# Michael Tabor, PhD, DSc Scientific and Professional Biography

**A brief scientific and professional biography**

I grew up in Cambridge, England, where I attended the Perse School. I then went to Bristol University with the intention of studying biochemistry, but became more interested in chemistry, and theoretical chemistry in particular. I had the opportunity to stay on in Bristol to pursue doctoral studies, and while the conventional wisdom is that one should not stay on in one’s undergraduate institution, this eventually turned out to be one of the best decisions I ever made. While struggling with a project in spectroscopy I attended, almost by chance, a talk by Michael V. Berry who was a member of the Physics Department. I went to his office the next morning, told him what I was interested in, and asked if I could work with him. He gave me a problem that one of his previous students had not been able to solve. He also told me that because he had just moved, he wouldn’t have very much time to talk me about the project. Not to be deterred, I offered to go to his new house on the weekends and help him paint it. And so began collaboration—up a ladder, as it were—that had a lasting impact on my scientific career. My official advisor at the time, Richard N. Dixon, graciously allowed me to complete my doctoral studies under Michael Berry’s supervision.

My dissertation research was in the field of semi-classical mechanics, the regime that explores the transition from quantum mechanics to classical mechanical behavior (asymptotically speaking, the limit *h *—> 0). Semi-classical approximations to quantum mechanical quantities, such as energy levels and scattering cross-sections are made, loosely speaking, by superimposing quantal effects on associated classical trajectories (“sewing quantal flesh on classical bones”). My dissertation project explored semi-classical approximations to the quantal energy spectrum and how that spectrum could be computed in terms of classical periodic orbits [1]. We also studied the distribution of energy level spacings [2], and this resulted in what is now known as the Berry-Tabor conjecture—a result that still continues to be a topic of interest. In addition, the new field of nonlinear science was just taking off, and it was an exciting time to be a graduate student—especially in the lively and collegial environment of Bristol University’s Physics Department. The remarkable discovery that relatively simple deterministic systems (such as coupled nonlinear oscillators) could exhibit chaotic behavior raised many fascinating questions including the possibility that some of the quantum mechanical properties of classically chaotic systems might be different from those of classically integrable systems [3]. This topic, sometimes termed “quantum chaology”, continues to be of considerable interest.

While completing my dissertation, I was awarded a Royal Society – Israel Academy fellowship that enabled me to take up a postdoctoral position to work with R. D. Levine at the Hebrew University of Jerusalem. It was a fascinating time: I learnt a lot of new things, and continued my studies of nonlinear dynamics. I then went on to another postdoctoral position at the University of Illinois with Rudolph A. Marcus who was interested in semi-classical approximations of molecular energy levels. Soon after arriving at the University of Illinois, Professor Marcus moved to Cal Tech and took me with him. After an enjoyable year and a half at Cal Tech I was offered a job in a small research institute affiliated with the Scripps Institution of Oceanography in La Jolla. The mission of the institute was to apply the ideas of nonlinear dynamics to problems in oceanography and fluid dynamics. It was here that my research program really took off.

One of the canonical models of nonlinear science was the Lorenz model—a deceptively simple system of three nonlinear differential equations that was derived as a highly simplified representation of atmospheric convection. Lorenz demonstrated that some initial conditions exhibited chaotic behavior and further studies showed that the system possessed a highly complex attracting set (a strange attractor). The sensitivity to initial conditions discovered by Lorenz led to the now popular notion of the “butterfly effect”. For various reasons, a colleague, John Weiss, and I started to wonder if the real time dynamics of the Lorenz system could be understood in terms of the positions of the complex time singularities possessed by the equations, and this led to a detailed analysis of the singularities of the Lorenz system [4]. One product of this analysis was the identification of system parameters for which the Lorenz equations were integrable. These special cases arose when the singularities were demonstrated to be simple poles – a property (meromorphicity) that is often referred to as the Painlevé property [5]. However, it should be noted that the connection between integrability and meromorphicity has its origins in the work of the great Russian mathematician, Sofia Kovalevskaya, in the late C19th [6].

Following our work on the Lorenz system we applied these ideas to another canonical model in nonlinear science, the Henon-Heiles system—a Hamiltonian system of two coupled nonlinear oscillators that was derived as a highly simplified representation of the motion of a star in a galactic potential. The Henon-Heiles system was shown to exhibit chaotic behavior sensitively dependent on initial conditions, with more and more orbits becoming chaotic as the energy of the system was increased [7]. An analysis of the singularities identified the known integrable cases and one new one. Detailed analysis of the nonintegrable cases revealed fascinating and exotic complex-time structures including self-similar natural boundaries [8].

