I will report the on joint work with Andras Nemethi
concerning the topological nature of certain   analytic invariants  of
 isolated surface singularities.
The geometric genus p_g is such an invariant, and the question is whether
one can express it  in terms of topological invariants of the link of the
singularity. For this to happen  the link has to be a rational
homology sphere (QHS), and the complex structure of the singularity
must satisfy some mild  rigidity condition (Gorenstein, or  complete
 intersection etc.)
 A decade ago  Fintushel-Stern, Neumann-Wahl gave an affirmative answer
 for many complete intersection singularities   whose links are
 INTEGRAL homology spheres (IHS) by proving a formula
 relating the Casson invariant of the link to  the signature of the Milnor
fiber. This signature  differs from p_g  by a topological
 The obvious  change Casson ---> Casson-Walker , when going from IHS to
 QHS links  leads to  false conclusions. Even worse, most singularities
 are not smoothable, and in these cases  there is no Milnor  fiber to speak
 In  my talk   I will   explain how to resolve this conundrum by relying on
 the  Seiberg-Witten invariants.