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构形。在调和分析和群表示中的经典结果表明，当在这种构型上求值时，满足某些正性约束的函数的存在。这些约束为配置的存在提供了一组必要条件。该项目研究了正性约束在多大程度上也是充分的，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课题是度量空间中距离较少的点集的最大尺寸。该项目的发展背景与最近建立的齐次空间中码的半定规划界有关。该项目还寻求均匀分布的点集(称为球面设计和欧氏设计)与更一般的立方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空间和函数空间中的立方公式。其中一个目标是在齐次空间中建立立方公式的新的普适界。空间点的有限集合在可靠的数值分析中得到了应用。研究点配置的结构有助于构建最优信号传输方案和蒙特卡罗积分的最优网络。

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.