At this point we started to wonder if there was a generalization of the Painlevé property that could be applied to nonlinear partial differential equations (pdes). The 1970s had seem some remarkable developments in which a few such equations, such as the Kortweg deVries (KdV) equation, were shown to exhibit special solutions, termed solitons, that were remarkably stable localized structures that could travel without loss of shape or stability – properties that were to become the basis of data propagation in optical fibers. Furthermore, the KdV equation was shown to be completely integrable, and a brilliantly original technique, the Inverse Scattering Transform (IST), was developed to solve the KdV equation and a few other integrable pdes. But a fundamental question remained: was there a simple way of testing to see if a given nonlinear pde was integrable? By generalizing the notion of poles to singular manifolds we were able to develop such a test [9] for pdes. Furthermore the technique also provided a route to actually solving the equation, and even finding special solutions in the cases when the system failed the generalized Painlevé test. The Painlevé property work for both ordinary and partial differential equations generated a lot of papers and wide interest.

After several years in La Jolla I was offered, in 1983, a faculty position at Columbia University in the Department of Applied Physics and Nuclear Engineering (now the Department of Applied Physics and Applied Mathematics) that was keen to develop research in nonlinear science. This was a very collegial department skillfully led by C.K. Chu who, among many actions supportive of my career, successfully nominated me for an Alfred P. Sloan Fellowship and oversaw my promotion through the ranks to full professor. I built up a research group of my own, developed lecture courses in nonlinear dynamics, and wrote lots of papers and research proposals. It was while at Columbia that I developed my research interests in fluid mechanics. A topic of interest at the time was the idea (developed by the late Hassan Aref) that very simple fluid flows could exhibit chaotic behavior. This was not only an idea of practical value (important for efficient mixing of fluids) but also one that was not without controversy since a more traditional school of thought believed that fluids could only behave chaotically if they were in a high Reynolds number (turbulent) regime [10]. A beautiful experiment by one of my research students, J. Chaiken, explicitly demonstrated how such chaotic behavior could arise in a very simple flow [11]. I also had the chance to work with P. G. deGennes (while he was on a visit to Columbia) on a theory of drag reduction—the remarkable phenomenon in which a miniscule addition of polymer can significantly reduce turbulent drag [12]. I also wrote “Chaos and Integrability in Nonlinear Dynamics” which was, at the time, one of the first textbooks in this field and still sells the occasional copy to this day [13].

In 1992 I was recruited to the University of Arizona to be the Head of their Program in Applied Mathematics - position I held from 1992 – 2015. I was greatly attracted by the highly interdisciplinary culture of the Program and the university as a whole, as well the opportunity to gain administrative experience and to expand my research interests—especially in biological and biomedical areas. One of the first people I met at Arizona was Neil Mendelson from the Department of Molecular Biology. He had spent many years studying a mutant of bacterium *bacillus subtilis* that grows into long filaments that, most strikingly, repeatedly winds up on itself to form rope-like threads. This was a problem that was begging for mathematical modeling. It was also a problem that proved to be immensely difficult but, in the end, one that led to new and productive lines of research. If the initial goal was to model the bacterial threads as some form of thin elastic filament it quickly became clear that the current theories of elastic rods, which assumed only small deformations, were inadequate for our problem. Working with my colleague (and former postdoc) Alain Goriely we developed a new nonlinear theory of elastic filaments in which the filament dynamics was represented by a system of coupled nonlinear pdes [14]. One application of this new technology was to model tendril perversion in climbing plants—a phenomenon originally studied by Charles Darwin [15]. Climbing plants often produce shoots (tendrils) that have a helical, spring-like, form that attach themselves to supporting structures. Darwin observed that the helical structure could undergo spontaneous changes of handedness, e.g. a right-handed helix could change to a left-handed one (he termed this effect “perversion”) and, over a hundred years later, our nonlinear equations were able to model this phenomenon [16]. One other biomechanical problem we studied, both mathematically and experimentally, was the growth of *phycomyces blakesleeanus* – a remarkable, rod-like fungus that slowly rotates during vertical growth and, furthermore, the direction of rotation changes at different stages of its growth (in effect, another form of perversion) [17]. Among other projects, I also undertook a study of the proliferation of hematopoietic stem cells—a topic of great clinical importance. However, this effort proved to be a failure—in part because crucial experimental results that were the basis of the mathematical models I was developing were found to have been falsified. All of which goes to prove that in research one can always learn new things—both good and bad.

While the application of mathematics to biological problems had its origins in population dynamics, and was generally considered to be a small subset of applied mathematics, it has now become a major endeavor as mathematical and computational techniques are being applied to a wide range of biomedical problems. With this in mind, my colleague Tim Secomb and I undertook a long-term effort to develop graduate training programs in this growing field, and we were successful in garnering a series of training grants over the years: from the Flinn Foundation, the NSF (IGERT), and currently from the NIH. At the same time, a separate collaboration with colleagues in the Mathematics department resulted in other large training grants (VIGRE) to support graduate training in the mathematical sciences as a whole. Big grants may be a headache to write and manage but they really do make life more pleasant for students and faculty.

In 2014 I decided it was time for a complete change and left the University of Arizona in 2015 to pursue a completely different interest of mine, namely writing literary fiction [18]. One mathematical endeavor continues in the background in the form of a textbook on applied mathematics (based on the course that I and other colleagues have taught over the years) that is being written with my former colleagues Alain Goriely and Shankar Venkatramani.

