Topology of Hyperplane Arrangements

DMS-0105342亚历山德鲁岛Suciu本项目围绕着复杂超平面排列的拓扑研究，目的是寻找其补集的有效可计算不变量，目标是确定给定的不变量是否是组合确定的，如果是，则用排列的相交格来显式表示。 具有局部系统系数的上同调的跳跃轨迹和相关的共振变体扮演着重要的角色。这些变体已经成为研究的中心对象。它们提供了关于排列补的同伦理论的深层信息，以及以通常意想不到的方式联系各种不变量阿萨桥梁。另一个关键角色是形式的理性同伦概念，这为许多遇到的现象提供了潜在的解释。只要有可能，研究都是在更一般的环境中进行的，其中包括某些类型的子空间排列，包括真实的和复杂的，以及三维球中的某些链接。这样的观点扩大了结果的适用范围，并有助于解释什么是真正独特的复杂超平面安排。在其最简单的表现形式，安排是有限的收集线在平面上。 这些线将平面切割成多个分量，理解补集的拓扑结构就等于计算这些分量。（或者，就此而言，复n-空间中的超平面），补集连通，及其拓扑（例如，反映在它的同伦群中）要有趣得多。排列理论是一个相对较新的数学分支，超平面理论始于20世纪60年代对纯辫群的分类空间的研究。该理论在拓扑学、代数学、代数几何学和组合学之间的界面上得到了发展。超平面排列和密切相关的配置空间被用于许多领域，包括机器人学、多维台球、图形学、分子生物学、计算机视觉、和数据库，用于表示以多个自由度为特征的系统的所有可能状态的空间。超平面排列、纽结理论、超几何函数、共形场论和量子上同调之间也有着深刻的联系。

DMS-0105342Alexandru I. Suciu This project is centered around a topological studyof complex hyperplane arrangements, with a view towardsfinding effectively computable invariants of their complements.The goal is to decide whether a given invariant iscombinatorially determined, and, if it is, to expressit explicitly in terms of the intersection lattice ofthe arrangement. An important role is played by thejumping loci for cohomology with coefficients in localsystems, and the related resonance varieties.These varieties have emerged as a central object of study.They provide deep information about the homotopy theoryof the complement of an arrangement, as well asa bridge relating various invariants, in often unexpected ways.Another key role is played by the rational-homotopynotion of formality, which provides the underlyingexplanation for many of the encountered phenomena.Whenever possible, the study is done in a more general setting,which includes certain types of subspace arrangements,both real and complex, as well as certain links in the 3-sphere.Such a point of view enlarges the range of applicability of theresults, and helps explain what is really peculiar to complexhyperplane arrangements.In its simplest manifestation, an arrangement is afinite collection of lines in the plane. These linescut the plane into components, and understanding the topologyof the complement amounts to counting those components.In the case of lines in the complex plane (or, for thatmatter, hyperplanes in complex n-space), the complementis connected, and its topology (as reflected, for example,in its homotopy groups) is much more interesting.The theory of arrangements is a relatively new branchof mathematics, started in the 1960's with a studyof the classifying space for the pure braid group.The theory has developed at the interface betweentopology, algebra, algebraic geometry, and combinatorics.Hyperplane arrangements, and the closely relatedconfiguration spaces, are used in numerous areas,including robotics, multi-dimensional billiards,graphics, molecular biology, computer vision, anddatabases for representing the space of all possiblestates of a system characterized by many degreesof freedom. There are also deep connections betweenhyperplane arrangements, knot theory, hypergeometricfunctions, conformal field theory, and quantum cohomology.