Mathematical Sciences: Topology of Complex Hyperplane Arrangements

数学科学：复杂超平面排列的拓扑

• 批准号: 9504833
• 负责人: Alexandru Suciu
• 金额: $ 5.22万
• 依托单位: Northeastern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-15 至 1997-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9504833&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Topology; Complex; Hyperplane

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. ***

小行星9504833 首席研究员亚历山德鲁一世。Suciu，将研究拓扑不变量的补充安排超平面在复杂的n-空间，以及它们的关系，组合不变量的交叉格的安排。 这个项目，继续与丹尼尔C。科恩，借鉴了各种技术，从代数拓扑，群论，代数几何，计算代数（Groebner基础），组合（定向拟阵）。 调查的出发点是介绍的基本组，密切反映了拓扑结构的补充。 从这个演示文稿，可计算的代数对象派生：陈群，亚历山大不变量，特征品种等，这些拓扑不变量中的一些可以直接计算从格，但有些可能需要更微妙的信息，从底层拟阵提取。 在某些假设下，这种方法允许计算的上同调的补充，在一个本地系统的系数确定的基本组的线性表示。 这种计算在奇点理论（非孤立奇点的米尔诺尔纤维化，复平面曲线的单值性），辫子理论（广义Burau和Gassner表示），微分方程（Knizhnik-Zamolodchikov方程，超几何函数）和低维拓扑中有应用。 在最简单的表现形式中，排列仅仅是平面上的线的集合。 这些线将真实的平面切割成碎片，理解补的拓扑结构是一个基本的练习，相当于计算这些碎片。 在复平面中的直线（或者，就此而言，复n-空间中的超平面）的情况下，补集是一片的。 但是，与圆盘不同的是，这并不意味着它可以被拉回到自身上，直到它缩小到一个点：测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，补中的环可以被视为辫子（相互编织的金属线串，没有备份），基本群可以与（纯）辫子群相识别。 对于任意排列，基本群的识别更加复杂，但它可以用算法的方式来完成，使用辫子理论。 反过来，这个理论又与三维空间中的节点和链接理论错综复杂地联系在一起，其丰富的代数和组合不变量，以及其在生物学，化学和物理学中的各种应用。 ***