Refined curve counting and refined Severi degrees

This is report on joint works with Vivek Shende, Florian Block and Sam Payne.
An old conjecture of mine gives a generating function for the numbers of
$\delta$-nodal curves in linear systems on surfaces. In this talk we want
to propose a refinement of the conjecture, where the numbers of curves are
replaced by polynomials in a variable $y$, which for $y=1$ specialize to
the numbers of curves.  For rational surfaces these refined invariants are
 related to Welschinger invariants and have an
interpretation in tropical geometry.