Mathematical Sciences: Problems in Discrete Geometry

这位研究员与纽约大学库兰特数学科学研究所的理查德·波拉克合作，计划通过推广最近关于有序构型的分类及其推广以及相关几何和组合问题的结果，继续他对欧几里德空间几何和拓扑之间相互作用的研究。古德曼和波拉克基于他们过去关于构形的允许序列和顺序类型的工作，以及他们最近关于拓扑线的有限排列到连续展开的可扩性的Grunbaum猜想的证明所产生的想法，计划继续他们关于构形、拓扑平的排列和几何横截理论的工作，研究(除其他问题之外)(1)将Grunbaum猜想的证明推广到中维平坦的问题，(2)为构形和多面体寻找紧凑表示的问题，以及反映其几何性质的中维平坦排列的分类，以及(3)通过识别实d维空间中具有适当Grassmann流形的开稠密子集的简单n点构形集来确定阶型的相对频率问题(经典的Sylvester问题，推广)。在过去的几十年里，高速计算的出现给这些问题中的许多问题带来了新的活力，因为人们已经可以相对轻松地调查一些不寻常的例子。有趣的有限几何集的大小随着维度的增加而呈指数增长。在最简单的例子中，依次考虑线段、正方形、立方体、方块(4维)等的顶点数。相应的数字是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，即使对于大的n，它们被称为n维立方体的性质仍然是可管理的。然而，如果没有机器的辅助计算，大小迅速增加的不太规则的集合变得更加难以处理。这是这种数学与计算关系的一个方面，但还有另一个互易关系。正如计算促进了似是而非的猜想和证明一样，一些证明也促进了计算的改进算法。我们这里有一种非常有用的共生关系，这种共生关系在未来几年肯定会更加蓬勃发展。

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.