Ai-Hua Fan is at present Professor at University of Picardie Jules Verne in Amiens, France. He will be Visiting Professor at the Centre for Mathematical Sciences, Lund University.

One of the subjects studied by the theory of dynamical systems is the exploration of mathematical models of chaotic phenomena, i.e. situations in which small changes in the initial conditions cause large deviations in the evolution of the system. This property makes it impossible, for example, to make long-term weather predictions, since weather is a chaotic phenomenon. Some degree of order exists in such systems though. We know with a high degree of confidence that a snowfall in Sweden is highly unlikely in July.

The study of regular patterns in chaotic systems is called ergodic theory. It was introduced over a hundred years ago by an Austrian physicist, Ludwig Boltzmann, who laid the foundations of statistical mechanics while studying the motion of gas particles.

Subsequently, the ergodic theory has been extended to the study of long-term evolution of ergodic averages in dynamical systems. For example, mathematicians were able to show that for a certain type of chaotic systems the initial conditions had no bearing on the development far into the future – the system “forgets” its past.

The methods needed to solve this type of problems are found in the intersection of many areas of mathematics, including harmonic analysis, probability theory, statistical mechanics, number theory, and the theory of fractals. The research group in Lund hopes to benefit from professor Fan’s broad mathematical expertise in order to make further progress in its studies of ergodic theory.

Photo Mathematisches Forschungsinstitut Oberwolfach/ Renate Schmid