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.
