 17 applicants for 3 model regions, only the Bremen proposal was selected by the evaluation committee. It was, however, the strangest “successful” proposal I have ever been involved in—because it came with no funding at all. Being chosen simply meant that we now had the right to go and find our own financing. I managed to orchestrate support from several local health insurance companies.
The next crucial step was to find and recruit an appropriate medical head for the program—someone who would remain truly faithful to the spirit and the quality standards of the proposal. This was mission-critical: I was convinced that a lip-service supporter, and potential reactionary under pressure, could destroy everything. My clear favorite was the very experienced and nationally recognized Hans Junkermann from Heidelberg.
My opponents in Bremen, however, argued: “Why bring someone from outside when we have good people here?” They fought my proposal tooth and nail, and some did not hesitate to fight dirty. I knew that if I lost this battle, the entire project would be doomed. A local candidate from the Bremen hospitals would have faced enormous pressure and, in my view, would have found it very difficult to resist. And in Bremen there was simply no one with national standing in breast cancer diagnosis.
During these weeks, Jan Hendriks was my most important advisor. He made me strong. He reminded me of the battles he himself had had to fight and kept saying: “If you want honest screening, now is the time. You must not cave in.”
So I persevered—and Hans actually left his position at the University Hospital in Heidelberg to take on this heavy responsibility in Bremen. To this day, I am full of admiration for him. What a great man. What a great attitude.
The official start of the screening model project in Bremen was July 2001. Hans implemented the program, made it workable in everyday practice, and kept it running despite considerable hostility and opposition in Bremen and beyond. I supported him as best I could – for example, by shielding him from the most absurd attacks and by serving on the steering committee.
Why, one might ask, all this struggle and drama when there were already excellent examples in Sweden, the Netherlands, and the UK? The answer is simple to state, but not so simple to live with: the healthcare systems in those countries differ fundamentally from the German system. The model projects in Germany were meant to find a structure that proved that the very demanding requirements of the European guidelines could actually be met within the German context.
There were regular monitoring visits by a group of European experts and, in the end, a formal evaluation. Their conclusion: Hans’s Bremen project had far exceeded expectations and would be a suitable template for rolling out breast cancer screening across Germany. And that is exactly what happened.
Two anecdotes illustrate the mixture of friendly and hostile fire. My older daughter, now a radiologist (as is my younger daughter), was in medical school at the time. As part of her training she chose a rotation in orthopaedic surgery. The chief surgeon treated her with respect, and she loved the work. Then, from one day to the next, she seemed to have fallen from grace. In the morning conference, in front of all surgeons, he announced that she was “one of Peitgen’s brood – the man who is destroying doctors’ business in Bremen.” This happened while the project was already running.
Around the same time, the Senator for Health of Bremen (the successor of the senator who had originally supported our proposal) summoned Hans and me to her office. She was furious and said: “If something like this happens again, I will shut the project down immediately. I cannot afford this.”
What had triggered such outrage? As part of the screening chain, there is an invitation system – in our case under the supervision of the Einwohnermeldeamt (the population registry office). In a defined sequence by city districts, all women between 50 and 70 receive an invitation letter, together with information material, every two years. A few women – if I remember correctly, three – replied that they did not wish to participate and requested no further invitations.
The staff in the invitation office, wanting to do right by these women, began keeping a handwritten list so they would not invite them again, because the computer system did not support such an exception – and, in fact, should not support it, in order not to violate German data protection regulations. During a routine audit, the auditor asked: “What do you do with women who do not want to be invited?” They answered: “We started a paper list.” The auditor classified this as a violation of data protection law. In other words, they had been trapped. They were not supposed to keep track of anyone – even if their intention was simply to respect the wishes of the women concerned.
These were the kinds of attacks we had to endure – and of course Hans much more than I. Almost every month brought a new incident, and sometimes the criticism even came from some women’s activists who were convinced that screening was just another male invention to exert control over women.
The contrast between my life in mathematics and my life in medicine could hardly have been harsher – and, at the same time, more rewarding. I am eternally grateful to Hans Junkermann for allowing me to share this experience.
Mathematics Education:
How I Was Lured into Education
During my years in Bonn, and later in Bremen up until 1991, I had no contact at all with mathematics education. Because the schools I attended were excellent and prepared me more than adequately for university, I simply assumed that everything was in good order when it came to training future mathematics teachers. How wrong and naive that assumption was, I only began to understand in 1988.
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Our book The Beauty of Fractals had been out for two years and had caused quite a stir in the mathematical community – raising some eyebrows, but mostly generating tremendous enthusiasm.
During Ronald Reagan’s first term as President, a national commission was tasked with analyzing the weaknesses and challenges of the American education system. Its 1983 report, A Nation at Risk: The Imperative for Educational Reform, had a profound impact on the mathematics community in the United States. One of its great achievements was that the major mathematical societies – AMS, MAA, SIAM, and NCTM – which had previously been very aware of their differences, came together under the umbrella of the Joint Policy Board for Mathematics. Their goal was to align around the central recommendations of the report and to coordinate their efforts in mathematics education reform.
One of the key figures in this process was the late Kenneth Hoffman, an algebraist and former head of the mathematics department at MIT. I knew him through Springer-Verlag in New York. One day he called me in Germany and said:
“I need your help. I want you to do three things for us. First, give a special evening lecture at the 100th Anniversary Meeting of the AMS in Atlanta. Second, speak to teachers at their NCTM Annual Meeting in Chicago. And third, come to Washington and give a keynote at the Presidential Awards for Excellence in Mathematics and Science Teaching (PAEMST).”
All three events took place in 1988: the AMS meeting in January, the NCTM meeting in April, and the PAEMST ceremony in October.
Museum of Modern Art, New York, June 1989; Ken Hoffman (right), next to Benoit Mandelbrot, next to the President of Springer-Verlag after my keynote lecture (still exhausted) on occasion of the 25th Anniversary of Springer-Verlag New York
100th Anniversary of the AMS in Atlanta
I was quite proud of the AMS assignment, which took place after dinner in a ballroom of the Atlanta Hyatt Regency Hotel. When Peter Hilton was about to introduce me, we suddenly realized we couldn’t start: there were still hundreds of mathematicians trying to get into the room. The talk had to be delayed by an hour so the hotel could open an adjacent ballroom and bring in more chairs. People had to leave, wait, and then come back. I will never forget what it felt like to speak to several hundred colleagues in a completely packed double ballroom.
