An interview with James A. Yorke

By Tim Sauer
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AN INTERVIEW WITH


JAMES A. YORKE

-- by Tim Sauer (George Mason University)
January 8, 2004 in Phoenix
Jim Yorke, 2003
Jim Yorke, 2003

A brief biography of Professor Yorke

James Alan Yorke was born August 3, 1941 in Plainfield, NJ. He went to Columbia University as an undergraduate, and received the Ph.D. degree in Mathematics from the University of Maryland in 1966. Yorke is a member and former director of the University of Maryland's Institute for Physical Sciences and Technology, has joint appointments with the Mathematics and Physics departments, and holds the title of Distinguished University Professor.

He has published over 300 papers, including the often-cited works Period Three Implies Chaos with T.Y. Li (1975), and Controlling Chaos with Ed Ott and Celso Grebogi (1990). He has supervised 30 Ph.D. students in mathematics and physics, and has co-authored three books on aspects of chaos and a monograph on epidemiology. In 2003 he received the Japan Prize, along with Benoit Mandelbrot, for fundamental work in the understanding of complex systems.

His current research topics include HIV epidemiology, better methods for determining genomes, weather prediction and data assimilation, topics in dynamical systems, and understanding computer network traffic patterns.

Yorke and Mandelbrot at the Japan Prize ceremony, 2003
Yorke and Mandelbrot at the Japan Prize ceremony, 2003

The interview

(Y = Jim Yorke, S = Tim Sauer)

S: How did you get interested in science?

Y: Well, I visited the Hayden Planetarium in fourth grade, and from then on I wanted to become an astronomer. It was hard to believe there was anything more interesting than what was going on out in the universe. I loved learning about planets and seeing the photos from the great observatories using long exposure times, though I had no interest in looking through small telescopes, which simply could not compete. Perhaps that was the beginning of a pattern: liking the results of physical experiments but not wanting to carry them out.

S: Who were your mentors?

Y: My mentors were not my teachers in school. They were great mathematicians and scientists whom I read about or who wrote wonderful books, like Norbert Wiener's popular books "Ex-Prodigy", "I Am a Mathematician", and "The Human Use of Human Beings". I read these in high school. In the latter Wiener talks about feedback control, entropy, non-equilibrium processes, all without equations. I learned about Maxwell's demon and his theory of gas, where colliding atoms play a key role. I think that was the beginning of my understanding of chaos. Certainly Maxwell understood the chaos. Brian Hunt and I wrote a paper entitled "Maxwell on Chaos". Another great book from my high-school days is "One Two Three ... Infinity" by George Gamow, with a lucid discussion of countable and uncountable sets.

High school yearbook photo Captain, Pingry High track team, 1958
High school yearbook photo Captain, Pingry High track team, 1958

S: How helpful was your early formal education?

Y: My elementary school teachers' primary interest was in accuracy of computation, which relegated me to the half of my eighth-grade class that learned to read gas meters and write checks. Somehow I caught up and managed to get accepted into Columbia anyway. Teachers can be really bad at evaluating a student's potential.

S: So how good are you at evaluating potential?

Y: No better than anyone else, but I am very concerned about potentially outstanding students who might fall through the cracks. I had a friend in high school who was a great problem solver, and he convinced me to go to Columbia. He was a year ahead of me though a bit younger. Throughout high school and college we discussed hundreds of problems and he was always faster at solving them than I was. To put that into context, I was pretty fast, since I was the top Columbia contestant on the Putnam exam two years in a row. My friend got straight A's in advanced math courses and went to graduate school at a famous east coast university. I went to Maryland and when I got my Ph.D., he was still struggling without a dissertation problem. I figured the reason he could solve all those problems faster than I could was that he thought like I did, but was better at it. I made one phone call to him, and told him about a problem that was my kind of problem: simple to state and probably very difficult to solve, but didn't require a whole lot of background.

He turned this one conversation into a Ph.D. dissertation and got his degree with the help of a very junior faculty member who got interested in the question. By the time he graduated, he was teaching four courses per semester at a local community college and continued to do so after getting his degree.

He has since died of cancer. So I have a question for you: What was that graduate school doing? Should we evaluate graduate schools by how many outstanding mathematicians they graduate, rather than how many potentially outstanding mathematicians they destroy? To this day I have a major concern about potentially outstanding students failing. People need help at various points in time and without it they can fail.

S: What made you want to become a math major?

Y: I hated doing physics lab reports, and once a week I had to stay up all night to get it done on time. I could deal with that, but the next semester they wanted two lab reports every two weeks. This was much worse. I couldn't stay up for two nights straight and so I dropped the course that was required for physics majors.

S: Why did you choose Maryland for graduate school?

