The Fred and Lois Gehring Special Year in Complex Analysis

． 提议活动的摘要。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雇用的初级教师，以及大量的短期访问者。此外，还将举行两次会议，重点介绍最近的发展情况，并邀请来自世界各地的专家。2. 拟研究摘要。自然界和人类社会中的许多现象，如天气或股票市场，都是混乱的，难以预测和分析。事实上，任何时候三个或更多的实体相互作用，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方的干扰，这些第三方干扰的影响累积并适得其反。大系统完全超出了我们的计算能力。人们只能完全精确地理解低维系统，然后人们可以希望从这些系统中推断出哪些现象可以发生在更大的系统中。复杂动力学为这种分析提供了低维设置和大多数可用工具。复杂分析理论为复杂动力学研究提供了有力的方法。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，让这些群体长时间聚在一起，应该会对这两个领域产生强烈的影响。复数势理论是研究复杂动力学的主要工具之一，是复杂分析的一个重要领域。利用势理论中的格林函数，通过复蒙日安培算符可以得到动力学的不变电流和测度。这里的一个基本问题是，在某些情况下，由于需要乘分布，很难定义这个算子。小林双曲线是另一个关键概念。小林双曲的不变区域会产生非混沌行为，因为迭代是一个正常族。然而，复杂流形中哪些区域是小林双曲是很难确定的。C^2中黎曼曲面嵌入问题的最佳结果使用了复杂的动力学技术，但仍有许多开放的情况。这些只是这个庞大的研究团队将要研究的主题中的一小部分

. 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