Mathematical Sciences: Problems in Discrete Geometry

研究者与Richard Pollack（纽约大学Courant数学科学研究所）合作，计划通过扩展最近关于有序构型分类及其推广的结果，以及相关的几何和组合问题，继续研究欧几里得空间的几何和拓扑之间的相互作用。古德曼和波拉克基于他们过去对允许序列和有序型构型的研究，以及他们最近对格伦鲍姆猜想的证明，即拓扑线的有限排列对连续扩展的可扩展性，他们计划继续他们在构型、拓扑平面的排列和几何截线理论方面的研究。研究（除其他问题外）(1)将Grunbaum猜想的证明推广到中维平面的问题，(2)寻找构型和多面体的紧表示问题，以及反映其几何性质的中维平面排列的分类问题，以及(3)确定有序类型的相对频率问题(经典的Sylvester问题，通过用合适的Grassmann流形的开密集子集识别实d维空间中的简单n点组形集（广义）。在过去的几十年里，高速计算的出现给许多这样的问题带来了新的生命，因为相对轻松地研究不平凡的例子已经成为可能。有趣的有限几何集的大小随着维数的增加呈指数增长。以最简单的例子为例，依次考虑线段、正方形、立方体、tesseract（四维）等的顶点数量。对应的数字是2、4、8、16、…这里的规则性是如此明显，以至于这些数字是可计算的，而且n维立方体的性质，就像它们被称为的那样，即使对于较大的n，也仍然是可管理的。然而，如果没有机器的辅助计算，尺寸迅速增加的不太规则的集合就变得更加难以处理。这是这种数学与计算关系的一个方面，但还有另一种互反关系。正如计算促进了貌似合理的猜想，从而促进了证明，一些证明也导致了计算算法的改进。我们这里有一个非常有用的共生关系，在未来的岁月里必将更加繁荣。

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.