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.