The National Council of Teachers of Mathematics
The NCTM lecture in Chicago did not mean much to me beforehand – but it meant a great deal afterward. Around two thousand mathematics teachers from across the United States were in the audience, many of them hearing about fractals and the Mandelbrot set for the very first time. The lecture was received with repeated bursts of enthusiasm, something I had never experienced in this way before. I was deeply moved.
Right after the talk, the president of NCTM came up to me and said that I now had an obligation to bring this material to more teachers. I immediately replied: “I would be glad to.”
Over the next years I gave keynote lectures at several national NCTM meetings (1989 Orlando, 1990 Salt Lake City, 1991 New Orleans, 1992 Nashville – where I brought Benoit Mandelbrot – and 1993 Seattle – where I brought the late Mitchell Feigenbaum), as well as at regional NCTM conferences throughout the US.
In 1989 I committed to write a small booklet for teachers that NCTM could distribute. By then I had met the late Lee E. Yunker, a high-school teacher from Chicago, and I realized I should seek real classroom advice for this project. When I saw Lee again in Orlando – he was an officer and director of NCTM at the time – he suggested involving Terry Perciante and Evan Maletsky. Together with Dietmar Saupe and Hartmut Jürgens back in Bremen, this became our core team.
The “small booklet,” however, steadily grew and eventually turned into the substantial two-volume set Fractals for the Classroom (published in 1991 and 1992) and the three-volume series Fractals for the Classroom: Strategic Activities for the Elementary, Middle, and Secondary Classroom (published in 1990, 1992, and 1999). All were published jointly by Springer-Verlag New York and NCTM and strongly supported by Rüdiger Gebauer, then mathematics editor at Springer and later its president.
The books and workbooks were eventually translated into several languages, including German, Polish, and Korean.
Lee was closely connected with Fermi Lab near Chicago. It is worth mentioning that, at the time, the US national laboratories were required to dedicate a portion of their budgets to educational outreach. As a result, we were able to run several summer institutes at Fermilab and later at the Princeton Plasma Physics Lab, aimed at regional and national mathematics and science teachers.
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The Presidential Award of Excellence in Mathematics and Science Teaching
​I looked forward to the Washington keynote for the PAEMST audience with great expectations. The prospect of meeting President Reagan — perhaps even speaking in his presence — felt very special. As it turned out, he wasn’t there. We later learned that he had chosen to attend a Dodgers game in Los Angeles instead, and we had to content ourselves with Vice President George H. W. Bush.
All of the award-winning teachers received gift packages, and Springer-Verlag had managed to include copies of The Beauty of Fractals. One of the honorees, Glenn Govertsen from Montana, later became a colleague and friend in my teacher enhancement programs in South Florida over the following two decades.
In Atlanta, Chicago, and Washington, I mostly used 35mm slides to present images — but I also did a bit of “magic.” I performed a live experiment I had first seen from my colleague and friend Ralph Abraham and Jim Crutchfield during my years at UC Santa Cruz and later refined: the now well-known video feedback demonstration. You take a video camera, point it at a TV screen, and feed the camera’s signal directly into the television. If the brightness and contrast on the TV and the zoom setting of the camera are just right, and you slowly rotate the camera around its long axis, something visually magical happens (see page 19 in Chaos and Fractals).
Videofeedback Composition by Japhy Riddle
From California to Florida
​In 1990, Arnold Mandell enticed me away from my position at UCSC to take a professorship at Florida Atlantic University (FAU). President Anthony Catanese essentially hired me on the spot, hoping I would attract substantial NSF funding for the department.
I took up the position in 1991 and began writing grant proposals for teacher enhancement projects in South Florida, designed to build and strengthen close collaborations between FAU and the surrounding school districts. The first grant was awarded in 1994, and we were able to maintain significant funding for regional and national projects until my retirement in 2012.
From 2004 to 2012, our projects were part of the national Math and Science Partnership (MSP) initiative (see MSP.NET Abstract and NSF Award Abstract #0412342).
South Florida and National Science Foundation (NSF)
To be eligible for funding, we had to build a strong partnership with a school district. In 1992/93 I approached both the School District of Palm Beach County and the School District of Broward County – with little success. Their attitude was essentially: How can a mathematician help us with our problems, when we know perfectly well how inadequate university-level preparation for math teachers is? My prior credentials meant very little to them.
Then, unexpectedly, rescue came from Marybeth Johnson, Vicky Quinn, and Nancy Barba. Marybeth and Vicky, both teachers in Broward County, had been in the audience at my NCTM lecture in Nashville and were enthusiastic. They were close friends with Nancy, who had influence in the district administration. Before committing, they decided to test us: we were asked to run a small summer program.
I brought in all the “weapons” I had: Evan Maletsky, Lee Yunker, and Terry Perciante, plus Dietmar Saupe – all co-authors of Fractals for the Classroom: Strategic Activities. The program was a success, and the district signed on to our NSF plans. That was the beginning of a partnership that lasted for two decades. Nancy, Vicky, Marybeth, and later Ana Escuder became my key partners and friends, carrying these highly complex and ambitious grants together with me. When I was able to attract Richard Voss to FAU, he joined our efforts and became co–principal investigator.
In the beginning, our work with math and science teachers revolved around discovery learning, using chaos and fractals as a central example. Later, we developed a full eight-course mathematics curriculum leading to a master’s degree for middle school teachers. More than 60 teachers completed the program and received their master’s degrees. Some of our grants funded summer institutes with an attached summer school for students; others supported summer institutes plus year-round coursework. From the very start of our work in South Florida, computers in the classroom played a major role – which was initially a struggle, because neither teachers nor schools were really ready for it.
