# 2015 Workshop on Combinatorics and Applications at SJTU

April 21 -- 27, Shanghai Jiao Tong University

Date: Friday, April 24, 2015

Time: 11:15 - 12:00

Venue: Large Meeting Room, Math Building

Title: Spherical Designs Over a Number Field

Abstract:

Let $k$ be a totally real number field. A spherical design over $k$ is just an ordinary spherical design in which the coordinates of all points are in $k$. One advantage of such designs is that there is no error while calculating something related to such a design in practice. The existence of spherical designs over $\mathbb{Q}$ is open in general. And the most difficult case is the existence of spherical designs over $\mathbb{Q}$ on the one dimensional unit sphere, namely the unit circle. In this talk, I will show that:

• There are many weighted spherical designs over $\mathbb{Q}$;
• A fast linear programming method to generate weighted spherical designs over $\mathbb{Q} in practice$;
• There are many spherical designs over $\mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5}, \dots)$;
• There are many interval designs over $\mathbb{Q}$.

The word "many" means that we have an asymptotic lower bound for the number of desgins.

This is joint work with Zhen Cui and Jiacheng Xia.

Slides: View slides

Webmaster