Description
Already in the 18th century, Lagrange proved that every positive integer can be expressed as the sum of four squares of integers. Nowadays we say that the sum of four squares is a universal quadratic form.
In the friendly and accessible talk, I'll start by discussing some results on universal quadratic forms over Z and over totally real number fields. Then I'll move on to the - markedly different! - situation over infinite degree extensions K of Q.
In particular, I'll show that if K doesn't have many small elements (i.e., "the ring of integers of K has the Northcott property"), then it admits no universal form. Further, almost all totally real fields do not have universal
forms or the Northcott property (in a suitable topological sense).
The talk is based on recent joint works with Nicolas Daans, Siu Hang Man, Pavlo Yatsyna, and Martin Widmer.