Two developments supported our efforts enormously. First, we hired several math graduate students and developed a set of around ten Java applets. They should still be available on this website – but they aren’t. Java has since evolved its runtime environment to the point that our applets no longer run because of network security restrictions. So much for web continuity! This is particularly frustrating for someone like me, who has been a passionate advocate of the internet age.
Computers into the Classroom
​The other development, fortunately, is still very much alive: between 2003 and 2005 we began using dynamic geometry software in our NSF programs. Several commercial products were available, and we initially preferred Cabri from France. But there was a serious problem: our teachers were enthusiastic and wanted to use the software in their own classrooms, yet district-wide licenses were prohibitively expensive.
We started looking for non-commercial alternatives. Most of them were far less capable than Cabri—until we discovered GeoGebra and its then young Austrian creator, Markus Hohenwarter,, who had developed the software as part of his PhD thesis at the University of Salzburg. One day I called him and learned that he was about to accept a position in Munich. I tried to lure him and his wife Judith, who was still working on her PhD, to Florida. Markus accepted, came to FAU, took a visiting position in the math department, and quickly became a key figure in our NSF projects. We were able to amend one grant on his behalf, and in the next one we included him as part of the project’s faculty.
This had a huge impact on our efforts to integrate computers into mathematics teaching. Our teachers were able to work directly with the creator of GeoGebra – an experience that inspired them beyond words. At the same time, GeoGebra itself found a temporary home in South Florida, supported by NSF funding (for example, the powerful spreadsheet feature was conceived during this period), and as part of the national MSP community we helped introduce and spread GeoGebra across the United States.
Markus is both an excellent math educator and a natural entrepreneur. From the beginning, he held the conviction that dynamic geometry and algebra software should be freely available to teachers and students. Today GeoGebra is available in an impressive range of languages – including Kazakh, Nepali, and multiple variants of Arabic – and Markus has built a global open-source community of GeoGebra developers. He later left South Florida for a professorship at the University of Linz in Austria. He is, in all likelihood, one of the most influential figures in mathematics education when it comes to the use of computers – and at the same time as modest as they come.
From South Florida to Bremen
Around 1991, it was time to bring what I had learned and developed about teacher education and teacher enhancement back to Bremen. The attention future mathematics teachers receive at German universities is no greater than at universities in the United States. By and large, mathematics professors believe that a teacher simply needs a solid dose of “real” mathematics. In Germany this attitude extends even to elementary school teachers, who were expected to complete a full calculus course.
At some point I was invited by German physicists to their annual spring meeting to talk about my experiences bringing fractals and chaos into high school classrooms. There I met a young physics teacher, Gisela Gründl. She offered to help organize similar efforts in Germany—on one condition: that I help her obtain a teaching position in Bremen. We managed that, and once she arrived, Gisela began organizing teacher enhancement activities in Bremen with remarkable efficiency.
Before long, we were hosting annual four-day teacher workshops in Bremen that attracted participants from Germany, Switzerland, and Austria. A small but very effective team of local teachers grew into expert multipliers: Reimund Albers, Heidi Christiansen, Claudia Homburg, and Klaus Lies. Between 1992 and 2012 we organized 25 workshops (two per year from 1992 to 1995), and from 1996 onward we also included activities for music and German teachers by involving the Shakespeare Company Bremen and the Deutsche Kammerphilharmonie Bremen.
As our work expanded, it became desirable to have a formal legal framework, and we founded Lehrerakademie Bremen e.V.
Around the year 2000, and for several years thereafter, we organized weekend workshops for mathematics teachers in Bremen, helping them learn how to enhance their teaching with effective use of computers in the classroom. This initiative became known as the Bremer Netzwerk Mathematik und Computer.
Our final major effort was within the university itself. Reimund and I developed and wrote a grant proposal to the Deutsche Telekom Stiftung, which funded our work from 2007 to 2010. The project, called Mathematik Neu Beginnen, aimed to design and implement an entirely new mathematics curriculum for future elementary school teachers at the University of Bremen. The idea was to present mathematics in a discovery-oriented style and to build a coherent body of content—viewed from a higher mathematical standpoint—that would form the backbone of elementary school mathematics.
Three Presidents
The late Johannes Rau, former President of Germany and a good friend of the surgeon Christoph Broelsch, once spent an evening in Bremen after attending the Bremer Schaffermahlzeit, playing cards. Christoph suggested it would be a good idea for me to meet him in person. I felt a bit awkward, but eventually agreed and joined them in a hotel room where they were playing cards, smoking large cigars, and clearly enjoying some excellent Bordeaux. The emphasis is on enjoying.
It was around midnight when I was introduced as a mathematician. Instantly, the President lit up and launched into a joke. He was well known for his jokes and for the elegant way he framed and delivered them. What followed was a wonderful mathematical joke, which I have since used in many of my lectures to teachers—always to illustrate that it is not enough to tell a student that an answer is wrong; what really matters is understanding the misconception or misunderstanding that led to the wrong answer.
The joke uses addition and multiplication in a peculiar number system that sounds perfectly logical on the surface but leads to very strange results. I always wondered how President Rau managed to remember all the details, because in this joke every single detail matters—otherwise it falls apart.
Later I discovered that the President’s joke goes back at least to the 1950s. Here is a wonderful rendering of it:
In 1995 I had the pleasure of meeting German President Roman Herzog. During his first official visit to the city-state of Bremen, it was decided that he should briefly visit the University of Bremen, where I was to demonstrate our work on liver surgery in our lab using live computer demonstrations.
At the last minute, however, security concerns intervened. Students had announced a protest against the President, and it was decided that I should give my presentation not in the lab, but in a large, windowless lecture hall with secured doors. So there I stood at the blackboard, with the President and his entourage in the front row—otherwise alone in a dark, empty hall. The atmosphere was almost ghostly, unreal.
We had an intense and extended discussion about our work, and the schedule went completely off track—something I am rather famous for. One year later, I met him again at his residence in Berlin, where he presented me with the Federal Cross of Merit, First Class.
