Mathematical Sciences: Problems in Discrete Geometry

调查员，在合作理查德波拉克（柯朗数学科学研究所，纽约大学），计划继续他的调查之间的相互作用的几何和拓扑结构的欧几里德空间的延伸最近的结果分类的有序配置和他们的推广，并对相关的几何和组合问题。基于他们过去关于配置的允许序列和序类型的工作，以及他们最近证明的关于拓扑线的有限排列到连续扩展的Grunbaum猜想的想法，古德曼和波拉克计划继续他们关于配置，拓扑平面的排列和几何横截理论的工作，检查（在其他问题中）（1）将Grunbaum猜想的证明推广到中间维平面的问题，（2）找到配置和多面体的紧致表示的问题，以及反映它们的几何特性的中维平面的排列的分类，（3）通过识别真实的d维空间中的简单n点构型的集合与适当的Grassmann流形的开稠密子集来确定序型的相对频率的问题（经典的西尔维斯特问题，推广）。在过去的几十年里，高速计算的出现为许多这些问题带来了新的生命，因为相对轻松地研究非平凡的例子已经成为可能。有趣的有限几何集的大小往往随着维数的增加而呈指数增长。举一个最简单的例子，依次考虑一条线段、一个正方形、一个立方体、一个超立方体（四维）等等的顶点数。对应的数字是2，4，8，16，...，这里的规律性是如此明显，以至于这些数字是可计算的，并且n维立方体的性质，正如它们所被称为的，即使对于大的n，也仍然是可管理的。然而，不太规则的集与快速增加的大小变得更加难以处理没有机器的帮助计算。这是这种数学与计算关系的一个方面，但还有另一种相互关系。正如计算促进了似是而非的猜想和证明一样，一些证明也导致了计算算法的改进。我们在这里有一种非常有益的共生关系，这种关系在今后几年必将更加蓬勃发展。

The investigator, in collaboration with Richard Pollack (Courant Institute of Mathematical Sciences, New York University), plans to continue his investigation of the interplay between the geometry and the topology of Euclidean spaces by extending recent results on the classification of ordered configurations and their generalizations, and on related geometric and combinatorial problems. Building on their past work on allowable sequences and order types of configurations, as well as on ideas coming out of their recent proof of the Grunbaum conjecture on the extendibility of finite arrangements of topological lines to continuous spreads, Goodman and Pollack plan to continue their work on configurations, arrangements of topological flats, and geometric transversal theory, examining (among other problems) (1) the problem of generalizing the proof of the Grunbaum conjecture to intermediate- dimensional flats, (2) the problem of finding a compact representation for configurations and polytopes, as well as a classification of arrangements of intermediate-dimensional flats which reflects their geometric properties, and (3) the problem of determining the relative frequency of order types (the classical Sylvester problem, generalized) via the identification of the set of simple n-point configurations in real d-dimensional space with an open dense subset of a suitable Grassmann manifold. The advent of high-speed computing has brought new life to many of these questions over the past few decades, since it has become possible to investigate non-trivial examples relatively painlessly. The size of interesting finite geometric sets tends to increase exponentially with the dimension. Consider, for the simplest example, the number of vertices of, in turn, a line segment, a square, a cube, a tesseract (4-dimensional), and so forth. The corresponding numbers are 2, 4, 8, 16, ..., and the regularity here is so pronounced that the numbers are computable and the properties of n-dimensional cubes, as they are called, remain manageable even for large n. However, less regular sets with rapidly increasing size become much more difficult to handle without a machine's assistance for computation. This is one aspect of the relation of this kind of mathematics to computing, but there is another reciprocal relation. Just as computing facilitates plausible conjecture and hence proof, so too do some of the proofs lead to improved algorithms for computing. We have here a very useful symbiosis that is bound to flourish even more in the years to come.