Mathematical Sciences: The Geometry of Configurations

小行星9400293 调查员和雅各布·E.古德曼（纽约市立大学城市学院）将继续他们的研究几何和拓扑之间的相互作用的欧氏空间，通过延长他们最近的结果（1）推广到格拉斯曼流形的标准凸结构的欧氏空间，（2）分类有序配置和他们的推广，（3）有关的几何和组合问题。 建立在他们过去的工作允许序列和秩序类型的配置，以及对思想出来的解决方案的Vincensini问题的超平面横的家庭凸体和他们最近的延伸凸格拉斯曼流形，古德曼和波拉克计划继续他们的工作配置，安排单位，和几何横向理论。 最优雅的事情之一，古德曼和波拉克最近所做的是扩大凸性，如上所述，从一套点在欧几里德空间的集合更高的维单位，例如线在普通的三维空间。 人们用这种方法得到“凸”的线集，例如通过一个特定点的所有线，或在一个特定平面上的所有线，这些没有什么特别令人兴奋的。 然而，人们也可以得到这样的集合，即被一个特定的双曲面所包围的所有直线，以及位于该双曲面表面上的一组划线。 由于凸性理论的暗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结构、证明和定理。 ***

9400293 Pollack The investigator and Jacob E. Goodman (CUNY City College) 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. ***