The Fred and Lois Gehring Special Year in Complex Analysis

.拟议活动的摘要。 2001/2002年密歇根大学的弗雷德和洛伊斯·格林特别年致力于复杂分析，复杂动力学及其相互作用。密歇根大学有几位复杂分析和复杂动力学方面的专家，此外，还将有几位资深长期访问者和几位在这些领域工作的初级教师，以及大量的短期访问者。此外，还将举行两次会议，突出最近的发展，并邀请来自世界各地的专家。 2.拟议研究摘要。 自然界和人类社会的许多现象，如天气、股市等，都是混沌的，难以预测和分析。事实上，任何时候三个或三个以上的实体相互作用，行为往往具有混乱的特征，因为其中任何两个实体之间的相互作用不断受到第三个实体的干扰，这些第三人称干扰的影响积累并适得其反。大型系统超出了我们完全计算的能力。人们只能完全精确地理解低维系统，然后才能希望从这些系统中推断出在更大的系统中可能发生的现象。复杂动力学为低维环境提供了可用于此类分析的大多数工具。复变分析理论为复杂动力学提供了强有力的方法。同样令人兴奋的是，复杂动力学为复杂分析提供了工具。因此，让这些群体在很长一段时间内聚集在一起，应该会对这两个领域产生强烈的影响。 复动力学的主要工具之一是（复数）势理论，它是复分析中的一个关键领域。从势流理论出发，利用绿色函数，通过复Monge安培算符，可以得到不变电流和动力学测度。这里的一个基本问题是，在某些情况下，由于需要乘以分布的事实，很难定义这个算子。小林双曲面是另一个关键概念。不变区域是小林双曲的，会产生非混沌行为，因为迭代是正规族。然而，很难确定复流形中哪些区域是小林双曲的。 关于黎曼曲面在C^2中嵌入问题的最好结果使用了复动力学技巧，但仍有许多开放的情况。而这些只是这个庞大的研究小组将要调查的几个课题

. 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