Mathematical Sciences: The Geometry of Configurations

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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. ***