Phenomena such as the weather are extremely complicated examples of dynamic systems, because countless factors influence each other there. In order to examine the subject, however, one can turn to even the simplest examples, because they also have exciting properties: for example, the dynamic system, which is determined by the equation f(x) = x2 – 1 arises. You choose a starting value x0 and insert this: f(x0). Now insert the result back into the equation: f(f(x0)), and this is repeated over and over again. When x0 = 2, the resulting values continue to increase. However, if you start with x0 = 1, we get a periodic sequence: 0, -1, 0, -1, 0, -1, … By marking the starting values that lead to periodic patterns, one can get amazing fractals like the Mandelbrot set.
Sullivan addressed the question for which systems one obtains periodic results and for which the values tend to infinity. To do this, he used methods from geometry, which he applied to the complex number plane – i.e. a kind of Cartesian coordinate system in which the y-component represents multiples of roots of negative numbers. He thus succeeded in proving a conjecture dating back to the 1920s.
Like a periodic pendulum
While working in this field, Sullivan constantly commuted between Paris and New York, as he was employed at universities in both cities. Perhaps this periodic back and forth inspired him in his work on dynamical systems. Eventually, in the late 1990s, he accepted a full-time position at the State University of New York, ending the commute. There he began to take an increasing interest in his original passion, topology.
In 1999, together with his colleague Moira Chas, he succeeded in defining a new topological invariant. This is a characteristic quantity (such as a number, polynomial, or matrix) that does not change when you deform a manifold. Such invariants help in categorizing these objects.
The question now arises as to whether “the true virtuoso” will remain true to the periodic pattern in his life and perhaps turn to dynamic systems again in the years to come. In any case, there is no lack of unsolved problems in this area.