Collaborative Research: Positive definite functions in distance geometry and combinatorics

该项目致力于研究度量空间中的有限点配置。调和分析和群表示的经典结果意味着存在的功能，满足一定的正性约束时，评估这样的配置。这些约束给出了一组必要条件的存在的配置。该项目研究积极性约束在多大程度上也是足够的，因为它们意味着存在具有所需属性的配置。该项目研究的一个相关主题是度量空间中距离很少的点集的最大大小。该项目的发展背景与最近建立的齐次空间中代码的半定编程界限有关。该项目还寻求被称为球形和欧几里得设计的均匀分布的点集与更一般的体积公式概念之间的联系，目的是使用代数组合学的方法来研究度量和函数空间中的体积公式。目标之一是建立齐次空间中求积公式的新的泛界。空间点的有限集合在可靠的数值分析中有应用。研究点配置的结构创建洞察建设的最佳信号传输方案和最佳网络的蒙特-卡罗集成。

The project is devoted to the study of finite point configurations in metric spaces. Classical results in harmonic analysis and group representations imply the existence of functions that satisfy certain positivity constraints when evaluated on such configurations. These constraints give a set of necessary conditions for the existence of the configuration. The project studies the extent to which the positivity constraints are also sufficient in that they imply that a configuration with desired properties exists. A related topic studied in the project is the maximum size of point sets with few distances in metric spaces. The context for the development of the project is related to the recently established semidefinite programming bounds on codes in homogeneous spaces. The project also pursues a link between uniformly distributed sets of points known as spherical and Euclidean designs and a more general concept of cubature formulas with the aim to use methods of algebraic combinatorics to study cubature formulas in metric and functional spaces. One of the goals is to establish new universal bounds on cubature formulas in homogeneous spaces. Finite collections of points in space find applications in reliable and numerical analysis. Studying the structure of point configurations creates insights into construction of optimal signal transmission schemes and of optimal nets for Monte-Carlo integration.