The Fred and Lois Gehring Special Year in Complex Analysis

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

. Abstract of the proposed activity. The Fred and Lois Gehring Special Year at the University of Michigan 2001/2002 is devoted to Complex Analysis, Complex Dynamics and their interaction. There are several Experts in Complex Analysis and in Complex Dynamics at the University of Michigan, and in addition there will be several senior long-term visitors and several junior faculty hired in these areas and a good number of short term visitors. Also there will be two conferences highlighting recent developments and bringing in experts from around the world. 2. Abstract of Proposed Research. Many phenomena in nature and human society, such as the weather, or the stock market, are chaotic and hard to predict and analyze. In fact any time three or more entities interact, the behaviour tends to have chaotic features as the interaction between any two of them is constantly interfered with by the third and the effects of these third person interferences accumulate and backfire. Large systems are beyond our ability to calculate completely. One can only understand with complete precision lower dimensional systems and then one can hope to infer from these which phenomena can happen in larger systems. Complex dynamics provides the low dimensional setting with the most tools available for such analysis. The theory of complex analysis provides powerful methods for complex dynamics. It is also exciting that complex dynamics provides tools back to complex analysis. So getting these groups together for an extended period should have strong impact on both areas. One of the main tools in complex dynamics is (pluri)potential theory, which is a key area in complex analysis. Using Green functions from potential theory one can get invariant currents and measures for the dynamics via the complex Monge ampere operator. A basic problem here is that in some cases it is difficult to define this operator due to the fact that one needs to multiply distributions. Kobayashi hyperbolicity is another key concept. Invariant regions which are Kobayashi hyperbolic gives rise to nonchaotic behaviour because iterates are then a normal family. It is however difficult to decide which regions in complex manifolds are Kobayashi hyperbolic. The best results on the embedding problem for Riemann surfaces in C^2 use complex dynamical techniques, but there are many open cases still. And these are only a few of the topics that will be investigated by this huge group of researchers