Y: When I was leaving Columbia, I already had a love for mathematics but wanted to be sure that the graduate school I went to avoided the severe, motivation-free abstraction that was characteristic of Columbia at the time. I got similar teaching assistant offers from Cornell and Maryland, but at the time, Cornell's program was much like Columbia's. Even then, Maryland was strong in interdisciplinary mathematics.

S: Did you find college and graduate school more helpful than your early education?

Y: I learned a great deal in college and graduate school, but I was never systematic about learning the material and in college may have had a B average in math and physics, and worse in other subjects. As a student on a full scholarship, that was dangerous. When I got to Maryland I immediately took and passed the Ph.D. qualifying exam. After that, grades were fairly irrelevant. It has always struck me that you might understand a course well enough to write a publishable paper about your ideas in the subject, but that doesn't mean you've learned the material systematically enough to get an A.

There's a great conflict for young people, whether to be a mathematician or a student. Give a student a problem that won't count for any grade, and he or she will say, I don't have time, I have so many other homework assignments to complete. So when I see a student with a 4.0 average I immediately worry that he or she might lack the motivation to be creative when grades are not in question. Of course, some 4.0 students are creative despite their GPAs.

It reminds me of something I heard when I was in high school, that excellent scientists generally have poor memory, and have to rethink the same questions over and over, and thereby learn to think. The really outstanding scientists learn to think despite having a great memory. Myself, I don't have the memory of a 20-year-old, and I never did.

Jim and one of his pet elephants 1998
Jim and one of his pet elephants 1998

S: Your collaborator Ed Ott says that sometimes you come up with a way of thinking about a problem that seems totally bizarre, but after a while it dawns on him that it's the only way to look at the problem. Is this a conscious effort on your part? And is this a hindrance in everyday life?

Y: I have often found it conversationally awkward that I make connections between ideas or events that other people feel are unrelated. They think I am changing the subject. Maybe they are right. But for mathematical or scientific questions I feel that whenever we have a question, or an answer, we have to try harder to ask whether we have the right question. If we really understand the question, we are a large part of the way to a solution. Understanding a question well is often making connections between apparently unrelated ideas. I tell the students they should spend half their time asking what the right question is, and not the first half. And even after they have a result, they must ask: what is the precise question that it answers?

S: Do you have a working style? If so, what is it?

Y: I'll give you an example of one of our styles. We published the first paper on fractal basin boundaries, aside from the Julia set results. But we were far from the first to understand them. People like Cartwright and Littlewood, the Berkeley and Moscow schools of dynamics, and people like Mark Levi could have given you a half-hour lecture on fractal basin boundaries with 5 minutes notice. Much of what they wrote about was dynamical systems that happened to have fractal basin boundaries. But they talked about dynamics in a more general way, in which closed invariant sets were the key concept, and basin boundaries are just one unmentioned example.

We felt that the basin boundaries would be very interesting to physicists, and our problem became how to write an original paper on this topic that people knew well. We wanted to reformulate ideas that mathematicians knew, in ways that were more useful to scientists. If they don't know what the boundary of a basin is like, it is impossible to ask how stable under perturbations an attractor is. So we created a concept of uncertainty dimension, a dimension of the boundaries that could actually be computed and measured by a physicist. Physicists want concepts that can be quantified.

Another example of reformulating ideas is illustrated in our most quoted paper. Our goal was to take the ideas of chaos and mix in well-known ideas of control theory. Ed Ott, as a part-time electrical engineer, was quite familiar with control theory, as was I from my college days. But we had to formulate the ideas in a manner that would be useful to scientists. We did so in a way in which equations were not written down, but yet scientists could carry out the ideas in the lab. The mathematicians couldn't see what it was about the paper that interested the physicists or how it was new. The subtleties of what makes a paper valuable in one field can be completely lost on experts in another. My efforts in physics have always been heavily dependent on the insights of my physics collaborators, especially Ed Ott and Celso Grebogi.

S: How do you write a paper for both mathematicians and physicists?

Y: You don't. One should pick an audience and write a paper for that audience and tell a story tuned to their ears. If the results are of interest to two audiences, then you might write the paper twice. But a paper that is aimed at two audiences is most likely to miss both audiences, and fail.

Jim with David Broomhead, 1997 Maryland-Penn State conference 2002
Jim with David Broomhead, 1997 Maryland-Penn State conference 2002

S: What was the origin of the "Period Three Implies Chaos" paper with T.Y. Li?

Y: It was totally inspired by an effort to explain the irregular behavior that Ed Lorenz was observing. While he mostly talked about a three-dimensional system of ordinary differential equations, he also showed how in some sense you could reduce the interesting part to a one-dimensional return map. He wrote down something like a tent map. So the paper we wrote explained how complicated behavior could be proved for such processes.

