Mathematical Sciences: The Geometry of Configurations

9322475 Goodman研究者和Richard Pollack（纽约大学Courant数学科学研究所）将继续研究欧几里得空间的几何和拓扑之间的相互作用，通过扩展他们最近的成果(1)欧几里得空间的标准凸性结构的Grassmann流形的推广，(2)有序构型的分类及其推广，以及(3)相关的几何和组合问题。古德曼和波拉克基于他们过去关于允许序列和有序型构型的工作，以及他们对凸体族的超平面截线的Vincensini问题的解决所产生的想法，以及他们最近对格拉斯曼流形的凸性的扩展，他们计划继续他们在构型、平面排列和几何截线理论方面的工作。古德曼和波拉克最近所做的最优雅的事情之一是将凸性从欧几里德空间中的点集扩展到高维平面的集合，例如普通三维空间中的线。通过这种方法，我们可以得到一组“凸”线，比如所有经过某一点的线，或者所有在某一平面上的线，这些都没有什么特别令人兴奋的。然而，我们也可以得到这样的集合，即由一张纸的特定双曲面所包围的所有直线，以及位于双曲面表面上的一组定则。由于凸性理论的提示性形式主义在这个新的和更广泛的背景下变得可用，它导致了一些令人惊讶的新的几何猜想、证明和定理。＊＊＊

9322475 Goodman The investigator and Richard Pollack (Courant Institute of the Mathematical Sciences, NYU) will continue their study of the interplay between the geometry and the topology of Euclidean spaces by extending their recent results (1) on a generalization to Grassmann manifolds of the standard convexity structure of Euclidean spaces, (2) on the classification of ordered configurations and their generalizations, and (3) 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 solution of the Vincensini problem for hyperplane transversals to families of convex bodies and their recent extension of convexity to Grassmann manifolds, Goodman and Pollack plan to continue their work on configurations, arrangements of flats, and geometric transversal theory. One of the most elegant things that Goodman and Pollack have done recently is to extend convexity, as noted above, from sets of points in Euclidean spaces to sets of higher dimensional flats, for example lines in ordinary three-space. One obtains in this way "convex" sets of lines such as all the lines through a particular point, or all the lines in a particular plane, and there is nothing especially exciting about these. However, one also obtains such sets as all the lines which are surrounded by a particular hyperboloid of one sheet together with one set of rulings lying on the surface of the hyperboloid. Since the suggestive formalism of convexity theory becomes available in this new and broader context, it leads to some surprising new geometric conjectures, proofs, and theorems. ***