具有少数对称性的光滑欧氏4-空间

Embedding compact surfaces into the 3-dimensional Euclidean space with maximum symmetry

The symmetries of surfaces which can be embedded into the symmetries of the 3-dimensional Euclidean space $\mathbb{R}^3$ are easier to feel by human's intuition. We give the maximum order of finite group actions on $(\mathbb{R}^3, Σ)$ among all possible embedded closed/bordered surfaces with given geometric/algebraic genus $>1$ in $\mathbb{R}^3$. We also identify the topological types of the bordered surfaces realizing the maximum order, and find simple representative embeddings for such surfaces.

关于光滑化立方体的离散对称性

On discrete symmetries of the cube of smoothings

研究带标记重心细分链环图赋予分辨率立方体的组合结构及对称集，基于三角剖分光滑化对称性构建组合链环不变量。

$K3$曲面上的对称性和奇异光滑结构

Symmetries and exotic smooth structures on a $K3$ surface

研究了一类奇异$K3$曲面的光滑对称性和辛对称性，并与标准$K3$曲面的相应对称性进行了比较.证明了光滑有限群作用在$K3$格上所引起的作用是强限制的，从而证明了素数阶全纯自同构$\geq 7$所引起的作用的非光滑性，并建立了若干$K3$群（其中包括阶数最小的二元四面体群$T_{24}$）的光滑作用的不存在性.关于辛对称，显式地确定了素数阶$\geq 5$的辛循环作用的不动点集结构，只要该作用是同调非平凡的。

欧几里得空间的保向同胚是换位子

Orientation-preserving homeomorphisms of Euclidean space are commutators

本文证明了欧几里得空间及环面的保向同胚均可表示为两个保向同胚的换位子，并将环面情形扩展至光滑范畴，推论得出实直线的保向微分同胚亦为换位子。

Symmetries of exotic aspherical space forms

We study finite group actions on smooth manifolds of the form $M\#Σ$, where $Σ$ is an exotic $n$-sphere and $M$ is a closed aspherical space form. We give a classification result for free actions of finite groups on $M\#Σ$ when $M$ is 7-dimensional. We show that if $\mathbb Z/p\mathbb Z$ acts freely on $T^n\#Σ$, then $Σ$ is divisible by $p$ in the group of homotopy spheres. When $M$ is hyperbolic, we give examples $M\#Σ$ that admit no nontrivial smooth action of a finite group, even though Isom($M$) is arbitrarily large. Our proofs combine geometric and topological rigidity results with smooth…

Reeb components of leafwise complex foliations and their symmetries II

We study the group of leafwise holomorphic smooth automorphisms of Reeb components of leafwise complex foliation which are obtained by a certain Hopf construction. In particular, in the case where the boundary holonomy is infinitely tangent to the identity, we determine the structure of the group of leafwise holomorphic automorphisms.