Invariants of Normal Surface Singularities

该提案预计将在三个主题方面取得重大进展。第一部分的最终目标是曲面奇点的拓扑分类；它的目的是通过识别椭圆奇点以外的自然子族，以及它们的解析不变量的拓扑确定，来推广Artin和Laufer(以及作者)关于有理奇点和椭圆奇点的经典工作。当作者(与R.Mendris)猜测甚至嵌入的拓扑型也可以从链接中恢复时，特别强调具有有理同调球(RHS)链接的超曲面(因此，预测了比Zariski的‘多重性猜想’更尖锐的性质)。第二部分是由L.I.Nicolaescu和作者的一个猜想所推动的，该猜想将RHS-链环的某个Seiberg-Witten不变量与奇点的几何亏格联系起来(作为Neumann和Wahl猜想对整数同调球环有效的推广)。该提案针对的是广义猜想的局限性。新成分由最近发明的Heegaard Floer同源化合物(Ozsvath和Szabo)提供。第三部分是第二部分的意外发展，它应用于有理(统一)尖顶射影平面曲线的分类这一经典公开问题。作者(在他的联合文章中)提出了非常强的判据，该判据猜想地满足(并在许多情况下验证了)奇点的局部拓扑类型，这些奇点可以表现为此类射影曲线的奇点。它似乎比现有的标准强得多，推测它甚至有分类能力。它与希加德·弗洛尔理论的联系是惊人的。描述现实生活或自然界中存在的几何对象的数学模型由函数或集合表示，其中函数为零。一般而言，一般点上的函数表现得很好，但在某些特殊点上，它可能会有一些反常现象：这些现象是用奇点理论来描述的。它的数学方法千差万别，它使用了拓扑学、代数、分析、组合学、数论等技术；这样，结果和理论变得非常有趣(也很难)，它融合了不同的方面。该提议针对的是表面奇点，所提出的问题存在于奇点理论的核心(奇点理论最近以很大的强度复兴)。该提案列出了一系列从概念上修改总体情况的新对象和新方法。在本科生、研究生和博士后培训过程中，将整合提出者的研究。

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.