I also have very fond memories of President Richard von Weizsäcker in 2003, shortly after the Iraq War had begun. I sat next to him at a lunch table, and when he heard about my close ties to the United States, he was eager to hear how the war was perceived in American society. In return, he shared his own deep and carefully considered view of President Bush’s war policy. He was profoundly concerned, and looking at the chaos in the Middle East by 2015, I have to admit that President von Weizsäcker seemed almost prophetic.
My meetings with Presidents Christian Wulff and Joachim Gauck were also interesting, as they offered insight into their personalities, but they do not really belong to this particular story.
Early Years and High School:
I was born and grew up in a tiny village east of Cologne. My early childhood was village life in its purest form. My world ended at the fields, the farms, and the neighboring villages. I went to a typical village school where the first four grades were taught together in a single classroom. Students were encouraged to try problems from higher grades. After school, pranks – some of them quite wild and dangerous – played a central role in our lives.
At that time, Germany had a strictly segregated school system: one track (Volksschule, grades 1–8) for the poor and/or less gifted, and another track (Gymnasium, grades 5–13) for the rich and/or gifted. Volksschule was free, while Gymnasium required substantial tuition, which was a burden even for many middle-class families. Around the time I was in seventh grade, things changed and Gymnasium became tuition-free. That was my opportunity.
The Wüllenweber-Gymnasium in Bergneustadt, about 2.5 hours away by bus, accepted students from Volksschule into special accelerated tracks after they passed a demanding entrance exam. I was admitted – the greatest adventure of my life up to that point and a huge window into the future.
At home, controversial political discussions were frequent, much to the dismay of my mother, Herta Peitgen. My father, Walter Peitgen, belonged to the FDP, while I eventually joined the SPD and became an ardent admirer of Willy Brandt. My father was quite popular in our municipality and served as mayor for 12 years, although his party was always in the minority. He was awarded the Federal Cross of Merit, and several years later the Federal Officer’s Cross of Merit. Sadly, he did not live to see that I followed in his footsteps in this respect in 1996.
My village Bruch
When I was 15, I had an experience that gave me nightmares for decades. I lost my best friend, Bodo Jürges, in a fatal car accident. Bodo’s brother owned a motor-assisted bicycle (a Moped), which Bodo occasionally borrowed without his brother’s knowledge. I had a regular street bike, which Bodo liked very much. Neither of us was allowed to ride the Moped—we were too young to have a license.
One day, I was driving the Moped and Bodo was riding my bike on a narrow, winding road. I towed him: he held onto my left shoulder while we drove. Suddenly, a car came toward us. Bodo let go of my shoulder, swerved toward the middle of the road, and the car hit him and pushed him into the curb. He died within minutes at the scene.
The driver of the car was under the influence of alcohol and was later convicted of manslaughter through culpable negligence. I also had to stand trial but was cleared of all charges. Still, the death of my friend—and my role in the accident—burdened and troubled me deeply for many years.
Germany at that time was post-war Germany, still deeply entangled in its horrific past, largely unwilling to talk about it, and even less willing to confront its enormous guilt. I still remember the lonely struggle of Fritz Bauer to bring about the Auschwitz trials, and how important it was for me that there were poets and playwrights like Bertolt Brecht, who had resisted the Nazis. This was a time when we had many questions, and far too many adults refused to speak.
Some of our teachers, however, were completely different. They understood that we needed to talk, and they talked. I know that their courage shaped me at least as much as the formal curriculum at the Wüllenweber Gymnasium. One of them, Herbert Heidtmann, our religion teacher, read excerpts from the diaries of Rudolf Höss with us. Here was a man who listened to Beethoven and Schubert in the evening after his day’s “work” as commandant of Auschwitz.
When I was sixteen, this remarkable teacher took a small group of us on a trip to Weimar. This was no small feat at the time: Weimar lay in the GDR, and such visits were rarely permitted. There we were confronted with the unbearable contrast between the city of Goethe and Schiller and the nearby concentration camp Buchenwald, just a few kilometers away. That experience left a deep and lasting mark on us. In Buchenwald we laid flowers in the cell where Dietrich Bonhoeffer had been held shortly before his execution on April 9, 1945.
We studied Bonhoeffer’s writings and discussed a particularly shocking fact: in 1956 the Bundesgerichtshof (German Federal Court of Justice) ruled that Bonhoeffer’s trial before an SS “stand court” — held the evening before his execution, by special order of Hitler, without defense counsel and without any prosecution or defense witnesses — was nonetheless a due process within the framework of the former law and remained legally valid.
This single, deeply troubling example explains better than anything the dark clouds still hanging over post-war Germany, and why it was so difficult for some of us for many years to develop a positive sense of identification with the country.
Academically, I was undergoing a profound transformation: from village boy to young adult, full of ambitions, with many new windows suddenly opening, and with rock-solid convictions shaped by my teachers. In particular, my mathematics teacher Gerhard Klawitter deserves credit for inspiring my ambition to become a professional mathematician — not exactly a modest goal. But passion carries far, and it was his passion for mathematics that he passed on to me with lasting effect.
For personal reasons my entry into university life was not without obstacles. When I arrived at the University of Bonn in the spring of 1965, the enrollment deadline for the summer semester had long passed. The semester had already been underway for a month, and the university’s admissions office told me: “There is no way we can enroll you unless the professors in charge of the courses request an exception in writing.”
With this seemingly hopeless condition, I went to see Friedrich Hirzebruch for the first time, in his office in Beringstraße 1. He listened, gave me a wonderful sense that he genuinely cared, asked a few questions, and then said:
„Ja, Herr Peitgen, was machen wir denn da? Geben Sie mir einen Tag, und ich suche eine Lösung.“
(“Well, Mr. Peitgen, what shall we do? Give me a day and I will try to find a solution.”)
The next day, he had already spoken with Jürgen Schmidt, who was responsible for Linear Algebra (Hirzebruch was responsible for Real Analysis). Hirzebruch proposed that two of his assistants, Klaus Lamotke (later a professor at the University of Cologne) and Klaus Jänich (later a professor at the University of Regensburg), would coach me for a week to help me catch up on the missed month. They would assess me, and if their judgment was positive, he would write a letter to the admissions office requesting an exception.
