Collaborative Research: Multivariate positive definite polynomials and their applications via SDP

本课题利用距离几何、编码理论和半定规划等方法，研究实球面上的点分配及其构型。所研究的点集的性质包括限制集合中任意对不同点之间的最小角距或它所支持的培养公式的程度。这些问题的最新进展包括研究者在四维中的接吻数问题的解决方案，三维中的接吻数问题的新证明，以及使用半定规划的码界的新方法，由于Schrijver， Bachoc和研究者。x维度上的接吻问题是多少个单位球体可以在x维度上接触（亲吻）一个单位球体的问题。在这些工作中发展起来的新思想为进一步解决编码大小的边界问题和距离几何中的许多其他问题铺平了道路。本课题主要研究球面上最优点集具有一定性质时的边界问题，以及边界问题与多元正定多项式之间的联系。点集的有趣性质包括在集合中任意两点之间具有超过给定值的最小不规则距离（这与信号处理问题相关），并且支持给定度的球调和函数的精确cubature公式。所研究的点集的应用包括通信理论、数值分析、网格问题、数据表示和传感器网络的定位。

MusinDMS-0807640BargDMS-0807411 This project is devoted to the study of point allocations onthe real sphere and related configurations using methods ofdistance geometry, coding theory, and semidefinite programming. The properties of the point sets studied include restricting theminimum angular separation between any pair of distinct points inthe set or the degree of the cubature formula supported by it. Recent advances in these problems include a solution by theinvestigator of the kissing number problem in 4 dimensions, a newproof for the kissing number problem in 3 dimensions, and a newapproach to bounds on codes using semidefinite programming, dueto Schrijver, Bachoc, and the investigator. The kissing problemin x dimensions is the question of how many unit spheres cantouch (kiss) a unit sphere in x dimensions. The new ideasdeveloped in these works pave the way for further advances in theproblems of bounding the size of codes and in a number of otherproblems in distance geometry. The main problems to be addressed in the project are relatedto bounding the size of optimal sets of points on a sphere whenthe sets have a certain property, and the links between thebounding problem and multivariate positive definite polynomials. Interesting properties of the point set include having a minimumangular separation that exceeds a given value between any twopoints in the set (this is relevant for signal processingproblems), and supporting an exact cubature formula for sphericalharmonic functions of a given degree. Applications of the pointsets studied include communication theory, numerical analysis,the meshing problem, data representation, and localization insensor networks.