Invariants of Normal Surface Singularities

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The proposal predicts substantial developments in three themes. The final goal of the first one is the topological classification of surface singularities; it aims to generalise the classical work of Artin and Laufer (and the author) about rational and elliptic singularities by identification of natural subfamilies beyond elliptic singularities, and topological determination of their analytic invariants. There is a special emphasise on hypersurfaces with rational homology sphere (RHS) links, when the author (with R. Mendris) conjectured that even the embedded topological type can be recovered from the link (hence, predicting an even sharper property than Zariski's `multiplicity conjecture'). The second part is motivated by a conjecture of L.I. Nicolaescu and the author which connects a certain Seiberg-Witten invariant of a RHS--link with the geometric genus of the singularity (as a generalisation of a conjecture of Neumann and Wahl valid for integer homology sphere links). The proposal targets the limits of the generalised conjecture. The new ingredients are provided by the recently invented Heegaard Floer homology (of Ozsvath and Szabo). The third part focuses an unexpected development of the second part applied in the classical open problem of classification of rational (uni)cuspidal projective plane curves. The author (in his joint article) formulates very strong criterion conjecturally satisfied (and in many cases verified) by the local topological types of singularities which can appear as singularities of such projective curves. It appears that it is much stronger than the existing criterions, conjecturally it even has a classification power. Its connection with Heegaard Floer theory is striking. Mathematical models which describe geometrical objects present in the real life or in the nature are represented by functions, or by the set where they are zero. In general, a function in a generic point behaves `nicely', but at some special points it may have some anomalies: these phenomena are described by singularity theory. Its mathematical methods vary rather diversely, it uses techniques from topology, algebra, analysis, combinatorics, number theory; in this way the results and the theory becomes very interesting (and difficult), it alloys different aspects. The proposal targets surface singularities, and the proposed problems lie in the core of singularity theory (which has revived recently with a large intensity). The proposal lines up a series of new objects and new methods which conceptually modify the general picture. The proposer's research will be integrated in the process of undergraduate, graduate and postdoctoral training.