Invariants of Normal Surface Singularities

该提案预测了三个主题的重大发展。第一种方法的最终目标是曲面奇异点的拓扑分类；它旨在通过识别超越椭圆奇点的自然亚族及其解析不变量的拓扑确定来推广Artin和Laufer（以及作者）关于理性和椭圆奇点的经典工作。当作者（与R. Mendris）推测甚至嵌入的拓扑类型也可以从连接中恢复（因此，预测了比Zariski的“多重猜想”更尖锐的性质）时，特别强调了具有理性同调球（RHS）链接的超曲面。第二部分是由L.I. Nicolaescu和作者的一个猜想推动的，该猜想将RHS-连杆的某个Seiberg-Witten不变量与奇点的几何属联系起来（作为对整数同调球连杆有效的Neumann和Wahl猜想的推广）。这个建议针对的是广义猜想的极限。新的成分是由最近发明的Heegaard花同源物（Ozsvath和Szabo）提供的。第三部分集中讨论了第二部分在有理（单）尖头投影平面曲线分类经典开放问题中的一个意外发展。作者（在他的联合文章中）提出了非常强的准则，推测满足（并在许多情况下验证）的局部拓扑类型的奇点，可以出现在这样的投影曲线的奇点。它似乎比现有的标准强得多，推测它甚至具有分类能力。它与Heegaard flower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用函数或函数为零的集合来表示。一般来说，一个函数在一般点上表现得“很好”，但在一些特殊点上可能会有一些异常：这些现象用奇点理论来描述。它的数学方法变化相当多样，它使用的技术从拓扑学，代数，分析，组合学，数论；这样的结果和理论变得非常有趣（和困难），它结合了不同的方面。该建议针对表面奇点，并且所提出的问题位于奇点理论的核心（最近以很大的强度复苏）。该提案列出了一系列从概念上改变总体情况的新对象和新方法。申请人的研究将整合到本科、研究生和博士后的培养过程中。

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.