That is how I was admitted. Thanks to the excellent preparation I had received from Gerhard Klawitter, and the intense coaching from Lamotke and Jänich, I was able to complete both courses with top grades.
When I first met Hirzebruch, I had no idea who he really was, or what a giant of mathematics he represented. I soon learned—and my respect and gratitude for his help only grew. But my attachment to Hirzebruch went much deeper. He was the best teacher I have ever seen. He became my role model for how to teach, and he firmly pulled me into mathematics, showing me from the very first days that becoming a research mathematician could be an exciting and deeply rewarding profession. Above all, he became a lifelong, almost fatherly friend.
PhD Students and Habilitations:
Advising students toward a doctoral degree is one of the great privileges of a university professor. To witness and support a student’s transition into an independent researcher is a source of deep joy and satisfaction. The topics of their theses trace the development and shifts of my own research interests over time.
After completing their doctorates, many of my students became co-authors of papers and books, co-founders of companies, partners in building institutions, collaborators in outreach projects such as breast cancer screening or teacher enhancement—and, most importantly, lifelong friends. The topics of their 31 theses are listed in the Mathematics Genealogy Project.
Among mathematicians it has become something of a sport to trace one’s “mathematical ancestry.” According to my genealogy, I am a great-great-great-grandstudent of David Hilbert and a great-great-great-grandstudent of Sophus Lie. This says nothing about my mathematical abilities, of course, but it is still fun to draw these lines.
In the 1990s, when I once proudly remarked that I could document a direct line of descent from Charlemagne, Dietmar Saupe announced a lecture on the statistics of descendants. He showed that a large fraction of all Germans share this genealogical origin with me—even if they cannot document it.
Dissertations in Mathematics, Computer Science and Math Education (1978 – 2015):
17 mathematics, 12 computer science, 2 mathematics education, *9 are serving as professors
Dr. rer. nat. Michael Prüfer, 1978, mathematics
Dr. rer. nat. Norbert Angelstorf, 1981, mathematics
Dr. rer. nat. Hubert Peters 1981, mathematics
Dr. rer. nat. Hans-Willi Siegberg, 1982, mathematics
Dr. rer. nat. Dietmar Saupe, 1982, mathematics, *now Professor of Computer Science University of Konstanz
Dr. rer. nat. Hartmut Jürgens, 1983, mathematics
Dr. rer. nat. Fritz von Haeseler, 1985, mathematics,
Dr.-Ing. Cornelia Zahlten, 1995, computer science
Dr. rer. nat. Jürgen Gerling, 1995, mathematics
Dr. rer. nat. Ehler Lange, 1996, mathematics
Dr. rer. nat. Kathrin Berkner, 1996, mathematics
Dr. rer. nat. Antje Ohlhoff, 1996, mathematics, * now Professor of Mathematics Hochschule Bielefeld
Dr. rer. nat. Anna Rodenhausen, 1996, mathematics, *now Professor of Mathematics Applied University of Hamburg
Dr. rer. nat. Thomas Netsch, 1998, computer science
Dr.-Ing. Mark Haidekker, 1998, computer science, *now Professor of Engineering University of Georgia Athens*
Dr. rer. nat. Wilhelm Berghorn, 1999, mathematics
Dr. rer. nat. Dirk Selle, 1999, computer science
Dr.-Ing. Christian Beck, 2002, computer science
Dr. rer. nat. Björn Engelke, 2002, mathematics
Dr. rer. nat. Ralf Hendrych, 2002, mathematics, * now Professor of Mathematics Applied University of Hamburg
Dr. rer. nat. Tobias Boskamp, 2003, mathematics
Dr. rer. nat. Martina Döhrmann, 2004, mathematics education,* now Professor of Didactics of Mathematics*
Dr.-Ing. Jens Breitenborn, 2004, computer science
Dr.-Ing. Horst Hahn, 2005, computer science, *now Professor of Digital Medicine and Director of Fraunhofer MEVIS*
Dr. rer. nat. Reimund Albers, 2006, mathematics education
Dr.-Ing. Jan-Martin Kuhnigk, 2008, computer science
Dr. rer. nat. Inga Altrogge, 2009, mathematics
Dr.-Ing. Tobias Böhler, 2011, computer science
Dr.-Ing. Anja Hennemuth, 2012, computer science, *now Professor of Digital Image Analysis and Modeling Charite and TU Berlin
Dr.-Ing. Andrea Schenk, 2012, computer science, *now Professor for Computer-Assisted Diagnosis and Therapy, Institute for Diagnostic and Interventional Radiology, Hannover Medical School
Dr.-Ing. Markus Wenzel, 2015, computer science, *now Professor of Medical Cognitive Computing, Constructor University
Habilitations: 5 mathematics, 1 computer science
Dr. Dietmar Saupe, now Professor of Computer Science University of Konstanz
Dr. Juri Suris, now Professor of Mathematics TU Berlin
Dr. Bernhard Preim, now Professor of Computer Science University of Magdeburg
Dr. Tobias Preußer, now Professor of Mathematics Constructor University, Bremen
Benoit B. Mandelbrot:
​Benoit Mandelbrot belongs to that rare class of scientists who make monumental contributions across many branches of science—mathematics, physics, chemistry, biology, medicine, engineering, geosciences, financial mathematics and economics—while at the same time becoming widely known and admired by the general public, yet not always warmly welcomed by many of their academic colleagues. Despite receiving major prizes and honors, he often remained something of an outsider.
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The common thread in Mandelbrot’s creation, fractal geometry, is the insight that much of the world is governed by power laws. Remarkably, this perspective can already be traced back to his 1952 dissertation: the first part dealt with Zipf’s Law and word frequency distributions, the second with aspects of statistical thermodynamics. Today, those same Zipfian word-frequency patterns are a cornerstone of how large language models represent and predict language, so his early interests turned out to be unexpectedly foundational for modern AI.
There is a wonderful video on Zipf’s Law by Michael Stevens on the Vsauce YouTube channel that vividly illustrates how fundamental this law is—it is well worth watching.
