至多有两个奇异点的正规3-伪流形的刻画

具有相对2-骨架的正规4-伪流形

Normal $4$-pseudomanifolds with a relative 2-skeleton

本文研究正规4-伪流形在g₂和g₃最优条件下的结构，证明其可由单形边界复形经顶点折叠、边折叠、连通和及单顶点悬浮等操作序列生成。

关于某些同调$d$-流形构造的研究

On a construction of some homology $d$-manifolds

本文刻画了$g_2=3$的同调$d$-流形，证明它们是由$g_2\leq2$的球面经联、重三角剖分和连通和得到，并给出素正规$d$-伪流形的结构结果。

具有有理同伦圆盘光滑化的正规复曲面奇点

Normal complex surface singularities with rational homology disk smoothings

本文证明含至少两个节点的正规曲面奇点无有理同伦圆盘光滑化，结合前期结果与J. Fowler的成果，完成了此类奇点的解图分类并验证了J. Wahl的相关猜想。

Lattice cohomology of normal surface singularities

For any negative definite plumbed 3-manifold M we construct from its plumbed graph a graded Z[U]-module. This, for rational homology spheres, conjecturally equals the Heegaard-Floer homology of Ozsvath and Szabo, but it has even more structure. If M is a complex singularity link then the normalized Euler-characteristic can be compared with the analytic invariants. The Seiberg--Witten Invariant Conjecture is discussed in the light of this new object.

配备半自由圆周作用的曲面上具有退化奇点的函数的规范型

Normal forms of functions with degenerate singularities on surfaces equipped with semi-free circle actions

本文研究了特定曲面上无鞍点Morse-Bott函数的推广类，证明了此类光滑圆周值函数可表示为由最简单Morse函数、微分同胚及光滑函数复合而成的规范型。

Symplectic surgeries and normal surface singularities

We show that every negative definite configuration of symplectic surfaces in a symplectic 4--manifold has a strongly symplectically convex neighborhood. We use this to show that, if a negative definite configuration satisfies an additional negativity condition at each surface in the configuration, and if the complex singularity with resolution diffeomorphic to a neighborhood of the configuration has a smoothing, then the configuration can be symplectically replaced by the smoothing of the singularity. This generalizes the symplectic rational blowdown procedure used in recent constructions of s…

The thick-thin decomposition and the bilipschitz classification of normal surface singularities

We describe a natural decomposition of a normal complex surface singularity $(X,0)$ into its "thick" and "thin" parts. The former is essentially metrically conical, while the latter shrinks rapidly in thickness as it approaches the origin. The thin part is empty if and only if the singularity is metrically conical; the link of the singularity is then Seifert fibered. In general the thin part will not be empty, in which case it always carries essential topology. Our decomposition has some analogy with the Margulis thick-thin decomposition for a negatively curved manifold. However, the geometric…

On Poincaré series associated with links of normal surface singularities

We study the counting function of topological Poincaré series associated with rational homology sphere plumbed 3-manifold with connected negative definite tree, interpreting as an alternating sum of coefficient functions associated with some Taylor expansions. It is motivated by a theorem of Szenes and Vergne which expresses these coefficient functions in terms of Jeffrey--Kirwan residues. This is used to prove the uniqueness of the quasipolynomiality inside a special cone, the structure of the counting function in terms of the graph and construction for a polynomial generalization of the Seib…

Fundamental groups of links of isolated singularities

We study fundamental groups of projective varieties with normal crossing singularities and of germs of complex singularities. We prove that for every finitely-presented group G there is a complex projective surface S with simple normal crossing singularities only, so that the fundamental group of S is isomorphic to G. We use this to construct 3-dimensional isolated complex singularities so that the fundamental group of the link is isomorphic to G. Lastly, we prove that a finitely-presented group G is Q-superperfect (has vanishing rational homology in dimensions 1 and 2) if and only if G is iso…

Line bundles associated with normal surface singularities

Recently L. Nicolaescu and the author formulated a conjecture which relates the geometric genus of a complex analytic normal surface singularity (whose link $M$ is a rational homology sphere) with the Seiberg-Witten invariant of $M$ associated with the ``canonical'' $spin^c$ structure of $M$. Since the Seiberg-Witten theory of the link $M$ provides a rational number for any $spin^c$ structure it was a natural challenge to search for a complete set of conjecturally valid identities, which involve all the Seiberg-Witten invariants (giving an analytic -- i.e. singularity theoretical -- interpreta…

二元正态混合中的奇点

Singularities in bivariate normal mixtures

本文利用映射奇点理论对二元正态密度映射进行分类，揭示三种类型及其几何性质，并给出各类混合众数数量的上界。