This proposal primarily concerns several aspects of geometry and topology. One of the projects proposes to study projective (and related) structures and combinatorial triangulations. A second project is to relate the algebraic theory of braids to the classification of surfaces (in either 3 or 4 dimensions) with boundary a given braid. In particular, to explore the connection between the gauge theory of 4-manifolds and the complexity of such surfaces via the braid group. A third project is to study the virtual Haken question for 3-manifolds which are bundles or semi-bundles.
Voronoi decompositions are of great importance in mathematics and computer science. One uses a finite set of points to subdivide Euclidean space into convex cells by assigning to each point the subset of space consisting of points closer to the given point than to any other point in the given set. Such decompositions are frequently used in computational situations to approximate a continuum by a finite grid. This theory is well known in Euclidean (flat) space, and we are extending it to the case of Riemannian (curved) geometry
