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Curtis T. McMullen has been awarded a medal primarily in recognition of his work in the fields of geometry and "complex dynamics," a branch of the theory of dynamic systems, better known perhaps as chaos theory. McMullen has made contributions in numerous fields of mathematics and fringe areas. He already provided one important result in his doctoral thesis. The question was how to calculate all the solutions of an arbitrary equation. For simple equations it is possible to obtain the solutions by simple rearrangement. For most equations, however it is necessary to use approximation. One well-known form is "Newton's method" -already known in a rudimentary form in ancient times. For second-degree polynomials this provides very good results without exception. A key question therefore was whether a comparable method - which happened not to have been discovered - also existed for equations of higher degrees. Curtis T. McCullen's conclusion was that there is definitely no such universal algorithm for equations above degree three; only a partially applicable method is possible. For degree-three equations he developed a "new" Newtonian method and could thus completely solve the question of approximation solutions.
A further result of McMullen relates to the Mandelbrot set. This set describes dynamic systems which can be used to model complicated natural phenomena such as weather or fluid flow. The point of interest is where a system drifts apart and which points move towards centers of equilibrium. The border between these two extremes is the so-called Julia set, named after the French mathematician Gaston Julia, who laid the foundations for the theory of dynamic systems early in the twentieth century. The Mandelbrot set shows the parameters for which the Julia set is connected, i.e. is mathematically attractive. This description is very crude, but a better characteristic of the boundary set was not available. Curtis T. McMullen made a major advance, however, when he showed that it is possible to decide in part on the basis of the Mandelbrot set which associated dynamic system is "hyperbolic" and can therefore be described in more detail. For these systems a well-developed theory is available. McMullen's results were suspected already in the sixties, but nobody had previously been able to prove this exact characterization of the Julia set.
Curtis T. McMullen (born 21 May 1958) is visiting professor at
Harvard University. He studied in Williamstown, Cambridge University
and Paris before gaining a doctorate in 1985 at Harvard. He lectured
at various universities before becoming professor at the University
of California in Berkeley. Since 1998 he has taught at Harvard. The
Fields Medal is his tenth major award. In 1998 he has been elected to
the American Academy of Arts and Sciences.
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