Martin Golubitsky is Distinguished Professor of Mathematics and Physical Sciences at Ohio State University, where he serves as Director of the Mathematical Biosciences Institute. He received his Ph.D. in Mathematics from M.I.T. in 1970 and has been Professor of Mathematics at Arizona State University and Cullen Distinguished Professor of Mathematics at the University of Houston. He is a former president of the Society for Industrial and Applied Mathematics.
Dr. Golubitsky works in the fields of nonlinear dynamics and bifurcation theory, studying the role of symmetry in the formation of patterns in physical systems and the role of network architecture in the dynamics of coupled systems. His recent research focuses on some mathematical aspects of biological applications: animal gaits, the visual cortex, the auditory system, and coupled systems. He has co-authored four graduate texts, one undergraduate text, two nontechnical trade books, and over 100 research papers.
Ivars Peterson: Did you become interested in mathematics at a young age? What attracted you to mathematics?
Martin Golubitsky: Yes. In school I liked the internal consistency of calculations
and the fact that math seemed like a game. Once I understood the structure of a calculation, which often took a while, I could set problems for myself and play with the ideas.
I was also lucky. My father owned a laundry and dry cleaning store where I worked as a clerk. When business was slow my father liked setting puzzle problems for me that he remembered from when he was a student.
IP: What people or experiences most influenced the direction of your studies? your subsequent career?
MG: I feel fortunate that at all stages of my student days I crossed paths with some very fine teachers and professors. The earliest of these that I remember clearly was in seventh grade. I failed my first quiz in middle school math. The problems were simple calculations and I had just written down the answers. The teacher, Cecile Brav, gave me a zero because I had not shown any of the intermediate calculations—basically accused me of cheating. Needless to say, I thought this was really unfair and I was quite upset. However, during the semester it became clear that I had not cheated, and Miss Brav became a great supporter. In ninth grade she arranged for me to work at my own pace, to do extra work and projects, and to enjoy what I was doing.
IP: How would you describe your main research areas? Why are these areas particularly exciting to you?
MG: Bifurcation theory studies the ways in which the number and type of solutions to an equation change as parameters are varied. The answer to this type of question depends on properties of the equations typified by the questions: How many parameters do the equations have, or do the equations have symmetries?
I am fascinated by the fact that, once the mathematics is understood, it can be used to determine the structure of solutions to mathematical models in many different applications. I was lucky to be a graduate student just when the mathematics needed to study these questions was being developed (by Thom, Malgrange, Mather, and others) and by the fact that Victor Guillemin decided to give an accessible course on this work.
Soon after this, the mid to late 1970s, the "catastrophe theory" controversy blew up. Christopher Zeeman and Rene Thom caught people's imagination by discussing some of the many ways the same piece of mathematics could be used to interpret, and even to anticipate, results in a variety of application areas. Zeeman, in particular, was a charismatic persuasive lecturer—and his claims for the theory made it into the popular press.
Many people, including many excellent mathematicians, decided
that the claims being made for catastrophe theory were too extreme and misleading, and a somewhat acrimonious public debate ensued. The mathematics behind catastrophe theory, called singularity theory, was never challenged, nor was the fact that the theory did provide new methods to analyze solutions to equations. I characterize the situation with the phrase "attempting to throw out the baby with the bath water." Around this time Dave Schaeffer, Ian Stewart, I, and many others began translating singularity theory ideas to the context of bifurcation theory and applying the new mathematics.
This sequence of events allowed us to find out about some of the fascinating areas that researchers in engineering, physics, and biology were exploring. We had techniques that could sometimes help the applied scientists understand solutions to their models. They provided us with glimpses into issues they found exciting and, by doing so, forced us to ask different mathematical questions—a very satisfying symbiotic relationship.
IP: Initially, was your interest in singularity theory and bifurcation theory mathematical, or were you always thinking of applications?
IP: Some mathematicians criticized catastrophe theory as being overblown and overhyped. Did the same thing happen in chaos theory, and how did that affect you?
IP: How did the aesthetic dimension of your work arise?
IP: How did your interest in biological systems come about?
IP: How do you envision the relationship between mathematics and biology?
MG: Mathematics and mathematicians have much to offer biology. Many people believe that as biology becomes more quantitative—and it is doing so at a great rate—mathematical-style thinking will become more important in biology. Mathematicians don't have to become biologists, but they do need to learn enough of the biology to be able to talk to and work with biologists, and to translate the types of questions the bioscientists are asking into mathematical models and mathematics. This is a project for a new generation of mathematicians.
It is a fact that the biological and medical sciences are huge (certainly in terms of the number of practitioners) compared to the mathematical sciences. The largest math meeting, the joint AMS and MAA winter meeting that is aimed at all mathematicians, attracts at most 5,000 people; whereas the typical annual neuroscience meeting attracts well over twice that number. So there is a need and an opportunity for many more mathematically trained researchers in the biological sciences.
It will be a slow process (just one small project at a time), but I am confident that the mathematical and computational sciences will continue to make significant contributions to the biosciences, and that this contact will lead to new areas for mathematical exploration.
My favorite example of this process occurred in neuroscience. Hodgkin and Huxley received a Nobel Prize for their mathematical model of electrical conduction in the giant axon of a squid, published in 1952. The model showed how voltage spiking and bursts of spiking occur in a neuron; it assumed the existence of ion channels that were not isolated experimentally until many years later. The Hodgkin-Huxley model has revolutionized (parts of) theoretical and experimental neuroscience. Also many years later, John Rinzel observed that bursting is due to the multiple time scales in the model that are introduced by the ion channels. The mathematical study of multiple time scale systems is now a popular and productive area of mathematics research, and it came to such a state because of the need to understand a basic issue in biology.
IP: Looking toward the future, what do you see as key questions or concerns that need to be addressed in your area of research?
MG: In the near term mathematical models need to be developed and analyzed in many areas of the life sciences, and the results of these models compared with experiment. In the longer term the structure of these models will need to be understood (many biological models have a feel to them that differs from models in physics and engineering). How
do we understand these differences?
How do we understand emergent phenomena? For example, neurons are not isolated—they occur in large networks that have a structure to them. How does network structure influence the types of dynamics that occurs in a brain, and how do the spiking and bursting of individual neurons lead to brain function? It's a wonderful puzzle.
IP: How important has writing and communicating your work been to you?