Mathematical Sciences: Non-Positive Curvature, Triangulations and Topology

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9505136 Ontaneda A basic problem in Geometry and Topology is to study the relation between curvature (local information about a space) and topology (global information about a space). The investigator studies a particular case of curvature: spaces of non-positive curvature, in the riemannian case and in the PL (piecewise linear) case. He also studies some problems concerning riemannian and PL rigidity (in the non-positively curved case); the relation between riemannian and PL non-positively curved geometry; and how the topology could determine the curvature: which spaces admit a non-positively curved geometry. A space with non-positively curved geometry is a space on which, essentially, an object, as it moves away from an observer, seems to decrease in size at a rate equal to or faster than the rate in the "real world" (euclidean geometry). In general, geometry determines some properties about the (global) shape of the space. For instance, if a space has a non-positively curved geometry, after "unwrapping" the space we obtain another space that, unlike a doughnut, can be shrunk within itself to a single point (Cartan-Hadamard theorem). This research project is oriented towards studying such relations on spaces with non-positive geometry: how the geometry (local data) determines the global shape (topology) and also how topology could determine the geometry. Although the relevance of this to the "real world" is not obvious if one considers only the apparently Euclidean environs of the earth, questions of cosmology -- the nature of the universe as a whole and not merely our small corner of it -- are intimately related to these mathematical questions. ***