Optimization Methods for Tackling Existence Proofs of Spherical t-Designs

Abstract

A finite set of points on the unit sphere in the Euclidean space R^{d+1} is called a spherical t-design if for any polynomial p of degree at most t, the average value of p on the set equals the average value of p on the whole sphere. The existence of a spherical t-design for any t≥1 and d≥1 was proved in 1984. We showed that the existence of a spherical t-design is equivalent to the existence of a solution of an optimization problem in R^{(t+1)^2} for t≥1 and d=2 in 2006. Moreover we proved the existence of spherical t-designs with (t+1)^2 points for t up to 100 and d=2 in 2011 by using the optimization formulation. Fields Medal 2022 winner, Viazovska and her collaborators proved the existence of a spherical t-design with O(t^d) points for any fixed d in 2013. However, the exact number of points for a spherical t-design is still unknown for any t≥1 and d≥1. We will review the development on existence proofs for spherical t-designs and present our recent results on spherical zone t-designs for d=2.

Joint works with An, Frommer, Lang, Li, Sloan and Womersley.

Biography

Xiaojun Chen is a Chair Professor of Department of Applied Mathematics, Hong Kong Polytechnic University. She is the Co-Director of CAS AMSS-PolyU Joint Laboratory of Applied Mathematics. Her research interests focus on mathematical optimization theory and algorithms for nonsmooth nonconvex optimization problems and stochastic variational inequalities with applications in data sciences. She is the PI of several large grants from Hong Kong Research Grant Council and Croucher Foundation. She published over 100 papers in top journals in applied mathematics. She served SIAM Journal on Numerical Analysis, SIAM Journal on Optimization and SIAM Journal on Control and Optimization as an Associate Editor. She is a fellow of Society for Industrial and Applied Mathematics (SIAM) and a fellow of American Mathematical Society (AMS). She is a Keynote speaker of the 25th International Symposium on Mathematical Programming in Canada 2024 and the 16th Viennese Conference on Optimal Control and Dynamic Games in Austria 2025, and an invited 45-minute speaker in the International Congress of Mathematics (ICM) in Philadelphia 2026.