Computation of capacity and automorphisms of the spectral ball

该研究程序由两个不同的子程序组成：容量的计算和谱球自同构的分类。 容量是位势理论的中心概念，在复分析、逼近论和数论中有许多重要的应用。计算一般紧集的容量是一个非常困难的问题。然而，最近的算法开发的兰斯福德和我提供了一个实用的方法来计算对数容量的一大类紧凑的子集的飞机。本研究计划的主要目标之一是开发，扩展和改进这种新方法。这里有一个简短的项目列表。 1-将该算法扩展到其他容量，例如Riesz容量，加权容量（在R^n中）。 2-放松算法收敛的假设，特别是HCP。 3-利用线性规划的特殊形式来提高算法的性能。 4-在不连通集的情形下，检验反例的强等周不等式。 5-为了提高特殊集合的容量的精度，例如Cantor集合。 6-开发了一个容量自动计算软件。 在Ransford-White猜想提出17年后，M_n的谱单位球自同构的分类仍未完成。虽然最近的进展已经取得了进展，该猜想仍然是开放的，即使在情况下n=2。我们建议一个实验的方法来解决一些基本的问题。此外，我们还想利用上同调理论中的问题的重新表述。 理解谱球的自同构的问题源于谱Nevanlinna-Pick问题。这一点反过来又来自控制理论领域的鲁棒稳定性问题。

This research program is made of two distinct subprograms: the computation of capacity and the classification of the automorphisms of the spectral ball. CAPACITY is a central notion in potential theory with many important applications in complex analysis, approximation theory and number theory. Computing the capacity of a general compact set is a very hard problem. However, the recent algorithm developed by Ransford and myself offers a practical way of computing the logarithmic capacity of a large class of compact subsets of the plane. One of the main objective of this research program is to exploit, extend and improve this new method. Here is a short list of projects. 1- To extend the algorithm to other capacities, for example, Riesz capacities, weighted capacities (in R^n). 2- To relax the hypotheses under which the algorithm converges, especially HCP. 3- To enhance the performance of the algorithm by exploiting the special form of the linear programs involved. 4- To test the strong isoperimetric inequality for counterexamples in the case of disconnected sets. 5- To improve the precision on the capacity of special sets, for example the Cantor set. 6- To develop a software for automatic computation of capacity. The classification of the AUTOMORPHISMS OF THE SPECTRAL UNIT BALL of M_n is still unfinished, 17 years after the Ransford-White conjecture was stated. Although recent progress have been made, the conjecture is still open even in the case n=2. We suggest an experimental approach to some of the underlying questions. Moreover, we would like to exploit the reformulation of the problem in the theory of cohomology. The problem of understanding the automorphisms of the spectral ball arises from the spectral Nevanlinna-Pick problem. This one in turn comes from a question of robust stability in the field of control theory.