How Math is Like a Ladder to the Moon

For unsolved time-dependent processes like the motion of fluids, I want to try to find a few important parameters and then successively add information to build up a better and better picture—and all of this using the methods of algebraic topology.”

Published May 1, 2010

By Dennis Sullivan
As told to Abigail Jeffries

My interest in mathematics began when I was nine or ten years old. I liked to think about ideas and do math puzzles, and I noticed there was some structure of prediction. Later on, I came to know this was called statistics, related to chance events. This ability to predict amazed me. I was a late bloomer academically in the sense that I didn’t have any pressure to study when I was growing up. In college I got back into academics again and made a fresh start. I was able to attend Rice University in Houston, which at the time was like a scaled-down Caltech. I rediscovered my academic self there after being a quasi-juvenile-delinquent, running around working on hotrods!

There’s an interesting theory that, among mathematicians for example, a person may discover they like mathematics and have a strong aptitude. They get so involved in it that their personality development is arrested at that point. They just stop caring about the finer points of their finishing, you might say. I’ve seen this in every one of my six children. They’re like little mathematicians or little scientists, then for some reason that usually washes out. They get interested in other things. For some of us, like me, it didn’t wash out.

After receiving my BA from Rice in 1963, I received my doctorate in 1966 from Princeton, where I wrote my thesis on triangulating homotopy equivalences. This work became part of surgery theory, which describes a way of manipulating mathematical spaces called manifolds.

Analyzing Mathematical Concepts

Almost everywhere you look, when you start to analyze a mathematical concept, it’s as if there’s this tightly woven Oriental rug covered by dried leaves. You sweep away the leaves, and you start finding out about it, and everywhere you look the beautiful tapestry is there to be uncovered. You can sweep anywhere and find it underneath, with all sorts of fantastic structure.

I was a member of the Institut des Hautes Études Scientifiques in France from 1974 until 1997. IHES was modeled after the Institute for Advanced Study in Princeton. It was a wonderful place for people like me who like to work on math all the time. I had no duties. I was given an office and a research environment with a library and a secretary. My colleagues were the best in the world, and I enjoyed the steady flow of visitors. I just worked on math. It was paradise.

While at IHES, I did some mentoring of people who had recently received their PhDs or who were on sabbatical and focusing on their research. The atmosphere was wonderfully collegial. Our lunches would start at one o’clock and we’d sometimes still be sitting there until tea at four o’clock, writing ideas on the backs of napkins.

During that time, and due to the six-month academic year in France, I was able to take advantage of an offer for the Einstein Chair at the City University of New York Graduate Center in 1981. So, I split my time between IHES and CUNY until 1997. I would move to each place for about six months. When my fourth child was in first grade, the back-and-forth schedule wasn’t tenable, so I substituted my current position as professor at SUNY Stony Brook for the position at IHES.

Receiving the National Medal of Science

The awards I have received have all meant a great deal, but the National Medal of Science in 2004 was special. At the White House we met George Lucas (which thrilled my 11-year-old son); the developer of the liver transplant which had saved the life of a relative one year before; and the inventor of the first computer games, Simon and Pong. All of us were receiving either science or technical awards from President Bush.

The Wolf Prize in Mathematics this year was for Shing-Tung Yau’s work on curved spaces and for my work in algebraic topology and conformal dynamics. Topology is an approach that allows one to ask scientific questions that are more qualitative in nature, such as whether or not a system would evolve and then come back to its original configuration, or whether there are cycles, and if so how many.

These are questions that can be expressed in words, without formulae, and they often involve integers or whole numbers or counting. By moving from formulaic considerations, which turn out to be very complicated, to a place where you try to define things in such a way that they can be counted, a problem becomes easier to understand.

You can study complicated spaces of many dimensions using algebraic topology. Think of the three-dimensional space we live in as a large hotel full of rooms, little boxes next to each other that fill up the entire space. Algebraic topology breaks down the hotel space we might otherwise think of as being continuous into all these little boxes put together.

Discrete Mathematical Descriptions

You can list these boxes on a computer or in your mind and give them names, and you can record how pairs of boxes relate to each other. If you give me the names of all the boxes and you tell me their relationships, then I can assign a purely algebraic description to each box and start applying algebraic topology to reconstruct many of the properties of the overall space.

We need discrete mathematical descriptions like these that can be inserted into a computer computation in an efficient way. Even though computers are very fast, it’s easy to generate problems that are much too big for them. If you’re trying to apply computers to study the flow of blood around the heart in the human body, this process is happening in space with a lot of little particles moving around. You cannot input an accurate assemblage of points and the way they all interact and ask a computer to compute that; it’s overwhelming.

In my work with conformal dynamics, I consider dynamical systems (processes that evolve in time) in small dimensions to make them more manageable. Some processes are reversible, meaning that they can run backward and forward, but others, such as a fire, are not. These non-reversible processes are more complex, but you can study them in very small dimensions. They can be studied in discrete time to reveal a very interesting structure that’s beautiful and that can also be analyzed. In the world of conformal geometry, you see the amazing fractal patterns of the Mandelbrot set, for example. It’s extremely interesting mathematically, and the incredible intricacy has an explanation via conformal dynamics.

Technology to Prove Math Theorems

I started studying this around 1980. There were primitive computers then that could draw figures, and you could plot this out and see all sorts of fantastic patterns. Then, we started trying to prove things about them. You could observe them, but to prove them as math theorems required technology, new ideas, and research.

That’s what I was working on at IHES. To draw a comparison to music, this structure is as breathtaking as if you had only known rhythmic drums and suddenly I show you Mozart. It was totally unexpected that such an incredibly beautiful structure should be there in such a simple problem.

Math is actually a very robust field. There are a lot of new ideas coming forth, and a lot of progress is being made. Yet in many ways we’re still at the beginning. Sometimes you use a problem as a North Star to guide you.

You don’t actually solve it because it’s often not that tractable yet, but the first steps you take to solve it lead you to other steps. You find other things, other structures. It’s like building a ladder to the moon: you have to build the steps of the ladder, always moving out and up.

The Next 15 Years

I expect many new developments in the next 15 years. Complex data will be attacked with all of the tools available, and we will see ideas from physics gain in influence. New technological developments have already impacted my work. The conformal dynamics used computer computations to find out what to prove. This was not really possible before 1980.

We still need ways to do fluid simulation, and this was pointed out at the White House event by George Lucas’ animators. For unsolved time-dependent processes like the motion of fluids, I want to try to find a few important parameters and then successively add information to build up a better and better picture—and all of this using the methods of algebraic topology.

I want to keep working on this algebraic model of space and its geometry. That’s my goal in a nutshell, and I hope my work will be useful in the sense that people can apply it.

This story originally appeared in the Spring 2010 issue of The New York Academy of Sciences Magazine.