Computation of capacity and automorphisms of the spectral ball

谱球容量和自同构的计算

• 批准号: 298440-2009
• 负责人: Rostand, Jérémie
• 金额: $ 1.02万
• 依托单位: Université Laval
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=571495
• 关键词: Computation; capacity; automorphisms; spectral; ball

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.

该研究程序由两个不同的子程序组成：容量计算和谱球自同构的分类。 容量是位势论的核心概念，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，兰斯福德和我自己最近开发的算法提供了一种计算平面的一大类紧凑子集的对数容量的实用方法。该研究计划的主要目标之一是开发、扩展和改进这种新方法。这是项目的简短列表。 1- 将算法扩展到其他容量，例如 Riesz 容量、加权容量（以 R^n 为单位）。 2-放宽算法收敛的假设，尤其是 HCP。 3- 通过利用所涉及的线性程序的特殊形式来增强算法的性能。 4-测试断开集情况下反例的强等周不等式。 5-提高特殊集容量的精度，例如康托集。 6-开发自动计算容量的软件。 在兰斯福德-怀特猜想提出 17 年后，M_n 光谱单位球的自同构分类仍未完成。尽管最近已经取得了进展，但即使在 n=2 的情况下，该猜想仍然是开放的。我们建议对一些根本问题采取实验方法。此外，我们希望利用上同调理论重新表述该问题。 理解谱球自同构的问题源于谱 Nevanlinna-Pick 问题。这又源于控制理论领域的鲁棒稳定性问题。