Mathematical Sciences: Topology of Complex Hyperplane Arrangements

首席研究员Alexandru I. Suciu将研究复n空间中超平面排列的补的拓扑不变量，以及它们与该排列的交格组合不变量的关系。这个项目继续与Daniel C. Cohen合作，借鉴了代数拓扑、群论、代数几何、计算代数（Groebner基）和组合学（定向拟阵）等多种技术。研究的起点是一个基本群的表示，它紧密地反映了补体的拓扑结构。在此基础上，导出了Chen群、Alexander不变量、特征变数等可计算代数对象。其中一些拓扑不变量可以直接从晶格中计算，但有些可能需要从底层矩阵中提取更微妙的信息。在某些假设下，这种方法允许补的上同调的计算，其系数由基本群的线性表示决定。这样的计算在奇点理论（非孤立奇点的Milnor振动、复杂平面曲线的单态）、辫形理论（广义Burau和Gassner表示）、微分方程（knizhnnik - zamolodchikov方程、超几何函数）和低维拓扑中都有应用。在其最简单的表现中，排列仅仅是平面上的线的集合。这些线将真实的平面切成小块，了解补面拓扑结构是一个基本的练习，相当于计算这些小块。对于复平面中的直线（或者，就此而言，复n空间中的超平面），补是一块的。但是，与圆盘不同的是，这并不意味着它可以收缩到自身，直到缩小到一个点：衡量这种失败的代数不变量是基本群，粗略地说，它由那些不能在补中收缩的环组成。一个特别重要的例子是复n空间中“对角”超平面的辫状排列。在这种情况下，补体中的循环可以被看作是辫子（一串一串的金属丝相互交织，没有备份），基本组可以被识别为（纯）辫子组。对于任意排列，基本群的识别更为复杂，但它可以用一种算法的方式来完成，使用辫的理论。反过来，这一理论又与三维空间中的结点和链接理论有着错综复杂的联系，具有丰富的代数和组合不变量，以及在生物学、化学和物理学中的各种应用。＊＊＊

9504833 Suciu The principal investigator, Alexandru I. Suciu, will study topological invariants of the complement of an arrangement of hyperplanes in complex n-space, and their relationship to combinatorial invariants of the intersection lattice of the arrangement. This project, continuing work with Daniel C. Cohen, draws on a variety of techniques from algebraic topology, group theory, algebraic geometry, computational algebra (Groebner basis), and combinatorics (oriented matroids). The starting point of the investigation is a presentation of the fundamental group that closely reflects the topology of the complement. From this presentation, computable algebraic objects are derived: Chen groups, Alexander invariants, characteristic varieties, etc. Some of these topological invariants can be computed directly from the lattice, but some probably require more subtle information, to be extracted from the underlying matroid. Under certain hypotheses, this approach permits the computation of the cohomology of the complement, with coefficients in a local system determined by a linear representation of the fundamental group. Such computations have applications in the theory of singularities (Milnor fibrations of non-isolated singularities, monodromies of complex plane curves), theory of braids (generalized Burau and Gassner representations), differential equations (Knizhnik-Zamolodchikov equations, hypergeometric functions), and low-dimensional topology. In its simplest manifestation, an arrangement is merely a collection of lines in the plane. These lines cut the real plane into pieces, and understanding the topology of the complement is an elementary exercise, which amounts to counting those pieces. In the case of lines in the complex plane (or, for that matter, hyperplanes in complex n-space), the complement is of one piece. But, unlike a disk, for example, this does not mean that it can be drawn back over itself until it shrinks down to a single poi nt: An algebraic invariant that measures this failure is the fundamental group, which, roughly speaking, consists of those loops that can not be shrunk in the complement. A particularly important example is the braid arrangement of "diagonal" hyperplanes in complex n-space. In that case, loops in the complement can be viewed as braids (strings of wire weaving around each other, without backing up), and the fundamental group can be identified with the (pure) braid group. For arbitrary arrangements, the identification of the fundamental group is more complicated, but it can be done in an algorithmic way, using the theory of braids. This theory, in turn, is intricately connected with the theory of knots and links in 3-space, with its wealth of algebraic and combinatorial invariants, and its varied applications to biology, chemistry, and physics. ***