Topology of Hyperplane Arrangements

DMS-0105342 Alexandru I.Suciu这个项目的中心是复杂超平面排列的拓扑研究，目的是寻找它们互补的有效可计算不变量。目标是确定给定的不变量是否组合确定，如果是，则用该排列的交格显式表示。在局部系统中，系数上同调的跳跃轨迹以及相关的共振变量起着重要的作用。这些变量已经成为研究的中心对象。它们以一种经常意想不到的方式提供了关于排列补语的同伦理论的深刻信息，以及连接各种不变量的桥梁。另一个关键作用是形式的有理同伦概念，它为许多遇到的现象提供了潜在的解释。只要有可能，研究是在更广泛的背景下进行的，其中包括某些类型的子空间排列，包括真实的和复杂的，这种观点扩大了结果的适用范围，并有助于解释复杂超平面排列的真正特殊之处。在其最简单的表现形式中，排列是平面中直线的有限集合。这些线把平面切成几个分量，而理解补的拓扑就等于计算这些分量。在复平面(或，就此而言，是复n-空间中的超平面)中的线的情况下，补是相连的，它的拓扑(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，始于20世纪60年代的S，研究纯辫子群的分类空间。该理论发展于拓扑学、代数、代数几何和组合学之间的交界处。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，用于表示具有多个自由度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、共形场理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.