Brownian motions and heat kernel lower bounds on Kähler and quaternion Kähler manifolds

四元数叶状结构与orbifolds的上同调

Cohomology of Quaternionic Foliations and Orbifolds

研究四元数叶状结构与orbifolds的上同调，先回顾四元数几何等定义横向四元数凯勒叶状结构，证明相关经典结果的叶状版本，最后为四元数orbifolds重新阐述结果。

Holomorphic Lagrangian fibrations of toric hyperkahler manifolds

For the sake of hyperk{ä}hler SYZ conjecture, finding holomorphic Lagrangian fibrations becomes an important issue. Toric hyperk{ä}hler manifolds are real dimension $4n$ non-compact hyperk{ä}hler manifolds which are quaternion analog of toric varieties. The $n$ dimensional residue circle action on it admitting a hyperk{ä}hler moment map. We use the complex part of this moment map to construct a holomorphic Lagrangian fibration with generic fiber diffeomorphic to $(\mathbb{C}^*)^n$, and study the singular fibers.

四元数随机面积在四元数全旗流形上的应用

Quaternionic stochastic areas on quaternionic full flag manifolds and applications

本文研究了四元数全旗流形上布朗运动的随机面积及其极限分布，通过矩阵值扩散过程和辛布朗运动的关系，推导出正态分布的协方差矩阵，并应用于四元数缠绕问题。

Positive Self Dual Einstein Orbifolds with One-Dimensional Isometry Group

The aim of this thesis is to construct new examples of compact orbifolds $\mathcal{O}^4(Θ)$ which admit a self dual Einstein (SDE) metric of positive scalar curvature $s>0$, with a one-dimensional group of isometries. In particular we want to prove that these examples are different from those described by Boyer, Galicki and Piccinni in \emph{$3-$Sasakian geometry, nilpotents orbits, and exeptional quotients}. We construct explicitly our new examples as quaternion-K$\ddot{\mathrm{a}}$hler reductions of the quaternion K$\ddot{\mathrm{a}}$hler Grassmannian $Gr_4(\mathbb{R}^8)$ by an isometric act…

Cotangent bundles of toric varieties and coverings of toric hyperkähler manifolds

Toric hyperk{ä}hler manifolds are quaternion analog of toric varieties. Bielawski pointed out that they can be glued by cotangent bundles of toric varieties. Following his idea, viewing both toric varieties and toric hyperk{ä}her manifolds as GIT quotients, we first establish geometrical criteria for the semi-stable points. Then based on these criteria, we show that the cotangent bundles of compact toric varieties in the core of toric hyperk{ä}hler manifold are sufficient to glue the desired toric hyperk{ä}hler manifold.

随机面积、水平布朗运动与亚椭圆热核

Stochastic areas, Horizontal Brownian Motions, and Hypoelliptic Heat Kernels

该专著研究李群与黎曼流形上布朗运动的随机面积泛函及相关热核，内容自足可作教材，通过实例阐释多数学领域间的深刻关联。