April 21 -- 27, Shanghai Jiao Tong University

**Home** || **Participants** || **Track A : Spherical Design and Numerical Analysis** || **Track B : Graph Homomorphisms and Dynamical Systems** || **Direction**

**Speaker**: Ziqing Xiang (向子卿), University of Georgia

**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