Title: Number fields with small discriminants

Abstract: The discriminant d_N of a number field N is a basic invariant of N. The smaller d_N is, the more elements there are in the ring of integers O_N of N that have a given bounded size. This is relevant, for example, to cryptography using elements of O_N. In this talk, I will first introduce the relevant definitions. Then I will describe work with Ted Chinburg on the explicit construction of an infinite family of number fields N having d_N^{1/[N:Q]} bounded by a constant times [N:Q]^e for some e < 1. This answers a question by Peikert and Rosen from 2006.