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

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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.