Abstract: Following the work of Polya and Szego in the 1950s, a lot of effort was put into the study of low eigenvalues of the Laplacian.

The purpose of this talk is to report on recent progress on these and related issues, showing that more than 50 years on there is still a lot we don't know, but that there are several directions that may still be explored at different levels.

This is based on work which includes collaborations with P. Antunes, D. Borisov, D. Krejcirik and B. Siudeja.