ИСТИНА

The Alexander polynomial in several variables is defined for links in three-dimensional homology spheres, in particular, in the Poincaré sphere: the intersection of the E_8-surface singularity (S,0) in (C^3,0) with the 5-dimensional sphere \S_{\epsilon}^5 of radius \epsilon in C^3. An algebraic link in the Poincaré sphere is the intersection of a germ of a complex analytic curve in (S,0) with the sphere \S_{\epsilon}^5 of radius \epsilon small enough. We discuss to which extend the Alexander polynomial in several variables of an algebraic link in the Poincaré sphere determines the topology of the link. It turns out that, if the strict transform of a curve in (S,0) does not intersect the component of the exceptional divisor corresponding to the end of the longest tail in the corresponding E_8-diagram, then its Alexander polynomial determines the combinatorial type of the minimal resolution of the curve and therefore the topology of the corresponding link. The Alexander polynomial of an algebraic link in the Poincaré sphere is determined by the Poincaré series of the filtration defined by the corresponding curve valuations. The Poincaré series of a filtration on the ring of germs of functions on a surface can be computed as a certain integral with respect to the Euler characteristic over the space of effective Cartier divisors. One can consider the corresponding integral over the space of effective Weil divisors. It is a power series with rational exponents: "the Weil Poincaré series". We describe to which extend the Weil Poincaré series of a collection of curve filtrations on a simple surface singularity determines the resolution of the curve and thus the topology of the corresponding algebraic link. The corresponding questions are considered for divisorial valuations on the simple surface singularities. The talk is based on a joint work with A.Campillo and F.Delgado.