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.
Research performed for this project was concerned with the investigation of configurations of points on the sphere in the n-dimensional space. Studies of such configurations are motivated by the design of communication systems, image processing, and signal representation, and rely on methods of discrete geometry and algebraic combinatorics. One of the oldest problems in this area is to determine the largest number of points such that the distance between any pair of them take one of only two values. Despite a simple formulation, this problem is very far from solution. In this research we advanced the state of the art, determining the exact maximum number of points that satisfy this condition for a large range of dimensions beyond what was known earlier. A particular case of this problem is concerned with determining the maximum number of lines that can be drawn through the origin so that the (acute) angles between them are all equal. For instance, on the plane there are three such lines that pass through the vertices of the regular hexagon with center at the origin. In higher-dimensional spaces the question of determining the maximum number of equiangular lines becomes much more difficult. Prior to this research this number was known only for dimensions n<=23 (with few exceptions). In a work performed for this project, we found the maximum number of lines for a range of dimensions between 24 and 41, again extending the state of the art. A unifying theme for these two projects is the use of the so-called positive definite functions, which give powerful constraints for point sets on the sphere. Manipulating these constraints makes it possible to rule out existence of certain point sets, and eventually leads to the described results. The problem of determining the maximum number of equiangular lines is also related to configurations of points on the sphere that approximate in some way a ``uniform’’ allocation of points, and are known as spherical designs. Spherical designs are characterized by a parameter called strength, which describes the type of approximation – the stronger the design, the larger the class of functions that it approximates. Two studies performed for this project are concerned with various types of spherical designs. In particular, a concept that is useful in representation of signals, called a frame, relates to sets of vectors in the n-dimensional space that can faithfully represent any vector (signal). In this project we found a new characterization for a class of frames with only two distances, connecting them with spherical designs of strength two. This project also motivated us to organize a Special Session at the National Meeting of the American Mathematical Society (January 2013, San Diego, CA). We also edited a collection of papers derived from the talks presented at the meeting. The collection was published by AMS in 2014.
