Mathematical Sciences: RUI: The Homotopy Theory of Arrangements

迈克尔·福尔克将继续研究 组合数学和代数的拓扑补 超平面在复域上的排列 号码 重点将放在两个相关的问题上： 分类问题和K（pi，1）问题。 的 分类集中在以下问题及其 匡威：是排列的补的拓扑 由原始的交叉模式决定 超平面？ K（pi，1）问题要求必要和/或 某些拓扑为零的充分条件 补充物中的障碍物。 这个简单的对象，是（n-1）集合的补数- n维空间中的线性子空间，显示一个 令人惊讶的丰富结构。 各种各样的方法 被发现对分析它有用来自组合学， 微分方程和K理论

Michael Falk will continue his research into the interplay of combinatorics and algebra with the topology of the complement of an arrangement of hyperplanes over the field of complex numbers. The emphasis will be on two related problems: the classification problem, and the K(pi,1) problem. The classification centers on the following question and its converse: is the topology of the complement of an arrangement determined by the intersection pattern of the original hyperplanes? The K(pi,1) problem asks for necessary and/or sufficient conditions for the vanishing of certain topological obstructions in the complement. This simple object, the complement of a collection of (n-1)- dimensional linear subspaces in n-dimensional space, displays a surprisingly rich structure. The variety of methods which have been found useful for analyzing it come from combinatorics, differential equations, and K-theory.