[1] Berry, M. V. & Tabor, M. “Closed Orbits and the Regular Bound Spectrum,” Proc. Roy. Soc. A349, 101-123 (1976).

[2] Berry, M. V. & Tabor, M. “Level Clustering in the Regular Spectrum,” Proc. Roy. Soc. A356, 375 – 394 (1977).

[3] Notions of integrability are not without debate, but for Hamiltonian systems the concept is rigorously clear-cut. A Hamiltonian system of n degrees of freedom that possesses n integrals of motion (conserved quantities) is integrable. All orbits are confined to n-dimensional tori (in the 2n-dimensional phase space). For non-Hamiltonian systems integrability is usually taken to mean that the equations of motion are exactly soluble in some sense. If nothing else such systems cannot exhibit chaotic motion.

[4] Tabor, M. & Weiss, J. “Analytic Structure of the Lorenz System,” Phys. Rev A. 24, 2157 (1981).

[5] Paul Painlevé (1863 – 1933) was a French mathematician and politician – serving two brief stints as prime-minister. He was also an enthusiastic proponent of flight, and Wilbur Wright’s first airplane passenger. He is now best known for his classification of second-order nonlinear differential equations whose only singularities are movable poles, and the special set of six such equations, the Painlevé transcendents, that cannot be solved in terms of known functions. These transcendents show up surprisingly often in a variety of problems in applied mathematics and mathematical physics.

[6] Sofia Kovalevskaya (1850 – 1891) was a student of Karl Weierstrass who supported her career at a time when women were not able to attend university. She was the first woman in Europe to receive a doctorate in mathematics (from the University of Gottingen) and the first woman to hold a professorship of mathematics (at Stockholm University). She was awarded the Bordin Prize for her work for finding a new integrable case of the Euler-Poisson equations—which she did by looking for solutions whose movable singularities were simple poles. A semi-fictional account of her life can be found in the short story “Too Much Happiness” by Alice Munro.

[7] When integrable Hamiltonians are subject to a perturbation (such as a nonlinear coupling between modes) that renders the system nonintegrable, the invariant tori break down in prescribed ways and the orbits are able to explore phase space in a chaotic manner. This result is explained by the famous Kolmogorv-Arnold-Moser theorem.

[8] Chang,Y. F., Greene, J.M., Tabor, M. & Weiss, J. “The Analytical Structure of Dynamical Systems and Self-Similar Natural Boundaries,” Physica D, 8, 183 (1983)

[9] Weiss, J., Tabor, M. & Carnevale, G. “The Painlevé Property for Partial Differential Equations,” J. Math. Phys. 24, 522-526 (1983).

[10] There is, in fact, no basis for controversy (even though academics enjoy them). The chaotic advection of fluid particles at low Reynolds number is easily understood when it is recognized that the particle motion is seen in the Lagrangian picture of fluid motion (following individual fluid particles) whereas fluid “turbulence” is seen in the Eulerian picture (following the space-time evolution of the fluid field in the laboratory frame). When I gave a talk about our work [11] at a research institute in Leningrad, a shouting match (in Russian) erupted between two members of the audience who had strong opinions on the topic. After a few minutes, one of them stormed out of the lecture theatre and I was able to resume my talk.

[11] Chaiken, J., Chevray, R., Tabor, M. & Tan, Q. M. Experimental Study of Lagrangian Turbulence in a Stokes Flow,” Proc. Roy. Soc. A408, 165 – 174 (1986).

[12] deGennes, P. G. & Tabor, M. “A Cascade Theory of Drag Reduction,” Europhys. Letts 2, 519 – 522 (1986).

[13] Michael Tabor, “Chaos and Integrability in Nonlinear Dynamics”, John Wiley, New York,1989. Translated into Russian in 2001.

[14] Goriely, A. & Tabor, M. “Nonlinear Dynamics of Filaments I: Dynamical Instabilities,” Physica D 105, 20 - 44 (1997); “Nonlinear Dynamics of Filaments II: Nonlinear Analysis,” Physica D 105, 45 - 61 (1997); “Nonlinear Dynamics of Filaments III: Instabilities of Helical Rods,” Proc. Roy. Soc. A453, 425-411 (1997); “Nonlinear Dynamics of Filaments IV: Spontaneous Looping of Twisted Elastic Rods,” Proc. Roy. Soc. A454, 3183 - 3202 (1998).

[15] Darwin, C. R. “The Movements and Habits of Climbing Plants,” John Murray, London, (1875).

[16] Goriely, A. & Tabor, M. “Spontaneous Helix-hand Reversal and Tendril Perversion in Climbing Plants,” Phys. Rev. Letts. 80, 1564 - 1567 (1998).

[17] Goriely, A. & Tabor, M. “Spontaneous Rotational Inversion in Phycomyces,” Phys. Rev. Letts. 106, 138103 (2011).

[18] “The Lost Heifetz and Other Stories,” S&S Bookends (2018). www.thelostheifetz.com

A full list of my scientific publications can be found elsewhere.