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I remember the days when Benoit would call me and we would talk for hours on the phone, discussing misunderstandings and misrepresentations of his work that had reached him and troubled him deeply.
For much of his life, he did not belong to the academic establishment, which usually finds its home in prestigious universities. Instead, he joined IBM Research and stayed there for most of his professional life, only later accepting positions at Harvard and Yale. At IBM he was an IBM Fellow—which essentially meant boundless scientific freedom, financial security, and almost complete independence, without the usual academic politics. It is quite likely that his maverick style and his habit of crisscrossing scientific disciplines were only possible in such an environment, one that did not cultivate rigid departmental boundaries the way universities typically do.
And if that is true, then fractal geometry itself might never have come into being in its full breadth—or it might have survived only as a minor subspecialty of mathematics or physics—had Mandelbrot been constrained by the walls of a traditional university department.
This aspect of Benoit’s life influenced me more than anything else when, in 1995, I decided to leave the mathematics department at the University of Bremen with my medical research ideas and to found MeVis Research GmbH as an independent center.
So what is it about Mandelbrot’s work that is so important—also for me personally?
Put simply, his work opened my eyes, both literally and philosophically. I was trained as a mathematician in Bonn under the influence of the Bourbaki School. In short, Bourbaki style stood for the absolute purity of mathematical exposition and a near total ban on pictures—on using the eye—as a tool in mathematical thinking. We would often say “as one can see,” but we did not mean seeing with the eyes.
Mandelbrot, by contrast, was an outspoken anti-Bourbakist. He repeatedly emphasized that the eye played a key role in his discoveries and in his understanding and solving of mathematical problems.
The Mandelbrot Set is a perfect example. His first experiments in 1980 were graphical experiments; visualizations of numerical results led him to realize that he was sitting on a gold mine. When we compare his early images with what we can generate today, we see not only the evolution of mathematical understanding but also a striking testimony to the revolution in computer graphics over the last decades.
Below, one of Mandelbrot’s original images (left) is shown next to one of our renderings of the electrostatic potential from 1990. Only ten years lie between these two images—yet they already reveal how dramatically the tools of visualization had improved in that short time.
Impacted by our own experiments in 1982/83 in Salt Lake City (make link) I moved from “eyes shut” to “eyes wide open” in my work. Incidentally, the well known movie Eyes Wide Shut, has no connection to what I am trying to say, except, that one of my most favorite compositions by my friend G. Ligeti, Musica Ricercata, is used in the movie "Eyes Wide Shut".
The Mandelbrot Set has many properties which are very difficult to accept. One of them is that if you zoom in onto details there is no limit, even if you zoom infinitely close. This movie zooms in by a magnification factor of an unbelievable 10 to the power of almost 1300! Look at it.
The other eye-opener was that there was something interesting in nature that I, and most of the science community, had missed, and which inspired my initial interest in medicine. The Fractal Geometry of Nature!
So what is it that Mandelbrot gave us? Foremost and most importantly he gave us a theory of roughness. What does that mean? From Newton and Leibniz to today we witness a huge and monumental strand in mathematics, the theory of smoothness. From calculus to differential equations and modern day differential manifolds the theme is the same: approximating reality and real world phenomena by models which are smooth, or are differentiable, or allow the application of differentiable calculus in some way or another. This has become so dominant that we were mislead to believe that reality indeed is smooth and can be perfectly modeled by mathematical objects, such as curves, differential equations, manifolds, etc. , which are smooth at heart. In this extremely successful approach which is at the heart of much of physics and engineering we have completely neglected objects and features of nature (and society) which are far from being smooth. Mandelbrot liked to say in his interviews: “Clouds are not spheres, coast lines are not straight and mountains are not cones.”
Let’s look at a chart of stock. The moving prizes over time generate a geometric shape (curve) which is as far from smooth as a rugged mountain range, and in order to be able to treat it mathematically, economists apply a simple trick, the method of moving averages, which results in a smoother shape (curve). And their belief is: “By filtering out noise and displaying key trends, moving averages can show you points at which a stock's price is likely to reverse course. That simple fact alone can help make you a better investor and improve your investing results!” I would say, however, if you force a chart into the smooth world, as you do with moving averages, you may have just lost what you were looking for. Incidentally, economics uses words like trend without caution. In fact, talking about trends means absolutely nothing unless you specify a time scale (meaning, at which detail are you obsevring the movement of prices) like a week, a month, or a year, and you specify a method of grabbing a trend. For example, you could pick a point on the chart, say every 10 days, and then connect these points with a straight line segment. This would be a piece-wise linear approximation of the chart. And now you could choose those segments, which pick up the movement of prizes neatly (just with some small variation). Maybe, none does: thus no trend; maybe a few do: thus a few trends with this crude method would be found. Or you could apply moving averages and whenever, say the 50 day moving average has a min or max (or vice versa) you got your start points and endpoints of trends.
It seems pretty obvious that this is completely arbitrary and cannot be consistently useful, because a trend, its existence and detection, depends on your choices of detail and measurement. And moreover, there would be as many differing trends as you want. There are two things, one demand and one question:
The demand: Whenever someone talks about a trend in moving prices of whatever, without specifying the time scale or detail of observation and the method of detection, this is totally worthless information.
The question: Why is that? Well the reason leads us to the heart of the matter, and that is: price movements are of a fractal nature. And it was Mandelbrot who told the world to pay attention to this fact, because it is not a negligible fact. Rather it matters a great deal if you want to understand stock and do anything about them, practically as an investor, or theoretically as a researcher. There is a whole array of fractal properties of stocks. The most fundamental one is that a chart is statistically self-similar, very much like coastlines and many patterns in nature.
In very popular science terms that means that the whole can be partitioned into parts, and each of the parts look like the whole (would be difficult to distinguish from the whole after proper scaling). (Note that I said “look like” and did not say “similar”. If the parts are similar to the whole, we have the more strict form of self-similarity”.) In other words, a chart over a short time span like a few days looks, or a few hours, very much like a chart over a month or a year. Obviously, if that is the case, then presenting a chart without reference to the time span is very useless information and the same then is true for trends in them. Or to put it differently, we can expect trends, short ones and long ones, for any time span. Another fractal property is hidden somewhat deeper and connects with the efficient market hypothesis. If the markets were indeed efficient, then stocks would be best represented by random walks and the mathematical treatment of stocks would be relatively simple. For example, we would know how to scale the 1-day volatility to a n-day volatilty by simply scaling with the square root of n.
