关于Neumann-Wahl的Casson不变量猜想

The Milnor fiber conjecture of Neumann and Wahl, and an overview of its proof

Splice type surface singularities, introduced in 2002 by Neumann and Wahl, provide all examples known so far of integral homology spheres which appear as links of complex isolated complete intersections of dimension two. They are determined, up to a form of equisingularity, by decorated trees called splice diagrams. In 2005, Neumann and Wahl formulated their Milnor fiber conjecture, stating that any choice of an internal edge of a splice diagram determines a special kind of decomposition into pieces of the Milnor fibers of the associated singularities. These pieces are constructed from the Mil…

某些曲面奇点的万能阿贝尔覆盖

Universal abelian covers of certain surface singularities

本文研究了某些曲面奇点的万能阿贝尔覆盖，证明其与完全交点奇点的等奇异变型关系，并引入了Neumann-Wahl系统作为核心方法。

拼接型曲面奇点的局部热带化

Local tropicalizations of splice type surface singularities

本文从热带几何视角研究拼接型曲面奇点，将其局部热带化刻画为拼接图上的锥，并由此重新证明早期结果、确认Newton非退化性及拼接图唯一恢复，还给出了平面曲线奇点嵌入消解定理的新证明。

Splice diagrams and splice-quotient surface singularities

The current work will appear in a Celebratio Mathematica volume in honor of Walter Neumann. We summarize results and methods from our long-time collaboration with Neumann, especially the motivation for the introduction of splice diagrams to define singularities of splice type and splice-quotient singularities. The Casson Invariant Conjecture and Milnor Fiber Conjectures are discussed, as well as some open questions about hypersurface and complete intersection singularities with integral homology sphere link.

splice quotient singularities的abelian covers的multiplicity

The multiplicity of abelian covers of splice quotient singularities

本文提出通过解析图和伽罗瓦群计算splice quotient奇点的abelian覆盖的multiplicity的方法。

Seiberg--Witten invariants and surface singularities

We formulate a very general conjecture relating the analytical invariants of a normal surface singularity to the Seiberg-Witten invariants of its link provided that the link is a rational homology sphere. As supporting evidence, we establish its validity for a large class of singularities: some rational and minimally elliptic (including the cyclic quotient and `polygonal') singularities, and Brieskorn-Hamm complete intersections. Some of the verifications are based on a result which describes (in terms of the plumbing graph) the Reidemeister-Turaev sign refined torsion (or, equivalently, the S…