Lagrangian Embeddings and Holomorphic Discs in the Complement of Complex Hypersurfaces

Families of monotone Lagrangians in Brieskorn--Pham hypersurfaces

We present techniques, inspired by monodromy considerations, for constructing compact monotone Lagrangians in certain affine hypersurfaces, chiefly of Brieskorn-Pham type. We focus on dimensions 2 and 3, though the constructions generalise to higher ones. The techniques give significant latitude in controlling the homology class, Maslov class and monotonicity constant of the Lagrangian, and a range of possible diffeomorphism types; they are also explicit enough to be amenable to calculations of pseudo-holomorphic curve invariants. Applications include infinite families of monotone Lagrangian $…

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. …

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.

全纯多边形与链环补空间的边缘Heegaard Floer同调

Holomorphic polygons and the bordered Heegaard Floer homology of link complements

本文提出利用Heegaard多图中全纯多边形计数构造 $\mathcal{A}_\infty$ 多模，并证明其与链环补空间的A型边缘缝合不变量拟同构，为后续基于链环Floer同调计算该不变量奠定基础。

复孤立超曲面奇点链上的可微映射

Differentiable maps on links of complex isolated hypersurface singularities

本文研究复孤立超曲面奇点链上由全纯函数限制诱导的可微映射，并给出一个奇异值集为同心圆的折叠映射显式例子。

Complements of connected hypersurfaces in $S^4$

We consider the possible Euler characteristics and fundamental groups of the complementary components $X$ and $Y$ of an embedding of a connected closed 3-manifold $M$ in $S^4$. We use a 2-knot satellite construction to change the fundamental groups, and Massey products to limit the values of $χ(X)$ and $χ(Y)$ when $M$ is the total space of an $S^1$-bundle with orientable base and Euler number 1.