Nested Spherical Designs: Theory, Computation, and Applications

Numerical integration, a vital mathematical technique for approximating functionintegrals, is commonly referred to as "quadrature" for one-dimensional cases and"cubature" for multidimensional cases. In ancient Greece, the Pythagorean doctrineunderstood the area calculations through the construction of squares (quadrature).Archimedes' work on the areas of spheres and parabolas are among the highestachievements of the antique analysis while figures like Galileo and de Roberval exploredquadratures in finding areas of cycloid arc and hyperbola in medieval Europe. In the late17th century, Newton and Leibniz independently developed integration principles,leading to the formulation of quadrature and cubature rules. One of the most importantquadrature rules was studied by Chebyshev in 1874, which is an equal-weightquadrature rule. A significant extension of equal-weight quadrature rules to the sphereled to the concept of spherical designs, which are related to many areas ranging frompure and applied mathematics, such as algebraic combinatorics, associated schemes,number theory, modular form, machine learning, and quantum computing.In this project, we shall investigate a special type of spherical designs: nested sphericaldesigns. We aim to provide a deep mathematical understanding of nested sphericaldesigns and significant improvement in the performance of spherical data analysis tasksas well as the cutting-edge techniques for geometric deep learning through varioustasks including (i) the existence and optimal asymptotic bound of nested sphericaldesigns, the well-separated property of the nested spherical designs, the numericalcomputations and applications of nested spherical designs, and many others. Sphericaldesigns provide an ever-growing vast horizon of theory and applications. The furtherdevelopment of spherical designs in terms of nested spherical designs can unlock newinsights into algebra, geometry, analysis, and beyond.