Complements of connected hypersurfaces in $S^4$

Higher-order Alexander Invariants of Hypersurface Complements

We define the higher-order Alexander modules $A_{n,i}(\mathcal{U})$ and higher-order degrees $δ_{n,i}(\mathcal{U})$ which are invariants of a complex hypersurface complement $\mathcal{U}$. These invariants come from the module structure of the homology of certain solvable covers of the hypersurface complement. Such invariants were originally developed by T. Cochran in [1] and S. Harvey in [8], and were used to study knots and 3-manifolds. In this paper, I generalize the result proved by C. Leidy and L. Maxim [22] from the plane curve complements to higher-dimensional hypersurface complements. …

Lagrangian Embeddings and Holomorphic Discs in the Complement of Complex Hypersurfaces

For a given embedded Lagrangian in the complement of a complex hypersurface we show existence of a holomorphic disc in the complement having boundary on that Lagrangian.

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.

关于具有任意大面积极小超曲面与Morse指数的存在性

On the Existence of Minimal Hypersurfaces with Arbitrarily Large Area and Morse Index

该研究证明了3到7维凹凸闭黎曼流形中存在面积和Morse指数均趋于无穷的连通闭嵌入双侧极小超曲面，改进了此前仅涉及任意大面积连通极小超曲面的相关结果。

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…

Index theory and partitioning by enlargeable hypersurfaces

In this paper we state and prove a higher index theorem for an odd-dimensional connected spin riemannian manifold $(M,g)$ which is partitioned by an oriented closed hypersurface $N$. This index theorem generalizes a theorem due to N. Higson and J. Roe in the context of Hilbert modules. Then we apply this theorem to prove that if $N$ is area-enlargeable and if there is a smooth map from $M$ into $N$ such that its restriction to $N$ has non-zero degree then the the scalar curvature of $g$ cannot be uniformly positive.