On the classification of certain hypersurfaces in CP^4

Twisted Alexander Polynomials of Hypersurface Complements

We define twisted Alexander polynomials of a complex hypersurface with arbitrary singularities. These generalize the classical Alexander polynomials of high dimensional hypersurfaces and the twisted Alexander polynomial of plane curves. We recover the classical torsionness and divisibility results, which say that, under certain assumptions, the twisted Alexander modules of a complex hypersurface are torsion modules, and that their orders, the twisted Alexander polynomials, divide the product of certain `local polynomials' defined in terms of the topology near singularities.

On the violation of Thurston-Bennequin inequality for a certain non-convex hypersurface

We show that any open subset of a contact manifold of dimension greater than three contains a certain non-convex hypersurface violating the Thurston-Bennequin inequality.

Homogeneous spaces, Tits buildings, and isoparametric hypersurfaces

We classify compact 2-connected homogeneous spaces with the same rational cohomology as a product of spheres. This classification relies on spectral sequences, homotopy theory, and representation theory. We then apply this classification to two geometric problems. The first problem is the classification of all isoparametric hypersurfaces which admit a transitive isometry group on at least one focal manifold. This generalizes the classification of homogeneous isoparametric hypersurfaces by Hsiang and Lawson and gives a new, independent proof of their result. Secondly, we classify certain compac…

具有预定径向平均曲率的曲面与超曲面

Surfaces and Hypersurfaces with Prescribed Radial Mean Curvature

本文研究了具有预定径向平均曲率的曲面与超曲面，提供了特殊曲面类及第二平均曲率为零的三维超曲面的局部分类，并提出递归方法将该曲率预定方法扩展至更高维度。

Pair-of-pants decompositions of 4-manifolds diffeomorphic to general type hypersurfaces

In this paper, we show that a smooth 4-manifold diffeomorphic to a complex hypersurface in $\mathbb{CP}^3 $ of degree $d\geq 5$ can be decomposed as the union of $d(d-4)^2$ copies of 4-dimensional pair-of-pants and certain subsets of K3 surfaces.