S: Why were you so fascinated by periodic orbits?

Y: One thing the period three implies chaos theorem says is that a continuous map on the line with a period-three orbit must have orbits of all other periods. This aspect of our paper was a special case of an earlier result of Sharkovsky. But actually I rather disliked periodic orbits as a way of understanding chaotic behavior. It's just one leg of the elephant, so to speak. Our paper spends much more time on the mixing behavior of one-dimensional maps. How you could follow initial conditions and they move apart and come closer repeatedly, that's another leg of the elephant. Mixing allowed us to talk about uncountable sets, while the periodic orbits were countable and, therefore, almost nothing.

S: Was that your first paper on chaos?

Y: Actually my first foray into the field of such maps was with Andy Lasota. We looked at invariant measures for piecewise-expanding maps. I remember feeling confused as to how would I explain what my area of research was: sometimes we looked at operators in Banach spaces to explain the measure theory of such maps, and other times at the point-theoretic properties of periodic orbits, which only depended on the continuity. Now there is a field of one-dimensional maps, but back then these results seemed unrelated. To me they all aimed at understanding complicated behavior. They were just different legs of the elephant. In these two works, the measure theory and topology of one-dimensional maps, I had two different collaborators. Throughout my career, I have found it immensely beneficial to work with people who had excellent ideas to mix with my ideas. Two good ideas on a problem are much better than two ideas on different problems. By the way, I think it's hard to tell how many legs the elephant has.

S: What makes a good question?

Y: From a trivial point of view, a good question is one for which you can give a good answer. But from another point of view, the point of research is to put questions together with answers, and evolve the questions and answers together until you get a great match: Co-evolution. Andrew Wiles had a monumental achievement in solving Fermat's Last Theorem. But I fear it will strengthen the concept that students have that you find a problem and then work on it for years.

That is rarely how research is done. Students must learn to co-evolve questions and answers. I look for questions that give an interesting answer, but then I hope to be able to build on the question with my collaborators and follow the question in new directions and let the problem grow. It may start out as a simple problem, but after several stages of evolution it becomes more interesting. As it evolves, I personally keep the audience in mind. What can I tell the audience about this problem that I feel they need to know, or that I feel will surprise them?

S: What do you tell a student who is stuck on a problem?

Y: I tell students that when they think about a problem, they shouldn't take notes. If they don't understand what's going on, they should start over from the beginning each time, hopefully having forgotten their wrong approaches. Taking notes allows them to remember their wrong approaches. Too much writing interferes with the thinking. I advise mathematicians to buy a dog who will need long walks, and in extreme cases, several dogs.

You should never work nonstop on a problem because you need time to forget your wrong approaches, which is another reason why people with lousy memories can be excellent scientists.

Maryland genome analysis group, 2003 Advisors Kalnay, Hunt, Ott, Yorke with new Ph.D. D.J. Patil
Maryland genome analysis group, 2003 Advisors Kalnay, Hunt, Ott, Yorke with new Ph.D. D.J. Patil

S: What is your advice to young people interested in mathematics?

Y: One view of mathematics it that it is based upon the three core legs of topology, analysis and algebra. That view is valid in a certain sense, but I prefer to ask how mathematics interacts with the world, and draw the core from that. The three legs of this interaction tripod are differential equations, numerical methods, and probability and statistics. Even for someone writing a dissertation in topology, algebra, or analysis I feel it is important to know about the legs of the interaction tripod. But we allow our graduate students to get PhD's without having taken any courses in the latter three areas. We require more of our undergraduates. We sometimes allow graduate students so much freedom that they can hurt their professional careers and their ability to get a job at universities where non-math students must be taught. I recommend to students that they take one course at least in each of the three areas of the second tripod.

S: What do you see as the future of dynamical systems?

Y: I think everybody, including me, is rather poor at foreseeing the future. Generally we can't even see the present. For example, there are many seminal papers in dynamical systems that had trouble getting published. Sheldon Newhouse tells the story that he refereed Hénon's original paper for David Ruelle, the editor, and thought it wasn't rigorous, it wasn't math, it wasn't physics, and he recommended rejection. But Ruelle liked the paper and fortunately overruled him. Newhouse has a record of wonderful discoveries in dynamical systems but missed the value of that paper. He couldn't see the future when it was in front of him. And I think we are all that way. Instead of trying to predict the future, I look for ideas that can surprise people, or interest them. I am not the kind of mathematician who tries to develop a general theory of something. Neat examples are good enough for me.

S: What is it with wearing the red socks?

Y: Nothing really, just a reminder to think differently.

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