Mathematical Sciences: RUI: The Homotopy Theory of Arrangements

迈克尔·福尔克将继续研究组合学和代数之间的相互作用，研究复数域上超平面排列的补集的拓扑结构。重点放在两个相关的问题上：分类问题和K(pi，1)问题。分类的核心是以下问题及其相反的问题：排列的补集的拓扑是否由原始超平面的交集模式决定？K(pi，1)问题给出了补中某些拓扑障碍消失的充要条件。这个简单的对象是n维空间中(n-1)维线性子空间集合的补充，显示了令人惊讶的丰富结构。已被发现用于分析它的各种方法来自组合学、微分方程式和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.