​There has been, and still is, a great deal of controversy about this issue. As a mathematician, I would put it this way: markets are “efficient” in the same sense that patterns in nature are “smooth.” When it is convenient and useful, we assume efficiency or smoothness as an approximation—especially when it leads to elegant theory or practical formulas. The Black–Scholes framework for option pricing, for example, relies on Einstein’s square-root law, a hallmark of random walks.
In my view, much of the efficiency debate is misguided. It is obvious that markets are not perfectly efficient, just as it is obvious that natural shapes are not perfectly smooth. That observation by itself is trivial—what matters is whether we can quantify how they deviate from the idealized models. This is precisely where fractal geometry becomes powerful: it provides a toolkit of measurements and models to capture roughness and long-term dependence. In that sense, price movements are better described by fractional Brownian motion than by classical random walks.
Fractional Brownian motion is characterized by the Hurst exponent H, derived from a power-law scaling relation and directly linked to the degree of long-term memory in the process.
Now, one observes that 0 < H < 1, where H=1/2 singles out Brownian motion or random walks. A general measure for roughness that Mandelbrot gave us is the Fractal Dimension D of a pattern, and D, unlike more familiar notions of dimension, typically is not an integer, but can be. The fractal dimension is related to the Hausdorff Dimension but has one important differing feature: it can be algorithmically computed. So for example, the geometric pattern of a chart has a distinguished D, where 1 < D < 2.
And for fractional Brownian motion there is a beautiful mathematical relationship:
H + D = 2.
We used precisely this perspective in the 1990s when we worked on generalizing the Black–Scholes formalism to obtain option pricing models that better reflect the rough, fractal nature of real markets.
That, in a nutshell, is the power of Mandelbrot’s contribution: he didn’t just say “things are rough.” He gave us a language, a geometry, and concrete tools to analyze roughness—whether in clouds, coastlines, or the chaos of financial markets.​
Mathematics and Music: György Ligeti, Iannis Xenakis and Jean-Claude Risset:
I met György Ligeti in 1985. Manfred Eigen had shown him our catalogues, and Ligeti wanted to meet. He was instantly captivated by the fractal images. From the very first moment he asked question after question—relentlessly, almost hungrily. He wanted to understand, really understand: the underlying mathematics, the precise meaning of chaos in rigorous terms.
From 1985 until our last meeting before his death, our conversations followed the same pattern. There was almost no small talk about life, family, or work; he wanted to talk about mathematics. Although he had no formal mathematical training beyond high school, his curiosity ran deep. It took me some time to realize that, without knowing about chaos or fractals, Ligeti had already been grappling in his compositions with phenomena that were not mathematically chaotic or fractal in the strict sense—but came remarkably close.
When he first saw our work in 1985, he must have sensed this kinship intuitively, and that is what fascinated him: the parallels between his musical world and the world of the sciences. For more details, I refer to my chapter in the book Gygöry Ligeti - Of Foreign Lands and Strange Sounds and to the interview that Volker Banfield and I conducted.
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​In the late 1980s and 1990s Ligeti’s reputation as a composer of contemporary music rose to almost Olympic heights. At the same time, chaos and fractals reached their own peak in public awareness. I don’t mean to suggest that the latter contributed to Ligeti’s fame, but the coincidence was striking. He enjoyed this parallel to some extent, yet it also made him a little uneasy.
Nevertheless, he and I gave several joint events with Volker Banfield: Volker would perform from the magnificent Études, and György and I would lecture and talk on stage about the connections between music and mathematics.
I remember one of these evenings especially well. The City of Bremen hosts an annual music festival, and I persuaded the artistic director to devote a night entirely to Ligeti. He agreed—on the condition that I convince Ligeti to attend and take part in a stage conversation that I would moderate. The expectation was that this would attract a large audience, and it did. The concert took place, quite deliberately, in the satellite assembly hall of what is now Airbus in Bremen. The program was built around Ligeti’s Violin Concerto, with the marvellous Christian Tetzlaff as soloist and the Deutsche Kammerphilharmonie Bremen as orchestra.
There was a final rehearsal late in the afternoon. I picked up Ligeti at the station and drove him to the hall. As soon as he walked in and heard Tetzlaff playing, he said without hesitation: “This will not work. The acoustics are terrible. We cannot perform my concerto here. Absolutely not.” Every attempt to persuade him failed. Around 7 p.m. we reached a compromise: he would skip the concert, but he would at least allow his Violin Concerto to be performed.
So the central attraction of the evening—the live presence of the great master on stage—had evaporated, but at least the concert was saved. I suddenly had about thirty minutes to rethink the entire moderation, which had originally relied on asking a few key questions and letting Ligeti speak in his wonderfully associative way.
The next day we met in his hotel room and spent our time, as so often, talking about different concepts of infinity in mathematics, one of his favourite topics. He did not even ask how the evening had gone. The concert itself had been a phenomenal success, despite its demanding contemporary program with works by Ligeti, Iannis Xenakis and Jean-Claude Risset. No one complained about the acoustics.
On one of my birthdays he phoned, apologized for not being able to come, and told me about the gift he had decided on: he would dedicate his 17th Étude for piano to me. It had not yet been written, but when it finally appeared it felt like a jewel.
In his last years it became increasingly difficult to communicate with him. He spoke less and less, and eventually stopped speaking long before he died. I was asked to speak at his funeral in Vienna. Although this was very hard for me, it helped me accept that we would never again have our wonderful conversations about music and mathematics.
It took me years after Ligeti’s death to understand how our conversations about Ed Lorenz and the discovery of chaos are woven into the 17th Étude. In 2023 I was invited to give the keynote lecture for Ligeti’s 100th birthday at a symposium in Cluj, Romania, the city where his musical journey began.
