A survey on spherical designs and Euclidean designs

Our aim is to study good subsets consisting of finitely many points of the sphere and/or of Euclidean space.
Sometimes we consider not only finite subsets but also finite subsets with weight functions, i.e., cubature formulas in analysis. In this talk, we start with the definition of spherical $t$-design due to Delsarte-Goethals-Seidel (1977). Then we study spherical $t$-designs from the viewpoint of algebraic combinatorics. Here, association schemes play an important role. There are natural lower bounds for the size of spherical $t$-designs, and those which attain one of such lower bounds are called tight spherical $t$-designs. We discuss the known examples of tight spherical $t$-designs, and survey the current status of the classification of tight spherical $t$-designs.
Another main purpose of this talk is to discuss the concept of Euclidean $t$-designs which are two step generalization of spherical $t$-designs. Natural lower bounds for the size of Euclidean $t$-designs, as well as the concept of tight Euclidean $t$-designs will be discussed. We review the examples and the current status of the classification problem of tight Euclidean $t$-designs. Some highlights will include our recent complete classification of tight Euclidean 9-designs on two concentric spheres (due to Etsuko Bannai and myself), as well as the new discovery of a tight 6-design on two concentric spheres (due to Etsuko Bannai, Junichi Shigezumi and myself). We discuss the connection between Euclidean designs and the theory of cubature formulas in analysis, and also discuss the role of coherent configurations (generalization of association schemes) in the study of Euclidean $t$-designs.