Invariants of Normal Surface Singularities

法向表面奇点的不变量

• 批准号: 0605323
• 负责人: C. Clemens
• 金额: --
• 依托单位: Ohio State University Research Foundation -DO NOT USE
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-08-15 至 2010-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0605323&HistoricalAwards=false
• 关键词: Invariants; Normal; Surface; Singularities

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.

该提案预计在三个主题上会有重大发展。最终目标的第一个是拓扑分类的表面奇点，它的目的是概括的经典工作的阿丁和劳弗（和作者）理性和椭圆奇点的确定自然亚科以外的椭圆奇点，和拓扑确定其解析不变量。 特别强调了具有有理同调球面（RHS）链的超曲面，当作者（与R。Mendris）指出，即使是嵌入的拓扑类型也可以从链接中恢复（因此，预测一个比Zebriki的“多重性”更尖锐的性质）。第二部分是由L. I.尼古拉斯库和作者，其中连接了一定的塞伯格-威滕不变的RHS-链接与几何属的奇异性（作为一个概括的猜想诺依曼和瓦尔有效的整数同源球链接）。 该提案的目标是广义猜想的极限。新的成分由最近发明的Heegaard Floer同源性（Ozsvath和Szabo）提供。第三部分集中了一个意想不到的发展，第二部分应用在经典的开放问题的分类合理（单）尖点射影平面曲线。作者（在他的联合文章）制定了非常强的标准，严格满足（并在许多情况下验证）的地方拓扑类型的奇点，可以出现作为奇点的这种投影曲线。它似乎比现有的标准强得多，甚至具有分类能力。它与Heegaard Floer理论的联系是惊人的。描述存在于真实的生活或自然界中的几何对象的数学模型由函数或由它们为零的集合表示。一般来说，一个函数在一般点上表现得“很好”，但在某些特殊点上它可能会有一些异常：这些现象由奇点理论描述。它的数学方法各不相同，而diligence，它使用的技术，从拓扑，代数，分析，组合学，数论;以这种方式的结果和理论变得非常有趣（和困难），它合金不同的方面。 该建议的目标是表面奇异性，所提出的问题位于奇异性理论的核心（最近以很大的强度复兴）。该提案提出了一系列新的目标和新的方法，从概念上改变了总体情况。 申请人的研究将被纳入本科，研究生和博士后培训过程中。