In preparing my lecture, I carried out extensive research and was fortunate to come across a musicology dissertation by Brian Lefresne from the University of Ottawa, which analyzes the 17th Étude in depth. There I found a reproduced sketch from Ligeti’s manuscript unmistakably depicting his version of the famous Lorenz attractor—about which he and I had spent many hours in intense discussion. The picture on the right shows the seminal Lorenz attractor; for details, see page 647 of my book Chaos and Fractals: New Frontiers of Science.
In fact, in his dissertation Lefresne argues that the central theme of chaos theory – sensitivity to initial conditions – lies at the very heart of the 17th Étude. Tiny variations in rhythmic patterns, dynamics, and timing continually reshape the musical texture, so that the piece behaves almost like a dynamical system in which the smallest perturbation can lead to a perceptibly different musical outcome.
The other great composers I had the privilege to meet were Iannis Xenakis, Jean-Claude Risset and Steve Reich. I managed to persuade Xenakis to come to Bremen for a concert devoted to some of his major works. The concert was co-funded by the TV station 3sat on the condition that I would interview him for broadcast together with the performance. A year earlier we had done a similar project with Ligeti (see video), and a year later another one with Detlev Müller-Siemens. When I say we, I mean my friend Volker Banfield, the pianist, who joined me in the interviews (and performed in the Ligeti event), and my friend Hans-Helmut Euler, who handled the negotiations with 3sat. My role was to design the concert programs, convince the composers to come, and conduct the on-stage conversations.
A few first-hand impressions: Reich’s works, such as Piano Phase, have deep connections to brain research, mathematics and acoustic physics. I had studied these links in some detail because I admire his music—Drumming is one of my favorites. When I met him briefly in Miami, I was eager to learn more. To my surprise, he seemed largely unaware of many of these connections and not particularly interested in them. What a contrast to Ligeti!
Xenakis, by comparison, had a solid education in mathematics, and much of his music is explicitly shaped by stochastic processes and probability theory. I also valued his anti-fascist resistance background during the Second World War. You don’t “hear” that biography in the music, of course, but knowing it adds an extra layer of respect when listening.
With Jean-Claude I could speak almost as with a fellow scientist. His thinking is rigorously structured, and I have often used his ingenious work on Shepard-tone glissandi in public lectures. His music illustrates beautifully that our perceptual system is an essential part of music—far beyond what is written in the score. I was astonished when I began to notice Risset-like glissandi in some of Bach’s organ works. A wonderful example is BWV 542 - Fantasia & Fugue in G Minor: listen to the glissando passages after about 3:40 minutes. With Risset’s idea of endlessly rising and falling glissandi in mind, I now hear Bach quite differently—enriched by an entirely new dimension.
Peter H. Richter:​
The late Peter Richter was a colleague and dear friend at the University of Bremen. A theoretical physicist by training, he and I did many things together that neither of us would likely have undertaken alone. Peter was always ready for scientific adventures, yet at the same time extremely modest and sincere. He kept my feet on the ground and, without him, I would probably never have entered the world of chaos theory.
We ran seminars to educate ourselves—on continuous (Hamiltonian) dynamical systems, Hamiltonian chaos, discrete dynamical systems, area-preserving diffeomorphisms, and eventually dynamical systems in the complex plane. On these journeys Peter always insisted on maintaining a link to real physics: classical mechanics and, in particular, the theory of phase transitions.
I would like to share one anecdote from 1993, connected with our bestselling book The Beauty of Fractals, which culminated in a rather triumphant victory over the German news magazine Der SPIEGEL. Peter Brügge, one of their longtime journalists, had published three consecutive articles under the headline “Der Kult um das Chaos” (“The Cult of Chaos”).
In the months before those articles appeared, Brügge had visited Peter and me several times in Bremen. He spent hours in our lab; we walked him through our experiments, explained what chaos means in strict mathematical terms, and where the limits of the theory lie. Again and again we tried to help him distinguish serious results from the more euphoric nonsense in parts of the media and popular-science scene. It was hard work; often he struggled to follow the mathematics, but he always seemed genuinely determined to understand what was substance and what was bluff.
That is why, when the series in DER SPIEGEL finally came out, we were stunned. The journalist who had gained our trust suddenly appeared in print as something like a prosecutor. On the one hand he quoted my appreciative remarks about him; on the other hand he insinuated that we had published bogus results on chaos in celestial mechanics in The Beauty of Fractals—results that, he claimed, a mathematician from Karlsruhe had exposed as non-chaotic. The tone was dismissive and sneering, as if we belonged to the camp of “chaos evangelists” peddling unfounded claims. His “revelation” was that chaos did not really exist at all, but was merely an artifact of sloppy computer simulations.
We were deeply disappointed. Together with the president of Springer-Verlag, our publisher, we decided not to let this pass. We engaged one of Hamburg’s leading media law firms and, on their advice, demanded a formal counterstatement (Gegendarstellung). At the court hearing DER SPIEGEL tried to avoid that outcome: they offered us the chance to publish three articles of our own, without editorial interference. Our lawyers took this as a clear sign of how badly DER SPIEGEL wanted to prevent a forced correction—and advised us to decline. We did.
The court then ruled in our favor: DER SPIEGEL had to print a counterstatement setting the record straight. Measured against the damage done by the original articles, this was a modest remedy—but in symbolic terms it was a triumph. For a magazine like DER SPIEGEL, there is hardly anything more painful than being legally compelled to acknowledge that they have misrepresented the truth.
The mathematician from Karlsruhe played a dubious role. In court Brügge invoked him as the authority who had allegedly refuted our results. Yet some time later the same mathematician published a paper confirming precisely those findings.
We later documented the entire affair, together with the mathematical details, in an article titled “Der SPIEGEL, das Chaos - und die Wahrheit,” published in Physikalische Blätter 4/1994.
In his later years Peter became vice-president of the University of Bremen and guided the university through the transition to bachelor’s and master’s programs. He had a huge heart for physics education; after retiring he devoted himself to improving the training of future physics teachers—a beautiful parallel to my own work in mathematics education.