Certain problems in mathematics, physics, and engineering are formulated as minimizing cost functions that take as input a set of points on a compact manifold. In applied and computational harmonic analysis one is usually interested in finding tight frames and equiangular tight frames, which are respectively minimizers of different cost functions. In quantum information theory, the study of SIC-POVMS is equivalent to the existence of a point configuration made of antipodal points on a complex sphere. There seems to be a phenomenon where highly symmetric configurations are optimizers and optimizers often exhibit (partial) symmetries. The theory of spherical designs in combinatorics and discrete geometry with applications in approximation theory in the form of cubature formulas is deeply related to point configurations and distributions. Training a neural network involves minimizing a cost function relating to the desired task; it was recently discovered that doing so often results in the last layer of the neural network corresponding to an optimal configuration of points on a sphere, a phenomenon called neural collapse.

Numerical solutions to optimization problems are used to study the putative geometric and arithmetic structure of the corresponding optimal configurations. Linear programming bounds have been a major tool in point configurations and led to the recent breakthrough in sphere packing problems in dimensions 8 and 24. Semidefinite programming is used to find exact solutions of maximal two-distance sets. Many spherical designs and point distribution problems are motivated by numerical integration and interpolation. Moreover, understanding neural collapse requires an understanding of the training of neural networks. As such, there is a need to expand the use of numerical and computational methods to solve these problems. Our goal is to provide a forum to explore problems in point configurations and distributions from the various aforementioned perspectives.