L1

Galois groups of p-class towers of number fields have long been a mystery,

but recent calculations have led to glimpses of a rich theory behind them,

involving Galois actions on trees, families of groups whose derived series

have finite index, families of deficiency zero p-groups approximated by

p-adic analytic groups, and so on.

We consider semilinear Sturm-Liouville and elliptic problems with jumping

nonlinearities. We show how `half-eigenvalues' can be used to describe the

solvability of such problems and consider the structure of the set of

half-eigenvalues. It will be seen that for Sturm-Liouville problems the

structure of this set can be considerably more complicated for periodic than

for separated boundary conditions, while for elliptic partial differential

operators only partial results are known about the structure in general.