Mathematical Sciences: Topology

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9505003 Davis The project of Professors Ruth Charney and Michael Davis involves problems in topology, geometry, and geometric group theory. The main focus of the project is the study of piecewise Euclidean polyhedra of nonpositive curvature (in the sense of Gromov's CAT(0) inequality). Understanding the local geometry of such spaces is equivalent to understanding when polyhedra with piecewise spherical metrics are "large" (i.e., satisfy the CAT(1) inequality). This project attempts to answer a number of questions concerning "large" piecewise spherical polyhedra, as well as the related question of when a negatively curved Riemannian manifold supports a piecewise hyperbolic CAT(0) metric (with B. Okun). Charney's research also includes the study of Artin groups via geometric techniques; Davis's includes the study of certain singular fiber bundles with applications to smooth torus actions on manifolds (with R. Scott). Groups are algebraic structures abstracting the concept of symmetry, which is pervasive in mathematics and mathematical physics. The field of geometric group theory has, historically, involved ideas from several areas of mathematics -- topology, combinatorics, and group theory. The recent introduction of techniques patterned on differential geometry has led to exciting new developments in the field as well as new applications to problems in topology. One of these techniques, a version of "nonpositive curvature" for topological metric spaces, has played a major role in these developments. The project of Charney and Davis aims at a better understanding of this notion of curvature and how it may be applied to problems in topology and group theory. ***