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.
The goals of the project included the analysis of optimal point allocations in metric spaces, study of extremal properties of point configurations, investigation of designs in polynomial spaces such as spherical designs, and studies of point sets with a small number of pairwise distances. Among the outcomes of this project were novel solutions for the Tammes problem in geometry, which asks how N identical non-overlapping circles can be distributed on a sphere when the radius of the circles has to be as large as possible. This question is also known as the problem of the "inimical dictators": Where should N dictators build their palaces on a planet so as to be as far away from each other as possible? The problem was first asked by the Dutch botanist Tammes, who was led to this problem by examining the distribution of openings on the pollen grains of different flowers. When N=13, the Tammes problem is also known as the strong thirteen spheres problem, which calls to find the maximum radius of, and an arrangement for, 13 identical non-overlapping spheres touching the unit sphere. This problem is a natural extension of the kissing number problem, the subject of the famous discussion between Sir Isaac Newton and David Gregory in 1694. The kissing number problem, also known as the thirteen spheres problem, asks how many white billiard balls can kiss (touch) a black ball. Most reports say that Newton believed the answer was 12 balls, while Gregory thought that the available extra space might make 13 possible. The solution (12 balls) was finally obtained in 1953 by Schutte and van der Waerden. Prior to this project, the Tammes problem had been solved for several values of N: for N=3,4,6,12 by L. Fejes Toth (1943); for N=5,7,8,9 by Schutte and van der Waerden (1951); for N=10,11 by Danzer (1963) and for N=24 by Robinson (1961). Working on this project we gave a solution of this long-standing open problem in geometry for N=13. Our computer-assisted proof is based on an enumeration of irreducible graphs. Recently, we completed our computations for N=14. Apart from solving this problem, we also obtained several interesting results on various types of point allocations. During the project period we organized three installments of the Discrete Geometry and Algebraic Combinatorics Conference. By continuing to host an established conference, we engaged young mathematicians – particularly students and recent graduates from diverse backgrounds – in research work on the problems of this project. The conference serves as a venue at which to present recent results related to the present topics, as well as longer survey lectures on geometry and combinatorics. Apart from raising awareness within the research community of the rich mathematical contents of the topics of interest to the present research, the conference also motivates new students to enter the area.
